> #Jeton Hida Project Math 336 > #Study numerically the ultimate periodic orbits for various parameters of the > generalized discrete Logistic Equation #x(n)=k x(n-1)(1-x(n-1)). #These > generalizations should have more parameters. For example: #x(n)=k x(n-1)^a > (1-x(n-1))^b , #that has three parameters (k, a, and b). Also study second- > (and higher-) order difference equations, e.g. #x(n)=k > x(n-1)(1-x(n-1))(1-x(n-2)) , #and more generally #x(n)=k > x(n-1)^a(1-x(n-1))^b(1-x(n-2))^c , #etc. > read "/Users/jeton/Desktop/Math 336/DMB.txt" First Written: Nov. 2021 This is DMB.txt, A Maple package to explore Dynamical models in Biology (both discrete and continuous) accompanying the class Dynamical Models in Biology, Rutgers University. Taught by Dr. Z. (Doron Zeilbeger) The most current version is available on WWW at: http://sites.math.rutgers.edu/~zeilberg/tokhniot/DMB.txt . Please report all bugs to: DoronZeil at gmail dot com . For general help, and a list of the MAIN functions, type "Help();". For specific help type "Help(procedure_name);" ------------------------------ For a list of the supporting functions type: Help1(); For help with any of them type: Help(ProcedureName); ------------------------------ For a list of the functions that give examples of Discrete-time dynamical systems (some famous), type: HelpDDM(); For help with any of them type: Help(ProcedureName); ------------------------------ For a list of the functions continuous-time dynamical systems (some famous) type: HelpCDM(); For help with any of them type: Help(ProcedureName); ------------------------------ > #Experimenting first different values of k only for the Discrete Logistic > Equation x(n) = k*x(n-1)*(1-x(n-1)) #We'll use an initial condition of x=.1 > Orb([.1*x*(1-x)],[x],[.1],1000,1010) [[ -1002] [ -1003] [ -1004] [[8.910162862 10 ], [8.910162862 10 ], [8.910162862 10 ], [ -1005] [ -1006] [ -1007] [8.910162862 10 ], [8.910162862 10 ], [8.910162862 10 ], [ -1008] [ -1009] [ -1010] [8.910162862 10 ], [8.910162862 10 ], [8.910162862 10 ], [ -1011] [ -1012]] [8.910162862 10 ], [8.910162862 10 ]] > Orb([.2*x*(1-x)],[x],[.1],1000,1010) [[ -701] [ -701] [ -702] [[9.428206632 10 ], [1.885641326 10 ], [3.771282652 10 ], [ -703] [ -703] [ -704] [7.542565304 10 ], [1.508513061 10 ], [3.017026122 10 ], [ -705] [ -705] [ -706] [6.034052244 10 ], [1.206810449 10 ], [2.413620898 10 ], [ -707] [ -708]] [4.827241796 10 ], [9.654483592 10 ]] > Orb([.3*x*(1-x)],[x],[.1],1000,1010) [[ -524] [ -525] [ -525] [[1.144769798 10 ], [3.434309394 10 ], [1.030292818 10 ], [ -526] [ -527] [ -527] [3.090878454 10 ], [9.272635362 10 ], [2.781790609 10 ], [ -528] [ -528] [ -529] [8.345371827 10 ], [2.503611548 10 ], [7.510834644 10 ], [ -529] [ -530]] [2.253250393 10 ], [6.759751179 10 ]] > Orb([.4*x*(1-x)],[x],[.1],1000,1010) [[ -400] [ -400] [ -400] [[9.733726568 10 ], [3.893490627 10 ], [1.557396251 10 ], [ -401] [ -401] [ -402] [6.229585004 10 ], [2.491834002 10 ], [9.967336008 10 ], [ -402] [ -402] [ -403] [3.986934403 10 ], [1.594773761 10 ], [6.379095044 10 ], [ -403] [ -403]] [2.551638018 10 ], [1.020655207 10 ]] > Orb([.5*x*(1-x)],[x],[.1],1000,1010) [[ -303] [ -303] [ -303] [[7.686291150 10 ], [3.843145575 10 ], [1.921572788 10 ], [ -304] [ -304] [ -304] [9.607863940 10 ], [4.803931970 10 ], [2.401965985 10 ], [ -304] [ -305] [ -305] [1.200982992 10 ], [6.004914960 10 ], [3.002457480 10 ], [ -305] [ -306]] [1.501228740 10 ], [7.506143700 10 ]] > Orb([.6*x*(1-x)],[x],[.1],1000,1010) [[ -223] [ -224] [ -224] [[1.118718071 10 ], [6.712308426 10 ], [4.027385056 10 ], [ -224] [ -224] [ -225] [2.416431034 10 ], [1.449858620 10 ], [8.699151720 10 ], [ -225] [ -225] [ -225] [5.219491032 10 ], [3.131694619 10 ], [1.879016771 10 ], [ -225] [ -226]] [1.127410063 10 ], [6.764460378 10 ]] > Orb([.7*x*(1-x)],[x],[.1],1000,1010) [[ -157] [ -157] [ -157] [[9.261905366 10 ], [6.483333756 10 ], [4.538333629 10 ], [ -157] [ -157] [ -157] [3.176833540 10 ], [2.223783478 10 ], [1.556648435 10 ], [ -157] [ -158] [ -158] [1.089653904 10 ], [7.627577328 10 ], [5.339304130 10 ], [ -158] [ -158]] [3.737512891 10 ], [2.616259024 10 ]] > Orb([.8*x*(1-x)],[x],[.1],1000,1010) [[ -99] [ -99] [ -99] [[8.058801616 10 ], [6.447041293 10 ], [5.157633034 10 ], [ -99] [ -99] [ -99] [4.126106427 10 ], [3.300885142 10 ], [2.640708114 10 ], [ -99] [ -99] [ -99] [2.112566491 10 ], [1.690053193 10 ], [1.352042554 10 ], [ -99] [ -100]] [1.081634043 10 ], [8.653072344 10 ]] > Orb([.9*x*(1-x)],[x],[.1],1000,1010) [[ -48] [ -48] [ -48] [[8.550337263 10 ], [7.695303537 10 ], [6.925773183 10 ], [ -48] [ -48] [ -48] [6.233195865 10 ], [5.609876278 10 ], [5.048888650 10 ], [ -48] [ -48] [ -48] [4.543999785 10 ], [4.089599806 10 ], [3.680639825 10 ], [ -48] [ -48]] [3.312575842 10 ], [2.981318258 10 ]] > #We see that for k values that less than 1, and greater than 0 our value of x > gets infinitely closer to 0 > #We've already tested to see the behavior of this equation when k is 1, and > we've noticed bifurcations into a two period orbit once k > 3. We move into a > four period orbit once the value of k is somewhere inbetween 3.4 & 3.5, we'll > check these again later, but first let's look at the behavior when k is a > negative number. > Orb([-.1*x*(1-x)],[x],[.1],1000,1010) [[ -1002] [ -1003] [ -1004] [[9.073501848 10 ], [-9.073501848 10 ], [9.073501848 10 ], [ -1005] [ -1006] [ -1007] [-9.073501848 10 ], [9.073501848 10 ], [-9.073501848 10 ], [ -1008] [ -1009] [ -1010] [9.073501848 10 ], [-9.073501848 10 ], [9.073501848 10 ], [ -1011] [ -1012]] [-9.073501848 10 ], [9.073501848 10 ]] > Orb([-.2*x*(1-x)],[x],[.1],1000,1010) [[ -701] [ -701] [ -702] [[9.787134624 10 ], [-1.957426925 10 ], [3.914853850 10 ], [ -703] [ -703] [ -704] [-7.829707700 10 ], [1.565941540 10 ], [-3.131883080 10 ], [ -705] [ -705] [ -706] [6.263766160 10 ], [-1.252753232 10 ], [2.505506464 10 ], [ -707] [ -707]] [-5.011012928 10 ], [1.002202586 10 ]] > Orb([-.3*x*(1-x)],[x],[.1],1000,1010) [[ -524] [ -525] [ -525] [[1.214127904 10 ], [-3.642383712 10 ], [1.092715114 10 ], [ -526] [ -527] [ -527] [-3.278145342 10 ], [9.834436026 10 ], [-2.950330808 10 ], [ -528] [ -528] [ -529] [8.850992424 10 ], [-2.655297727 10 ], [7.965893181 10 ], [ -529] [ -530]] [-2.389767954 10 ], [7.169303862 10 ]] > Orb([-.4*x*(1-x)],[x],[.1],1000,1010) [[ -399] [ -400] [ -400] [[1.058949590 10 ], [-4.235798360 10 ], [1.694319344 10 ], [ -401] [ -401] [ -401] [-6.777277376 10 ], [2.710910950 10 ], [-1.084364380 10 ], [ -402] [ -402] [ -403] [4.337457520 10 ], [-1.734983008 10 ], [6.939932032 10 ], [ -403] [ -403]] [-2.775972813 10 ], [1.110389125 10 ]] > Orb([-.5*x*(1-x)],[x],[.1],1000,1010) [[ -303] [ -303] [ -303] [[8.635546880 10 ], [-4.317773440 10 ], [2.158886720 10 ], [ -303] [ -304] [ -304] [-1.079443360 10 ], [5.397216800 10 ], [-2.698608400 10 ], [ -304] [ -305] [ -305] [1.349304200 10 ], [-6.746521000 10 ], [3.373260500 10 ], [ -305] [ -306]] [-1.686630250 10 ], [8.433151250 10 ]] > Orb([-.6*x*(1-x)],[x],[.1],1000,1010) [[ -223] [ -224] [ -224] [[1.313350619 10 ], [-7.880103714 10 ], [4.728062228 10 ], [ -224] [ -224] [ -224] [-2.836837337 10 ], [1.702102402 10 ], [-1.021261441 10 ], [ -225] [ -225] [ -225] [6.127568646 10 ], [-3.676541188 10 ], [2.205924713 10 ], [ -225] [ -226]] [-1.323554828 10 ], [7.941328968 10 ]] > Orb([-.7*x*(1-x)],[x],[.1],1000,1010) [[ -156] [ -157] [ -157] [[1.161932006 10 ], [-8.133524042 10 ], [5.693466829 10 ], [ -157] [ -157] [ -157] [-3.985426780 10 ], [2.789798746 10 ], [-1.952859122 10 ], [ -157] [ -158] [ -158] [1.367001385 10 ], [-9.569009695 10 ], [6.698306786 10 ], [ -158] [ -158]] [-4.688814750 10 ], [3.282170325 10 ]] > Orb([-.8*x*(1-x)],[x],[.1],1000,1010) [[ -98] [ -99] [ -99] [[1.136048928 10 ], [-9.088391424 10 ], [7.270713139 10 ], [ -99] [ -99] [ -99] [-5.816570511 10 ], [4.653256409 10 ], [-3.722605127 10 ], [ -99] [ -99] [ -99] [2.978084102 10 ], [-2.382467282 10 ], [1.905973826 10 ], [ -99] [ -99]] [-1.524779061 10 ], [1.219823249 10 ]] > Orb([-.9*x*(1-x)],[x],[.1],1000,1010) [[ -47] [ -47] [ -47] [[1.584710705 10 ], [-1.426239634 10 ], [1.283615671 10 ], [ -47] [ -47] [ -48] [-1.155254104 10 ], [1.039728694 10 ], [-9.357558246 10 ], [ -48] [ -48] [ -48] [8.421802421 10 ], [-7.579622179 10 ], [6.821659961 10 ], [ -48] [ -48]] [-6.139493965 10 ], [5.525544568 10 ]] > #Nothing too different compared to the positive counterparts of these k > values. It seems that as long as the absolute value of k is between 0 & 1, the > value tends to 0 > Orb([-1*x*(1-x)],[x],[.1],1000,1010) [[0.02196474294], [-0.02148229301], [0.02194378192], [-0.02146225236], [0.02192288063], [-0.02144226794], [0.02190203880], [-0.02142233950], [0.02188125614], [-0.02140246677], [0.02186053236]] > Orb([-1.1*x*(1-x)],[x],[.1],10000,10010) [[0.3365056467], [-0.2455965560], [0.3365056467], [-0.2455965560], [0.3365056467], [-0.2455965560], [0.3365056467], [-0.2455965560], [0.3365056467], [-0.2455965560], [0.3365056467]] > Orb([-1.2*x*(1-x)],[x],[.1],1000,1010) [[0.4652146415], [-0.2985479746], [0.4652146415], [-0.2985479746], [0.4652146415], [-0.2985479746], [0.4652146415], [-0.2985479746], [0.4652146415], [-0.2985479746], [0.4652146415]] > Orb([-1.3*x*(1-x)],[x],[.1],1000,1010) [[0.5522237189], [-0.3214544882], [0.5522237189], [-0.3214544882], [0.5522237189], [-0.3214544882], [0.5522237189], [-0.3214544882], [0.5522237189], [-0.3214544882], [0.5522237189]] > Orb([-1.4*x*(1-x)],[x],[.1],1000,1010) [[0.6166606851], [-0.3309463984], [0.6166606837], [-0.3309463989], [0.6166606851], [-0.3309463984], [0.6166606837], [-0.3309463989], [0.6166606851], [-0.3309463984], [0.6166606837]] > Orb([-1.5*x*(1-x)],[x],[.1],1000,1010) [[0.4979368430], [-0.3749936151], [0.7734207396], [-0.2628616488], [0.4979368430], [-0.3749936151], [0.7734207396], [-0.2628616488], [0.4979368430], [-0.3749936151], [0.7734207396]] > Orb([-1.6*x*(1-x)],[x],[.1],1000,1010) [[0.6488739829], [-0.3645384595], [0.7958827970], [-0.2599253926], [0.5239785638], [-0.3990800456], [0.8933518858], [-0.1524388703], [0.2810823670], [-0.3233201120], [0.6845696109]] > Orb([-1.7*x*(1-x)],[x],[.1],1000,1010) [[-0.09445843651], [0.1757474158], [-0.2462624447], [0.5217429819], [-0.4241963126], [1.027036001], [0.04720380880], [-0.07645853569], [0.1399175538], [-0.2045790742], [0.4189338421]] > Orb([-1.8*x*(1-x)],[x],[.1],1000,1010) [[-0.1096721859], [0.2190603137], [-0.3079312069], [0.7249551032], [-0.3589113627], [0.8779117124], [-0.1929289277], [0.4142708980], [-0.4367709380], [1.129571623], [0.2634487713]] > Orb([-1.9*x*(1-x)],[x],[.1],1000,1010) [[0.8374467582], [-0.2586464023], [0.6185342908], [-0.4483042816], [1.233633920], [0.5476155843], [-0.4706922366], [1.315262495], [0.7878405780], [-0.3175808232], [0.7950329646]] > Orb([-2*x*(1-x)],[x],[.1],1000,1010) [[1.493187711], [1.472843659], [1.392849570], [1.094360709], [0.2065293048], [-0.3277499022], [0.8703398010], [-0.2256968636], [0.5532718758], [-0.4943242144], [1.477361286]] > Orb([-2.1*x*(1-x)],[x],[.1],1000,1010) [[Float(infinity)], [Float(infinity)], [Float(infinity)], [Float(infinity)], [Float(infinity)], [Float(infinity)], [Float(infinity)], [Float(infinity)], [Float(infinity)], [Float(infinity)], [Float(infinity)]] > #There are several things to take notice of here. We see that once we got to > negative values starting from -1 and going lower by increments of .1, we see > that the behavior does not work the same as the positive counterparts. When k > took on values between 1 and 3 we notice that we do have a stable fixed point > that the value of x eventually reaches. In the case of negative values of k > between -1.4 & -1 we seem to have orbit of period 2 behavior with our value of > x. Once we go from -1.4 and beyond it seems we are in a period of 4 orbit for > our value of x. This period 4 orbit disappears quickly as when we reach -1.6 > we see no immediate pattern amongst our values of x. We can try and find near > what value of k does this period doubling occur. > Orb([-1.51*x*(1-x)],[x],[.1],1000,1010) [[0.4843954804], [-0.3771323135], [0.7842352544], [-0.2555075835], [0.4843954804], [-0.3771323135], [0.7842352544], [-0.2555075835], [0.4843954804], [-0.3771323135], [0.7842352544]] > Orb([-1.52*x*(1-x)],[x],[.1],1000,1010) [[0.4720336884], [-0.3788111858], [0.7939098326], [-0.2486978557], [0.4720336884], [-0.3788111858], [0.7939098326], [-0.2486978557], [0.4720336884], [-0.3788111858], [0.7939098326]] > Orb([-1.53*x*(1-x)],[x],[.1],1000,1010) [[0.4606202804], [-0.3801273337], [0.8026749091], [-0.2423334861], [0.4606202770], [-0.3801273332], [0.8026749076], [-0.2423334875], [0.4606202804], [-0.3801273337], [0.8026749091]] > Orb([-1.54*x*(1-x)],[x],[.1],1000,1010) [[0.4499929531], [-0.3811489148], [0.8106920516], [-0.2363444916], [0.4499929440], [-0.3811489134], [0.8106920474], [-0.2363444956], [0.4499929531], [-0.3811489148], [0.8106920516]] > Orb([-1.55*x*(1-x)],[x],[.1],1000,1010) [[0.4861881835], [-0.3872043122], [0.8325538118], [-0.2160823416], [0.4072995760], [-0.3741802787], [0.7969962977], [-0.2507794588], [0.4861881835], [-0.3872043122], [0.8325538118]] > #It is evident that once we hit -1.53 for k we lose our period of 4 orbit, > though we may have another period doubling at this value. For that we'll need > to expand the range we print out from K1 to K2, we'll now have Orb print 30 > values. > Orb([-1.53*x*(1-x)],[x],[.1],1000,1030) [[0.4606202804], [-0.3801273337], [0.8026749091], [-0.2423334861], [0.4606202770], [-0.3801273332], [0.8026749076], [-0.2423334875], [0.4606202804], [-0.3801273337], [0.8026749091], [-0.2423334861], [0.4606202770], [-0.3801273332], [0.8026749076], [-0.2423334875], [0.4606202804], [-0.3801273337], [0.8026749091], [-0.2423334861], [0.4606202770], [-0.3801273332], [0.8026749076], [-0.2423334875], [0.4606202804], [-0.3801273337], [0.8026749091], [-0.2423334861], [0.4606202770], [-0.3801273332], [0.8026749076]] > #Now we can see that there is indeed an orbit of period 8 present at -1.53. So > somewhere between -1.53 & -1.52 we have another period doubling occurring. > Let's see if we can approximate where that happens. > Orb([-1.521*x*(1-x)],[x],[.1],1000,1030) [[0.4708527771], [-0.3789578183], [0.7948241731], [-0.2480427133], [0.4708527771], [-0.3789578183], [0.7948241731], [-0.2480427133], [0.4708527771], [-0.3789578183], [0.7948241731], [-0.2480427133], [0.4708527771], [-0.3789578183], [0.7948241731], [-0.2480427133], [0.4708527771], [-0.3789578183], [0.7948241731], [-0.2480427133], [0.4708527771], [-0.3789578183], [0.7948241731], [-0.2480427133], [0.4708527771], [-0.3789578183], [0.7948241731], [-0.2480427133], [0.4708527771], [-0.3789578183], [0.7948241731]] > Orb([-1.522*x*(1-x)],[x],[.1],1000,1030) [[0.4696811092], [-0.3791009242], [0.7957296579], [-0.2473919214], [0.4696811092], [-0.3791009242], [0.7957296579], [-0.2473919214], [0.4696811092], [-0.3791009242], [0.7957296579], [-0.2473919214], [0.4696811092], [-0.3791009242], [0.7957296579], [-0.2473919214], [0.4696811092], [-0.3791009242], [0.7957296579], [-0.2473919214], [0.4696811092], [-0.3791009242], [0.7957296579], [-0.2473919214], [0.4696811092], [-0.3791009242], [0.7957296579], [-0.2473919214], [0.4696811092], [-0.3791009242], [0.7957296579]] > Orb([-1.523*x*(1-x)],[x],[.1],1000,1030) [[0.4685185095], [-0.3792405788], [0.7966264651], [-0.2467454033], [0.4685185060], [-0.3792405783], [0.7966264635], [-0.2467454048], [0.4685185095], [-0.3792405788], [0.7966264651], [-0.2467454033], [0.4685185060], [-0.3792405783], [0.7966264635], [-0.2467454048], [0.4685185095], [-0.3792405788], [0.7966264651], [-0.2467454033], [0.4685185060], [-0.3792405783], [0.7966264635], [-0.2467454048], [0.4685185095], [-0.3792405788], [0.7966264651], [-0.2467454033], [0.4685185060], [-0.3792405783], [0.7966264635]] > Orb([-1.5221*x*(1-x)],[x],[.1],1000,1030) [[0.4695644448], [-0.3791150438], [0.7958197265], [-0.2473270783], [0.4695644434], [-0.3791150436], [0.7958197260], [-0.2473270788], [0.4695644448], [-0.3791150438], [0.7958197265], [-0.2473270783], [0.4695644434], [-0.3791150436], [0.7958197260], [-0.2473270788], [0.4695644448], [-0.3791150438], [0.7958197265], [-0.2473270783], [0.4695644434], [-0.3791150436], [0.7958197260], [-0.2473270788], [0.4695644448], [-0.3791150438], [0.7958197265], [-0.2473270783], [0.4695644434], [-0.3791150436], [0.7958197260]] > Orb([-1.52201*x*(1-x)],[x],[.1],1000,1030) [[0.4696694388], [-0.3791023377], [0.7957386689], [-0.2473854350], [0.4696694383], [-0.3791023376], [0.7957386686], [-0.2473854353], [0.4696694388], [-0.3791023377], [0.7957386689], [-0.2473854350], [0.4696694383], [-0.3791023376], [0.7957386686], [-0.2473854353], [0.4696694388], [-0.3791023377], [0.7957386689], [-0.2473854350], [0.4696694383], [-0.3791023376], [0.7957386686], [-0.2473854353], [0.4696694388], [-0.3791023377], [0.7957386689], [-0.2473854350], [0.4696694383], [-0.3791023376], [0.7957386686]] > Orb([-1.52202*x*(1-x)],[x],[.1],1000,1030) [[0.4696577696], [-0.3791037509], [0.7957476787], [-0.2473789492], [0.4696577685], [-0.3791037508], [0.7957476784], [-0.2473789496], [0.4696577696], [-0.3791037509], [0.7957476787], [-0.2473789492], [0.4696577685], [-0.3791037508], [0.7957476784], [-0.2473789496], [0.4696577696], [-0.3791037509], [0.7957476787], [-0.2473789492], [0.4696577685], [-0.3791037508], [0.7957476784], [-0.2473789496], [0.4696577696], [-0.3791037509], [0.7957476787], [-0.2473789492], [0.4696577685], [-0.3791037508], [0.7957476784]] > Orb([-1.52203*x*(1-x)],[x],[.1],1000,1030) [[0.4696461006], [-0.3791051637], [0.7957566878], [-0.2473724638], [0.4696460997], [-0.3791051635], [0.7957566874], [-0.2473724643], [0.4696461006], [-0.3791051637], [0.7957566878], [-0.2473724638], [0.4696460997], [-0.3791051635], [0.7957566874], [-0.2473724643], [0.4696461006], [-0.3791051637], [0.7957566878], [-0.2473724638], [0.4696460997], [-0.3791051635], [0.7957566874], [-0.2473724643], [0.4696461006], [-0.3791051637], [0.7957566878], [-0.2473724638], [0.4696460997], [-0.3791051635], [0.7957566874]] > Orb([-1.52204*x*(1-x)],[x],[.1],1000,1030) [[0.4696344328], [-0.3791065761], [0.7957656955], [-0.2473659794], [0.4696344328], [-0.3791065761], [0.7957656955], [-0.2473659794], [0.4696344328], [-0.3791065761], [0.7957656955], [-0.2473659794], [0.4696344328], [-0.3791065761], [0.7957656955], [-0.2473659794], [0.4696344328], [-0.3791065761], [0.7957656955], [-0.2473659794], [0.4696344328], [-0.3791065761], [0.7957656955], [-0.2473659794], [0.4696344328], [-0.3791065761], [0.7957656955], [-0.2473659794], [0.4696344328], [-0.3791065761], [0.7957656955]] > Orb([-1.52205*x*(1-x)],[x],[.1],1000,1030) [[0.4696227664], [-0.3791079883], [0.7957747029], [-0.2473594947], [0.4696227656], [-0.3791079882], [0.7957747025], [-0.2473594951], [0.4696227664], [-0.3791079883], [0.7957747029], [-0.2473594947], [0.4696227656], [-0.3791079882], [0.7957747025], [-0.2473594951], [0.4696227664], [-0.3791079883], [0.7957747029], [-0.2473594947], [0.4696227656], [-0.3791079882], [0.7957747025], [-0.2473594951], [0.4696227664], [-0.3791079883], [0.7957747029], [-0.2473594947], [0.4696227656], [-0.3791079882], [0.7957747025]] > #To 5 decimal places we see that the period doubling occurs somewhere around > -1.52205. Now that we have this value for where we reach a period 8 orbit, > let's approximate the two other values of k to five decimal places and see if > we will observe Feigenbaum's first constant, which we should expect. > Orb([-1.0001*x*(1-x)],[x],[.1],1000,1010) [[0.02314502080], [-0.02261158974], [0.02312518603], [-0.02259267084], [0.02310540993], [-0.02257380711], [0.02308569221], [-0.02255499830], [0.02306603261], [-0.02253624415], [0.02304643086]] > Orb([-1.00001*x*(1-x)],[x],[.1],1000,1010) [[0.02208182567], [-0.02159443458], [0.02206097480], [-0.02157450393], [0.02204018355], [-0.02155462940], [0.02201945163], [-0.02153481073], [0.02199877880], [-0.02151504768], [0.02197816474]] > Orb([-1.00002*x*(1-x)],[x],[.1],1000,1010) [[0.02219912698], [-0.02170675987], [0.02217838686], [-0.02168693975], [0.02215770626], [-0.02166717564], [0.02213708488], [-0.02164746729], [0.02211652244], [-0.02162781443], [0.02209601869]] > Orb([-1.01000*x*(1-x)],[x],[.1],1000,1010) [[0.1040840788], [-0.09418308917], [0.1040840788], [-0.09418308917], [0.1040840788], [-0.09418308917], [0.1040840788], [-0.09418308917], [0.1040840788], [-0.09418308917], [0.1040840788]] > Orb([-1.00100*x*(1-x)],[x],[.1],1000,1010) [[0.03423353842], [-0.03309466487], [0.03422411164], [-0.03308587464], [0.03421473030], [-0.03307712661], [0.03420539415], [-0.03306842055], [0.03419610294], [-0.03305975622], [0.03418685639]] > Orb([-1.00200*x*(1-x)],[x],[.1],1000,1010) [[0.04596963746], [-0.04394414275], [0.04596698090], [-0.04394172561], [0.04596434608], [-0.04393932822], [0.04596173276], [-0.04393695038], [0.04595914078], [-0.04393459196], [0.04595657001]] > Orb([-1.00300*x*(1-x)],[x],[.1],1000,1010) [[0.05617013003], [-0.05317409166], [0.05616958043], [-0.05317360235], [0.05616903742], [-0.05317311889], [0.05616850097], [-0.05317264127], [0.05616797095], [-0.05317216938], [0.05616744731]] > Orb([-1.00400*x*(1-x)],[x],[.1],1000,1010) [[0.06502293725], [-0.06103813470], [0.06502284376], [-0.06103805304], [0.06502275175], [-0.06103797267], [0.06502266123], [-0.06103789362], [0.06502257217], [-0.06103781583], [0.06502248452]] > Orb([-1.00500*x*(1-x)],[x],[.1],1000,1010) [[0.07289109594], [-0.06791587400], [0.07289108214], [-0.06791586215], [0.07289106860], [-0.06791585052], [0.07289105537], [-0.06791583916], [0.07289104236], [-0.06791582799], [0.07289102962]] > Orb([-1.00600*x*(1-x)],[x],[.1],1000,1010) [[0.08003758954], [-0.07407336324], [0.08003758772], [-0.07407336170], [0.08003758599], [-0.07407336024], [0.08003758426], [-0.07407335878], [0.08003758261], [-0.07407335739], [0.08003758096]] > Orb([-1.00700*x*(1-x)],[x],[.1],1000,1010) [[0.08663275632], [-0.07968141450], [0.08663275608], [-0.07968141430], [0.08663275585], [-0.07968141412], [0.08663275566], [-0.07968141395], [0.08663275547], [-0.07968141380], [0.08663275531]] > Orb([-1.00800*x*(1-x)],[x],[.1],1000,1010) [[0.09278979974], [-0.08485329163], [0.09278979974], [-0.08485329163], [0.09278979974], [-0.08485329163], [0.09278979974], [-0.08485329163], [0.09278979974], [-0.08485329163], [0.09278979974]] > Orb([-1.00710*x*(1-x)],[x],[.1],1000,1010) [[0.08726665902], [-0.08021671269], [0.08726665889], [-0.08021671258], [0.08726665877], [-0.08021671247], [0.08726665857], [-0.08021671231], [0.08726665840], [-0.08021671217], [0.08726665824]] > Orb([-1.00720*x*(1-x)],[x],[.1],1000,1010) [[0.08789629805], [-0.08074776672], [0.08789629794], [-0.08074776663], [0.08789629785], [-0.08074776656], [0.08789629777], [-0.08074776649], [0.08789629761], [-0.08074776636], [0.08789629747]] > Orb([-1.00730*x*(1-x)],[x],[.1],1000,1010) [[0.08852176056], [-0.08127466377], [0.08852176047], [-0.08127466370], [0.08852176039], [-0.08127466364], [0.08852176033], [-0.08127466359], [0.08852176027], [-0.08127466354], [0.08852176022]] > Orb([-1.00740*x*(1-x)],[x],[.1],1000,1010) [[0.08914313103], [-0.08179748831], [0.08914313093], [-0.08179748823], [0.08914313084], [-0.08179748815], [0.08914313076], [-0.08179748807], [0.08914313067], [-0.08179748800], [0.08914313059]] > Orb([-1.00750*x*(1-x)],[x],[.1],1000,1010) [[0.08976049078], [-0.08231632166], [0.08976049074], [-0.08231632164], [0.08976049072], [-0.08231632162], [0.08976049070], [-0.08231632160], [0.08976049067], [-0.08231632157], [0.08976049064]] > Orb([-1.00760*x*(1-x)],[x],[.1],1000,1010) [[0.09037391865], [-0.08283124268], [0.09037391863], [-0.08283124266], [0.09037391861], [-0.08283124265], [0.09037391860], [-0.08283124264], [0.09037391859], [-0.08283124263], [0.09037391858]] > Orb([-1.00770*x*(1-x)],[x],[.1],1000,1010) [[0.09098349091], [-0.08334232760], [0.09098349089], [-0.08334232758], [0.09098349087], [-0.08334232757], [0.09098349086], [-0.08334232756], [0.09098349085], [-0.08334232756], [0.09098349085]] > Orb([-1.00780*x*(1-x)],[x],[.1],1000,1010) [[0.09158928106], [-0.08384964999], [0.09158928102], [-0.08384964996], [0.09158928099], [-0.08384964994], [0.09158928097], [-0.08384964992], [0.09158928095], [-0.08384964990], [0.09158928093]] > Orb([-1.00790*x*(1-x)],[x],[.1],1000,1010) [[0.09219136106], [-0.08435328170], [0.09219136105], [-0.08435328170], [0.09219136105], [-0.08435328170], [0.09219136105], [-0.08435328170], [0.09219136105], [-0.08435328170], [0.09219136105]] > Orb([-1.00791*x*(1-x)],[x],[.1],1000,1010) [[0.09225136778], [-0.08440344465], [0.09225136777], [-0.08440344464], [0.09225136776], [-0.08440344463], [0.09225136775], [-0.08440344462], [0.09225136774], [-0.08440344462], [0.09225136774]] > Orb([-1.00792*x*(1-x)],[x],[.1],1000,1010) [[0.09231133824], [-0.08445357152], [0.09231133824], [-0.08445357152], [0.09231133824], [-0.08445357152], [0.09231133824], [-0.08445357152], [0.09231133824], [-0.08445357152], [0.09231133824]] > #We reach a solid orbit of period 2 at around -1.00792 > Orb([-1.39000*x*(1-x)],[x],[.1],1000,1010) [[0.6109612926], [-0.3303857522], [0.6109612914], [-0.3303857526], [0.6109612926], [-0.3303857522], [0.6109612914], [-0.3303857526], [0.6109612926], [-0.3303857522], [0.6109612914]] > Orb([-1.38000*x*(1-x)],[x],[.1],1000,1010) [[0.6051145920], [-0.3297522730], [0.6051145916], [-0.3297522732], [0.6051145920], [-0.3297522730], [0.6051145916], [-0.3297522732], [0.6051145920], [-0.3297522730], [0.6051145916]] > Orb([-1.37000*x*(1-x)],[x],[.1],1000,1010) [[0.5991145359], [-0.3290415431], [0.5991145359], [-0.3290415431], [0.5991145359], [-0.3290415431], [0.5991145359], [-0.3290415431], [0.5991145359], [-0.3290415431], [0.5991145359]] > Orb([-1.36000*x*(1-x)],[x],[.1],1000,1010) [[0.5929546995], [-0.3282488164], [0.5929546983], [-0.3282488168], [0.5929546995], [-0.3282488164], [0.5929546983], [-0.3282488168], [0.5929546995], [-0.3282488164], [0.5929546983]] > Orb([-1.35000*x*(1-x)],[x],[.1],1000,1010) [[0.5866282475], [-0.3273689880], [0.5866282471], [-0.3273689882], [0.5866282475], [-0.3273689880], [0.5866282471], [-0.3273689882], [0.5866282475], [-0.3273689880], [0.5866282471]] > Orb([-1.34000*x*(1-x)],[x],[.1],1000,1010) [[0.5801278995], [-0.3263965564], [0.5801278995], [-0.3263965564], [0.5801278995], [-0.3263965564], [0.5801278995], [-0.3263965564], [0.5801278995], [-0.3263965564], [0.5801278995]] > Orb([-1.33000*x*(1-x)],[x],[.1],1000,1010) [[0.5734458847], [-0.3253255837], [0.5734458845], [-0.3253255838], [0.5734458847], [-0.3253255837], [0.5734458845], [-0.3253255838], [0.5734458847], [-0.3253255837], [0.5734458845]] > Orb([-1.32000*x*(1-x)],[x],[.1],1000,1010) [[0.5665738926], [-0.3241496502], [0.5665738926], [-0.3241496502], [0.5665738926], [-0.3241496502], [0.5665738926], [-0.3241496502], [0.5665738926], [-0.3241496502], [0.5665738926]] > Orb([-1.31000*x*(1-x)],[x],[.1],1000,1010) [[0.5595030225], [-0.3228618013], [0.5595030225], [-0.3228618013], [0.5595030225], [-0.3228618013], [0.5595030225], [-0.3228618013], [0.5595030225], [-0.3228618013], [0.5595030225]] > Orb([-1.31900*x*(1-x)],[x],[.1],1000,1010) [[0.5658759073], [-0.3240260212], [0.5658759073], [-0.3240260212], [0.5658759073], [-0.3240260212], [0.5658759073], [-0.3240260212], [0.5658759073], [-0.3240260212], [0.5658759073]] > Orb([-1.31990*x*(1-x)],[x],[.1],1000,1010) [[0.5665041839], [-0.3241373377], [0.5665041837], [-0.3241373378], [0.5665041839], [-0.3241373377], [0.5665041837], [-0.3241373378], [0.5665041839], [-0.3241373377], [0.5665041837]] > Orb([-1.31980*x*(1-x)],[x],[.1],1000,1010) [[0.5664344548], [-0.3241250141], [0.5664344548], [-0.3241250141], [0.5664344548], [-0.3241250141], [0.5664344548], [-0.3241250141], [0.5664344548], [-0.3241250141], [0.5664344548]] > Orb([-1.31970*x*(1-x)],[x],[.1],1000,1010) [[0.5663647062], [-0.3241126793], [0.5663647062], [-0.3241126793], [0.5663647062], [-0.3241126793], [0.5663647062], [-0.3241126793], [0.5663647062], [-0.3241126793], [0.5663647062]] > Orb([-1.31960*x*(1-x)],[x],[.1],1000,1010) [[0.5662949376], [-0.3241003332], [0.5662949375], [-0.3241003333], [0.5662949376], [-0.3241003332], [0.5662949375], [-0.3241003333], [0.5662949376], [-0.3241003332], [0.5662949375]] > Orb([-1.31950*x*(1-x)],[x],[.1],1000,1010) [[0.5662251492], [-0.3240879760], [0.5662251492], [-0.3240879760], [0.5662251492], [-0.3240879760], [0.5662251492], [-0.3240879760], [0.5662251492], [-0.3240879760], [0.5662251492]] > Orb([-1.31952*x*(1-x)],[x],[.1],1000,1010) [[0.5662391083], [-0.3240904483], [0.5662391083], [-0.3240904483], [0.5662391083], [-0.3240904483], [0.5662391083], [-0.3240904483], [0.5662391083], [-0.3240904483], [0.5662391083]] > Orb([-1.31954*x*(1-x)],[x],[.1],1000,1010) [[0.5662530668], [-0.3240929202], [0.5662530668], [-0.3240929202], [0.5662530668], [-0.3240929202], [0.5662530668], [-0.3240929202], [0.5662530668], [-0.3240929202], [0.5662530668]] > Orb([-1.31956*x*(1-x)],[x],[.1],1000,1010) [[0.5662670247], [-0.3240953916], [0.5662670247], [-0.3240953916], [0.5662670247], [-0.3240953916], [0.5662670247], [-0.3240953916], [0.5662670247], [-0.3240953916], [0.5662670247]] > Orb([-1.31958*x*(1-x)],[x],[.1],1000,1010) [[0.5662809817], [-0.3240978627], [0.5662809817], [-0.3240978627], [0.5662809817], [-0.3240978627], [0.5662809817], [-0.3240978627], [0.5662809817], [-0.3240978627], [0.5662809817]] > Orb([-1.31959*x*(1-x)],[x],[.1],1000,1010) [[0.5662879596], [-0.3240990980], [0.5662879596], [-0.3240990980], [0.5662879596], [-0.3240990980], [0.5662879596], [-0.3240990980], [0.5662879596], [-0.3240990980], [0.5662879596]] > #-1.31960 is our next bifurcation value where we begin our period of 4 orbit. > #Will see if this matches Feigenbaum's constant using the ratio of > (k(n-1)-k(n-2))/(k(n)-k(n-1)) > evalf((-1.31960+1.00792)/(-1.52205+1.31960)) 1.539540627 > #Did not get the Feigenbaum constant as we expected, but we'll see if this > 1.539540627 number is a constant ratio that will appear further. I will now > look for where, if at all, we see another period doubling to a period of 16. > Orb([-1.53*x*(1-x)],[x],[.1],1000,1010) [[0.4606202804], [-0.3801273337], [0.8026749091], [-0.2423334861], [0.4606202770], [-0.3801273332], [0.8026749076], [-0.2423334875], [0.4606202804], [-0.3801273337], [0.8026749091]] > Orb([-1.54*x*(1-x)],[x],[.1],1000,1010) [[0.4499929531], [-0.3811489148], [0.8106920516], [-0.2363444916], [0.4499929440], [-0.3811489134], [0.8106920474], [-0.2363444956], [0.4499929531], [-0.3811489148], [0.8106920516]] > Orb([-1.55*x*(1-x)],[x],[.1],1000,1010) [[0.4861881835], [-0.3872043122], [0.8325538118], [-0.2160823416], [0.4072995760], [-0.3741802787], [0.7969962977], [-0.2507794588], [0.4861881835], [-0.3872043122], [0.8325538118]] > Orb([-1.56*x*(1-x)],[x],[.1],1000,1020) [[0.5125741976], [-0.3897533478], [0.8449911912], [-0.2043304817], [0.3838870269], [-0.3689677329], [0.7879636766], [-0.2606399968], [0.5125741995], [-0.3897533476], [0.8449911907], [-0.2043304821], [0.3838870277], [-0.3689677332], [0.7879636772], [-0.2606399961], [0.5125741976], [-0.3897533478], [0.8449911912], [-0.2043304817], [0.3838870269]] > Orb([-1.55900*x*(1-x)],[x],[.1],1000,1020) [[0.5104084902], [-0.3895811031], [0.8439717263], [-0.2052945009], [0.3857594791], [-0.3694036522], [0.7886399655], [-0.2598649867], [0.5104084902], [-0.3895811031], [0.8439717263], [-0.2052945009], [0.3857594791], [-0.3694036522], [0.7886399655], [-0.2598649867], [0.5104084902], [-0.3895811031], [0.8439717263], [-0.2052945009], [0.3857594791]] > Orb([-1.55990*x*(1-x)],[x],[.1],1000,1020) [[0.5123607112], [-0.3897366672], [0.8448907224], [-0.2044255027], [0.3840712292], [-0.3690107553], [0.7880298026], [-0.2605638753], [0.5123607112], [-0.3897366672], [0.8448907224], [-0.2044255027], [0.3840712292], [-0.3690107553], [0.7880298026], [-0.2605638753], [0.5123607112], [-0.3897366672], [0.8448907224], [-0.2044255027], [0.3840712292]] > Orb([-1.55995*x*(1-x)],[x],[.1],1000,1020) [[0.5124675382], [-0.3897450222], [0.8449409967], [-0.2043779549], [0.3839790459], [-0.3689892286], [0.7879966991], [-0.2606019731], [0.5124675382], [-0.3897450222], [0.8449409967], [-0.2043779549], [0.3839790459], [-0.3689892286], [0.7879966991], [-0.2606019731], [0.5124675382], [-0.3897450222], [0.8449409967], [-0.2043779549], [0.3839790459]] > Orb([-1.55998*x*(1-x)],[x],[.1],1000,1020) [[0.5125315541], [-0.3897500211], [0.8449711229], [-0.2043494622], [0.3839238151], [-0.3689763276], [0.7879768762], [-0.2606247959], [0.5125315541], [-0.3897500211], [0.8449711229], [-0.2043494622], [0.3839238151], [-0.3689763276], [0.7879768762], [-0.2606247959], [0.5125315541], [-0.3897500211], [0.8449711229], [-0.2043494622], [0.3839238151]] > Orb([-1.55999*x*(1-x)],[x],[.1],1000,1020) [[0.5125528796], [-0.3897516849], [0.8449811583], [-0.2043399707], [0.3839054183], [-0.3689720298], [0.7879702751], [-0.2606323976], [0.5125528796], [-0.3897516849], [0.8449811583], [-0.2043399707], [0.3839054183], [-0.3689720298], [0.7879702751], [-0.2606323976], [0.5125528796], [-0.3897516849], [0.8449811583], [-0.2043399707], [0.3839054183]] > #We say the period of 16 begins at -1.56000 > evalf((-1.52205+1.31960)/(-1.56000+1.52205)) 5.334650856 > #Did not find anything in terms of Feigenbaum's constants or any sort of > pattern with the length between bifurcation values so we'll move onto testing > different variations of this logistic equation. #We'll now play around with > the equation > #x(n)=k*x(n-1)*(1-x(n-1))*(1-x(n-2)) > Help(ToSys) ToSys(k,z,f): converts the kth order difference equation x(n)=f(x[n-1],x[n-2],\ ...x[n-k]) to a first-order system x1(n)=F(x1(n-1),x2(n-1), ...,xk(n-1)), it gives the unerlying transormation, followed by the set of variables Try: ToSys(2,z,z[1]+z[2]); > F:=.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -1002 -1001] [[8.582526351 10 , 8.582526351 10 ], [ -1003 -1002] [8.582526351 10 , 8.582526351 10 ], [ -1004 -1003] [8.582526351 10 , 8.582526351 10 ], [ -1005 -1004] [8.582526351 10 , 8.582526351 10 ], [ -1006 -1005] [8.582526351 10 , 8.582526351 10 ], [ -1007 -1006] [8.582526351 10 , 8.582526351 10 ], [ -1008 -1007] [8.582526351 10 , 8.582526351 10 ], [ -1009 -1008] [8.582526351 10 , 8.582526351 10 ], [ -1010 -1009] [8.582526351 10 , 8.582526351 10 ], [ -1011 -1010] [8.582526351 10 , 8.582526351 10 ], [ -1012 -1011]] [8.582526351 10 , 8.582526351 10 ]] > F:=.2*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -701 -700] [[8.903851992 10 , 4.451925996 10 ], [ -701 -701] [1.780770398 10 , 8.903851992 10 ], [ -702 -701] [3.561540796 10 , 1.780770398 10 ], [ -703 -702] [7.123081592 10 , 3.561540796 10 ], [ -703 -703] [1.424616318 10 , 7.123081592 10 ], [ -704 -703] [2.849232636 10 , 1.424616318 10 ], [ -705 -704] [5.698465272 10 , 2.849232636 10 ], [ -705 -705] [1.139693054 10 , 5.698465272 10 ], [ -706 -705] [2.279386108 10 , 1.139693054 10 ], [ -707 -706] [4.558772216 10 , 2.279386108 10 ], [ -708 -707]] [9.117544432 10 , 4.558772216 10 ]] > F:=.3*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -524 -524] [[1.056539898 10 , 3.521799660 10 ], [ -525 -524] [3.169619694 10 , 1.056539898 10 ], [ -526 -525] [9.508859082 10 , 3.169619694 10 ], [ -526 -526] [2.852657725 10 , 9.508859082 10 ], [ -527 -526] [8.557973175 10 , 2.852657725 10 ], [ -527 -527] [2.567391952 10 , 8.557973175 10 ], [ -528 -527] [7.702175856 10 , 2.567391952 10 ], [ -528 -528] [2.310652757 10 , 7.702175856 10 ], [ -529 -528] [6.931958271 10 , 2.310652757 10 ], [ -529 -529] [2.079587481 10 , 6.931958271 10 ], [ -530 -529]] [6.238762443 10 , 2.079587481 10 ]] > F:=.4*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -400 -399] [[8.741837592 10 , 2.185459398 10 ], [ -400 -400] [3.496735037 10 , 8.741837592 10 ], [ -400 -400] [1.398694015 10 , 3.496735037 10 ], [ -401 -400] [5.594776060 10 , 1.398694015 10 ], [ -401 -401] [2.237910424 10 , 5.594776060 10 ], [ -402 -401] [8.951641696 10 , 2.237910424 10 ], [ -402 -402] [3.580656678 10 , 8.951641696 10 ], [ -402 -402] [1.432262671 10 , 3.580656678 10 ], [ -403 -402] [5.729050684 10 , 1.432262671 10 ], [ -403 -403] [2.291620274 10 , 5.729050684 10 ], [ -404 -403]] [9.166481096 10 , 2.291620274 10 ]] > F:=.5*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -303 -302] [[6.677587845 10 , 1.335517569 10 ], [ -303 -303] [3.338793922 10 , 6.677587845 10 ], [ -303 -303] [1.669396961 10 , 3.338793922 10 ], [ -304 -303] [8.346984805 10 , 1.669396961 10 ], [ -304 -304] [4.173492402 10 , 8.346984805 10 ], [ -304 -304] [2.086746201 10 , 4.173492402 10 ], [ -304 -304] [1.043373100 10 , 2.086746201 10 ], [ -305 -304] [5.216865500 10 , 1.043373100 10 ], [ -305 -305] [2.608432750 10 , 5.216865500 10 ], [ -305 -305] [1.304216375 10 , 2.608432750 10 ], [ -306 -305]] [6.521081875 10 , 1.304216375 10 ]] > F:=.6*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -224 -223] [[9.320715384 10 , 1.553452564 10 ], [ -224 -224] [5.592429230 10 , 9.320715384 10 ], [ -224 -224] [3.355457538 10 , 5.592429230 10 ], [ -224 -224] [2.013274523 10 , 3.355457538 10 ], [ -224 -224] [1.207964714 10 , 2.013274523 10 ], [ -225 -224] [7.247788284 10 , 1.207964714 10 ], [ -225 -225] [4.348672970 10 , 7.247788284 10 ], [ -225 -225] [2.609203782 10 , 4.348672970 10 ], [ -225 -225] [1.565522269 10 , 2.609203782 10 ], [ -226 -225] [9.393133614 10 , 1.565522269 10 ], [ -226 -226]] [5.635880168 10 , 9.393133614 10 ]] > F:=.7*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -157 -156] [[7.301017206 10 , 1.043002458 10 ], [ -157 -157] [5.110712044 10 , 7.301017206 10 ], [ -157 -157] [3.577498431 10 , 5.110712044 10 ], [ -157 -157] [2.504248902 10 , 3.577498431 10 ], [ -157 -157] [1.752974231 10 , 2.504248902 10 ], [ -157 -157] [1.227081962 10 , 1.752974231 10 ], [ -158 -157] [8.589573734 10 , 1.227081962 10 ], [ -158 -158] [6.012701614 10 , 8.589573734 10 ], [ -158 -158] [4.208891130 10 , 6.012701614 10 ], [ -158 -158] [2.946223791 10 , 4.208891130 10 ], [ -158 -158]] [2.062356654 10 , 2.946223791 10 ]] > F:=.8*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -99 -99] [[5.871238031 10 , 7.339047539 10 ], [ -99 -99] [4.696990425 10 , 5.871238031 10 ], [ -99 -99] [3.757592340 10 , 4.696990425 10 ], [ -99 -99] [3.006073872 10 , 3.757592340 10 ], [ -99 -99] [2.404859098 10 , 3.006073872 10 ], [ -99 -99] [1.923887278 10 , 2.404859098 10 ], [ -99 -99] [1.539109822 10 , 1.923887278 10 ], [ -99 -99] [1.231287858 10 , 1.539109822 10 ], [ -100 -99] [9.850302864 10 , 1.231287858 10 ], [ -100 -100] [7.880242291 10 , 9.850302864 10 ], [ -100 -100]] [6.304193833 10 , 7.880242291 10 ]] > F:=.9*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -48 -48] [[5.489299059 10 , 6.099221177 10 ], [ -48 -48] [4.940369153 10 , 5.489299059 10 ], [ -48 -48] [4.446332238 10 , 4.940369153 10 ], [ -48 -48] [4.001699014 10 , 4.446332238 10 ], [ -48 -48] [3.601529113 10 , 4.001699014 10 ], [ -48 -48] [3.241376202 10 , 3.601529113 10 ], [ -48 -48] [2.917238582 10 , 3.241376202 10 ], [ -48 -48] [2.625514724 10 , 2.917238582 10 ], [ -48 -48] [2.362963252 10 , 2.625514724 10 ], [ -48 -48] [ -48 -48] [2.126666927 10 , 2.362963252 10 ], [1.914000234 10 , 2.126666927 10 ] ] ] > F:=1.0*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.0004943899675, 0.0004948799013], [0.0004939010033, 0.0004943899675], [0.0004934130060, 0.0004939010033], [0.0004929259727, 0.0004934130060], [0.0004924399005, 0.0004929259727], [0.0004919547865, 0.0004924399005], [0.0004914706280, 0.0004919547865], [0.0004909874221, 0.0004914706280], [0.0004905051661, 0.0004909874221], [0.0004900238571, 0.0004905051661], [0.0004895434922, 0.0004900238571]] > F:=1.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082], [0.04653741082, 0.04653741082]] > F:=1.2*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.08712907085, 0.08712907086], [0.08712907085, 0.08712907085], [0.08712907086, 0.08712907085], [0.08712907086, 0.08712907086], [0.08712907085, 0.08712907086], [0.08712907085, 0.08712907085], [0.08712907086, 0.08712907085], [0.08712907086, 0.08712907086], [0.08712907085, 0.08712907086], [0.08712907085, 0.08712907085], [0.08712907086, 0.08712907085]] > F:=1.3*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806], [0.1229419806, 0.1229419806]] > F:=1.4*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452], [0.1548457452, 0.1548457452]] > F:=1.5*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190], [0.1835034190, 0.1835034190]] > F:=1.6*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850], [0.2094305850, 0.2094305850]] > F:=1.7*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113], [0.2330350113, 0.2330350113]] > F:=1.8*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076], [0.2546440076, 0.2546440076]] > F:=1.9*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498], [0.2745237498, 0.2745237498]] > F:=2.0*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188], [0.2928932188, 0.2928932188]] > F:=2.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407], [0.3099344407, 0.3099344407]] > F:=2.2*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375], [0.3258001375, 0.3258001375]] > F:=2.3*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3406195267, 0.3406195267], [0.3406195265, 0.3406195267], [0.3406195265, 0.3406195265], [0.3406195267, 0.3406195265], [0.3406195267, 0.3406195267], [0.3406195265, 0.3406195267], [0.3406195265, 0.3406195265], [0.3406195267, 0.3406195265], [0.3406195267, 0.3406195267], [0.3406195265, 0.3406195267], [0.3406195265, 0.3406195265]] > F:=2.4*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758], [0.3545027758, 0.3545027758]] > F:=2.5*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680], [0.3675444680, 0.3675444680]] > F:=2.6*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3798263271, 0.3798263271], [0.3798263269, 0.3798263271], [0.3798263269, 0.3798263269], [0.3798263271, 0.3798263269], [0.3798263271, 0.3798263271], [0.3798263269, 0.3798263271], [0.3798263269, 0.3798263269], [0.3798263271, 0.3798263269], [0.3798263271, 0.3798263271], [0.3798263269, 0.3798263271], [0.3798263269, 0.3798263269]] > F:=2.7*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804], [0.3914193804, 0.3914193804]] > F:=2.8*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4023856953, 0.4023856956], [0.4023856950, 0.4023856953], [0.4023856953, 0.4023856950], [0.4023856956, 0.4023856953], [0.4023856956, 0.4023856956], [0.4023856953, 0.4023856956], [0.4023856950, 0.4023856953], [0.4023856953, 0.4023856950], [0.4023856956, 0.4023856953], [0.4023856956, 0.4023856956], [0.4023856953, 0.4023856956]] > F:=2.9*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4127797804, 0.4127797807], [0.4127797804, 0.4127797804], [0.4127797807, 0.4127797804], [0.4127797807, 0.4127797807], [0.4127797804, 0.4127797807], [0.4127797804, 0.4127797804], [0.4127797807, 0.4127797804], [0.4127797807, 0.4127797807], [0.4127797804, 0.4127797807], [0.4127797804, 0.4127797804], [0.4127797807, 0.4127797804]] > F:=3.0*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308], [0.4226497308, 0.4226497308]] > F:=3.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4320381659, 0.4320381662], [0.4320381653, 0.4320381659], [0.4320381656, 0.4320381653], [0.4320381662, 0.4320381656], [0.4320381659, 0.4320381662], [0.4320381653, 0.4320381659], [0.4320381656, 0.4320381653], [0.4320381662, 0.4320381656], [0.4320381659, 0.4320381662], [0.4320381653, 0.4320381659], [0.4320381656, 0.4320381653]] > F:=3.2*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4409830054, 0.4409830058], [0.4409830054, 0.4409830054], [0.4409830058, 0.4409830054], [0.4409830058, 0.4409830058], [0.4409830054, 0.4409830058], [0.4409830054, 0.4409830054], [0.4409830058, 0.4409830054], [0.4409830058, 0.4409830058], [0.4409830054, 0.4409830058], [0.4409830054, 0.4409830054], [0.4409830058, 0.4409830054]] > F:=3.3*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4495181173, 0.4495181176], [0.4495181173, 0.4495181173], [0.4495181176, 0.4495181173], [0.4495181176, 0.4495181176], [0.4495181173, 0.4495181176], [0.4495181173, 0.4495181173], [0.4495181176, 0.4495181173], [0.4495181176, 0.4495181176], [0.4495181173, 0.4495181176], [0.4495181173, 0.4495181173], [0.4495181176, 0.4495181173]] > F:=3.4*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4576738548, 0.4576738558], [0.4576738552, 0.4576738548], [0.4576738562, 0.4576738552], [0.4576738558, 0.4576738562], [0.4576738548, 0.4576738558], [0.4576738552, 0.4576738548], [0.4576738562, 0.4576738552], [0.4576738558, 0.4576738562], [0.4576738548, 0.4576738558], [0.4576738552, 0.4576738548], [0.4576738562, 0.4576738552]] > F:=3.5*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4654775153, 0.4654775164], [0.4654775160, 0.4654775153], [0.4654775170, 0.4654775160], [0.4654775164, 0.4654775170], [0.4654775153, 0.4654775164], [0.4654775160, 0.4654775153], [0.4654775170, 0.4654775160], [0.4654775164, 0.4654775170], [0.4654775153, 0.4654775164], [0.4654775160, 0.4654775153], [0.4654775170, 0.4654775160]] > F:=3.6*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4729537220, 0.4729537238], [0.4729537228, 0.4729537220], [0.4729537246, 0.4729537228], [0.4729537238, 0.4729537246], [0.4729537220, 0.4729537238], [0.4729537228, 0.4729537220], [0.4729537246, 0.4729537228], [0.4729537238, 0.4729537246], [0.4729537220, 0.4729537238], [0.4729537228, 0.4729537220], [0.4729537246, 0.4729537228]] > F:=3.7*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4801247546, 0.4801247558], [0.4801247543, 0.4801247546], [0.4801247554, 0.4801247543], [0.4801247561, 0.4801247554], [0.4801247550, 0.4801247561], [0.4801247539, 0.4801247550], [0.4801247550, 0.4801247539], [0.4801247561, 0.4801247550], [0.4801247554, 0.4801247561], [0.4801247543, 0.4801247554], [0.4801247546, 0.4801247543]] > F:=3.8*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4870108234, 0.4870108207], [0.4870108272, 0.4870108234], [0.4870108245, 0.4870108272], [0.4870108207, 0.4870108245], [0.4870108234, 0.4870108207], [0.4870108272, 0.4870108234], [0.4870108245, 0.4870108272], [0.4870108207, 0.4870108245], [0.4870108234, 0.4870108207], [0.4870108272, 0.4870108234], [0.4870108245, 0.4870108272]] > F:=3.9*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],100000,100010) [[0.4936303144, 0.4936303180], [0.4936303148, 0.4936303144], [0.4936303184, 0.4936303148], [0.4936303184, 0.4936303184], [0.4936303148, 0.4936303184], [0.4936303144, 0.4936303148], [0.4936303180, 0.4936303144], [0.4936303187, 0.4936303180], [0.4936303152, 0.4936303187], [0.4936303141, 0.4936303152], [0.4936303176, 0.4936303141]] > F:=4.0*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],100000,100020) [[0.4720172340, 0.4992379740], [0.4991935688, 0.4720172340], [0.5279813928, 0.4991935688], [0.4992379888, 0.5279813928], [0.4720175108, 0.4992379888], [0.4991935852, 0.4720175108], [0.5279811160, 0.4991935852], [0.4992380036, 0.5279811160], [0.4720177876, 0.4992380036], [0.4991936016, 0.4720177876], [0.5279808392, 0.4991936016], [0.4992380180, 0.5279808392], [0.4720180644, 0.4992380180], [0.4991936180, 0.4720180644], [0.5279805624, 0.4991936180], [0.4992380328, 0.5279805624], [0.4720183412, 0.4992380328], [0.4991936344, 0.4720183412], [0.5279802856, 0.4991936344], [0.4992380476, 0.5279802856], [0.4720186180, 0.4992380476]] > F:=4.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.2316917428, 0.4219131174], [0.4219131183, 0.2316917428], [0.7683082570, 0.4219131183], [0.4219131174, 0.7683082570], [0.2316917428, 0.4219131174], [0.4219131183, 0.2316917428], [0.7683082570, 0.4219131183], [0.4219131174, 0.7683082570], [0.2316917428, 0.4219131174], [0.4219131183, 0.2316917428], [0.7683082570, 0.4219131183]] > F:=4.2*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.1882241457, 0.3908910548], [0.3908910547, 0.1882241457], [0.8117758543, 0.3908910547], [0.3908910548, 0.8117758543], [0.1882241457, 0.3908910548], [0.3908910547, 0.1882241457], [0.8117758543, 0.3908910547], [0.3908910548, 0.8117758543], [0.1882241457, 0.3908910548], [0.3908910547, 0.1882241457], [0.8117758543, 0.3908910547]] > F:=4.3*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.1614357146, 0.3679323644], [0.3679323644, 0.1614357146], [0.8385642854, 0.3679323644], [0.3679323644, 0.8385642854], [0.1614357146, 0.3679323644], [0.3679323644, 0.1614357146], [0.8385642854, 0.3679323644], [0.3679323644, 0.8385642854], [0.1614357146, 0.3679323644], [0.3679323644, 0.1614357146], [0.8385642854, 0.3679323644]] > F:=4.4*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.1421894363, 0.3492443278], [0.3492443278, 0.1421894363], [0.8578105637, 0.3492443278], [0.3492443278, 0.8578105637], [0.1421894363, 0.3492443278], [0.3492443278, 0.1421894363], [0.8578105637, 0.3492443278], [0.3492443278, 0.8578105637], [0.1421894363, 0.3492443278], [0.3492443278, 0.1421894363], [0.8578105637, 0.3492443278]] > F:=4.5*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],10000,10030) [[0.2052537030, 0.4859531780], [0.3773416172, 0.2052537030], [0.8402829908, 0.3773416172], [0.3760443734, 0.8402829908], [0.1686384039, 0.3760443734], [0.3936521801, 0.1686384039], [0.8929701756, 0.3936521801], [0.2607810930, 0.8929701756], [0.09284670464, 0.2607810930], [0.2801771781, 0.09284670464], [0.8232875824, 0.2801771781], [0.4712558553, 0.8232875824], [0.1981444501, 0.4712558553], [0.3780385919, 0.1981444501], [0.8484147850, 0.3780385919], [0.3599490438, 0.8484147850], [0.1571538167, 0.3599490438], [0.3815050771, 0.1571538167], [0.8949469640, 0.3815050771], [0.2616703767, 0.8949469640], [0.09133263238, 0.2616703767], [0.2757361543, 0.09133263238], [0.8165973496, 0.2757361543], [0.4881158312, 0.8165973496], [0.2062114200, 0.4881158312], [0.3770524538, 0.2062114200], [0.8390167119, 0.3770524538], [0.3786303287, 0.8390167119], [0.1704349893, 0.3786303287], [0.3953409025, 0.1704349893], [0.8923706559, 0.3953409025]] > F:=4.6*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.8779791049, 0.3803416771], [0.3053715024, 0.8779791049], [0.1190619909, 0.3053715024], [0.3351420463, 0.1190619909], [0.9029443104, 0.3351420463], [0.2680209028, 0.9029443104], [0.08758831600, 0.2680209028], [0.2690875011, 0.08758831600], [0.8254819549, 0.2690875011], [0.4843632045, 0.8254819549], [0.2004994639, 0.4843632045]] > F:=4.7*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.8909176514, 0.3806141395], [0.2829118824, 0.8909176514], [0.1040102290, 0.2829118824], [0.3140866578, 0.1040102290], [0.9072346912, 0.3140866578], [0.2713140922, 0.9072346912], [0.08619779870, 0.2713140922], [0.2697656219, 0.08619779870], [0.8460556621, 0.2697656219], [0.4470157129, 0.8460556621], [0.1788533826, 0.4470157129]] > F:=4.8*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.9346805674, 0.6303970056], [0.1083134366, 0.9346805674], [0.03028155680, 0.1083134366], [0.1256832245, 0.03028155680], [0.5114851373, 0.1256832245], [1.048626548, 0.5114851373], [-0.1195675436, 1.048626548], [0.03124483851, -0.1195675436], [0.1626611547, 0.03124483851], [0.6333450158, 0.1626611547], [0.9333411778, 0.6333450158]] > F:=4.9*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)]] > F:=5.0*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)], [Float(infinity), -Float(infinity)], [-Float(infinity), Float(infinity)]] > #We see that when k is between 0 and 1, it exhibits the same behavior as the > original discrete logistic equation of first order. Though we see that for > this case where we include 1 in our bounds, our equation has no stability. > Once k reaches 1.1 we reach a stable equilibrium which is not 0. We'll check > with SFP what these stable equilibrium are. > F:=1.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > FP(A,[z[1],z[2]]) {[0., 0.], [0.04653741075, 0.04653741075], [1.953462589, 1.953462589]} > SFP(A,[z[1],z[2]]) {[0.04653741075, 0.04653741075]} > F:=1.0*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > FP(A,[z[1],z[2]]) {[0., 0.], [2., 2.]} > SFP(A,[z[1],z[2]]) {} > F:=4.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > FP(A,[z[1],z[2]]) {[0., 0.], [0.5061352017, 0.5061352017], [1.493864798, 1.493864798]} > SFP(A,[z[1],z[2]]) {} > F:=.8*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > FP(A,[z[1],z[2]]) {[-0.1180339888, -0.1180339888], [0., 0.], [2.118033989, 2.118033989]} > SFP(A,[z[1],z[2]]) {[0., 0.]} > #It seems that even though it appears that our Orb function shows that our > values of z[1] & z[2] get lower thus closer and closer to 0 when k = 1.0, the > SFP function shows that there is no stable fixed point at that value! Let's > try again, once more with the Orb function to see what happens when we check > the activity of this equation many time steps in the future. > F:=1.0*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000000,1000010) [[ -7 -7] [ -7 -7] [[4.999901750 10 , 4.999906750 10 ], [4.999896750 10 , 4.999901750 10 ], [ -7 -7] [ -7 -7] [4.999891750 10 , 4.999896750 10 ], [4.999886750 10 , 4.999891750 10 ], [ -7 -7] [ -7 -7] [4.999881750 10 , 4.999886750 10 ], [4.999876750 10 , 4.999881750 10 ], [ -7 -7] [ -7 -7] [4.999871750 10 , 4.999876750 10 ], [4.999866750 10 , 4.999871750 10 ], [ -7 -7] [ -7 -7] [4.999861750 10 , 4.999866750 10 ], [4.999856750 10 , 4.999861750 10 ], [ -7 -7]] [4.999851750 10 , 4.999856750 10 ]] > #Even when going one million timesteps in the future, we do not reach 0, the > number is in fact small, but it took us a long time to get from our starting > values to this point. Going to modify the initial conditions and see if this > same activity is still seen one million timesteps forward. > Orb(A,[z[1],z[2]],[.001,.002],1000000,1000010) [[ -7 -7] [ -7 -7] [[4.997452787 10 , 4.997457781 10 ], [4.997447793 10 , 4.997452787 10 ], [ -7 -7] [ -7 -7] [4.997442799 10 , 4.997447793 10 ], [4.997437805 10 , 4.997442799 10 ], [ -7 -7] [ -7 -7] [4.997432811 10 , 4.997437805 10 ], [4.997427817 10 , 4.997432811 10 ], [ -7 -7] [ -7 -7] [4.997422823 10 , 4.997427817 10 ], [4.997417829 10 , 4.997422823 10 ], [ -7 -7] [ -7 -7] [4.997412835 10 , 4.997417829 10 ], [4.997407841 10 , 4.997412835 10 ], [ -7 -7]] [4.997402847 10 , 4.997407841 10 ]] > #Even with initial conditions much closer to 0 than our previous initial > conditions, we still do not get that much closer to 0. It seems there really > is no stability at [0,0] which I was expecting based on my initial analysis. > Orb(A,[z[1],z[2]],[.9,.99],1000000,1000010) [[ -7 -7] [ -7 -7] [[4.972308280 10 , 4.972313224 10 ], [4.972303336 10 , 4.972308280 10 ], [ -7 -7] [ -7 -7] [4.972298392 10 , 4.972303336 10 ], [4.972293448 10 , 4.972298392 10 ], [ -7 -7] [ -7 -7] [4.972288504 10 , 4.972293448 10 ], [4.972283560 10 , 4.972288504 10 ], [ -7 -7] [ -7 -7] [4.972278616 10 , 4.972283560 10 ], [4.972273672 10 , 4.972278616 10 ], [ -7 -7] [ -7 -7] [4.972268728 10 , 4.972273672 10 ], [4.972263784 10 , 4.972268728 10 ], [ -7 -7]] [4.972258840 10 , 4.972263784 10 ]] > #Using initial conditions closer to 1, in turn make our spare capacity of the > (n-1) & (n-2) generations a smaller number, I thought this may show us numbers > even closer to 0 after one million iterations. Though it did show that, it was > nothing that was significantly closer to 0 like I hoped. The equation just has > a really small rate of decay it seems. > #Since I finished playing around with that, I will now use FP & SFP to check > for stability of fixed points when k=2.3 and above. As stability isn't so > clear through the Orb function when we reach that 2.3 value. > F:=2.3*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > FP(A,[z[1],z[2]]) {[0., 0.], [0.3406195266, 0.3406195266], [1.659380473, 1.659380473]} > SFP(A,[z[1],z[2]]) {[0.3406195266, 0.3406195266]} > #We'll say that the not obvious case of stability with the Orb funciton is a > result of digit erros in Maple. > #It also is not too apparent if our bifurcation in a periodic orbit begins at > 4.0 for a value of k or prior to that so more testing with SFP function will > allow us to really see what the case is. Since we still have stability at k > value of 2.3 we'll see if this equation exhibits the same behavior as the > original at a k value of 3.0 as that was when we really start to see stability > of a fixed point be lost. > F:=3.0*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.4226497308, 0.4226497308]} > F:=3.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.4320381658, 0.4320381658]} > F:=3.2*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.4409830056, 0.4409830056]} > #Seems we still have stability when the original equation would already be > going into a period 4 orbit, our clearest indication of a move away from this > stability is when k=4 so we will see what SFP has to say about it. > F:=4.0*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {} > #It is apparent that we do in fact lose stability here. > F:=3.9*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.4936303165, 0.4936303165]} > #Somewhere between 3.9 & 4 we have our first bifurcation, through rigorous > testing with SFP we'll be able to find it. Could use Orb function but that > will take much more time since the stability isn't all that clear at some > points. > F:=3.99000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.4993738257, 0.4993738257]} > F:=3.99900*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.4999374883, 0.4999374883]} > F:=3.99990*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.4999937499, 0.4999937499]} > F:=3.99999*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.4999993750, 0.4999993750]} > F:=3.999999*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.4999999375, 0.4999999375]} > F:=3.9999999999*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {} > F:=3.9999999994*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0.5000000000, 0.5000000000]} > F:=3.9999999995*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {} > #Looks like we finally lost our stability at 3.9999999995, this is our > bifurcation value of k from when we go to one stable fixed point to a period > of 2 orbit. > #Let's look back at our case when k=1 and check for the eigenvalues of the > corresponding Jacobian Matrix > a1:=diff(z[1]*(1-z[1])*(1-z[2]),z[1]) > a2:=diff(z[1]*(1-z[1])*(1-z[2]),z[2]) > a21:=diff(z[1],z[1]) > a22:=diff(z[1],z[2]) > a11:=subs({z[1]=0,z[2]=0},a1) > a12:=subs({z[1]=0,z[2]=0},a2) > with(LinearAlgebra): > J:=Matrix([[a11,a12],[a21,a22]]) > evalf(Eigenvalues(J)) > #We may able to say that the point at (0,0) is a semistable fixed point for > the equation x(n)=x(n-1)*(1-x(n-1))*(1-x(n-2)) > #Now we'll try negative values of k for this second order version. > F:=-.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -1002 -1001] [[9.029000687 10 , -9.029000687 10 ], [ -1003 -1002] [-9.029000687 10 , 9.029000687 10 ], [ -1004 -1003] [9.029000687 10 , -9.029000687 10 ], [ -1005 -1004] [-9.029000687 10 , 9.029000687 10 ], [ -1006 -1005] [9.029000687 10 , -9.029000687 10 ], [ -1007 -1006] [-9.029000687 10 , 9.029000687 10 ], [ -1008 -1007] [9.029000687 10 , -9.029000687 10 ], [ -1009 -1008] [-9.029000687 10 , 9.029000687 10 ], [ -1010 -1009] [9.029000687 10 , -9.029000687 10 ], [ -1011 -1010] [-9.029000687 10 , 9.029000687 10 ], [ -1012 -1011]] [9.029000687 10 , -9.029000687 10 ]] > F:=-.5*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[ -303 -302] [[9.014732655 10 , -1.802946531 10 ], [ -303 -303] [-4.507366328 10 , 9.014732655 10 ], [ -303 -303] [2.253683164 10 , -4.507366328 10 ], [ -303 -303] [-1.126841582 10 , 2.253683164 10 ], [ -304 -303] [5.634207910 10 , -1.126841582 10 ], [ -304 -304] [-2.817103955 10 , 5.634207910 10 ], [ -304 -304] [1.408551978 10 , -2.817103955 10 ], [ -305 -304] [-7.042759890 10 , 1.408551978 10 ], [ -305 -305] [3.521379945 10 , -7.042759890 10 ], [ -305 -305] [-1.760689972 10 , 3.521379945 10 ], [ -306 -305]] [8.803449860 10 , -1.760689972 10 ]] > F:=-1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.02183846301, -0.02184912152], [-0.02182827553, 0.02183846301], [0.02181764771, -0.02182827553], [-0.02180748912, 0.02181764771], [0.02179689184, -0.02180748912], [-0.02178676199, 0.02179689184], [0.02177619512, -0.02178676199], [-0.02176609389, 0.02177619512], [0.02175555728, -0.02176609389], [-0.02174548453, 0.02175555728], [0.02173497804, -0.02174548453]] > SFP(A,[z[1],z[2]]) {} > F:=-1.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3015113443, -0.3015113443], [-0.3015113443, 0.3015113443], [0.3015113443, -0.3015113443], [-0.3015113443, 0.3015113443], [0.3015113443, -0.3015113443], [-0.3015113443, 0.3015113443], [0.3015113443, -0.3015113443], [-0.3015113443, 0.3015113443], [0.3015113443, -0.3015113443], [-0.3015113443, 0.3015113443], [0.3015113443, -0.3015113443]] > SFP(A,[z[1],z[2]]) {} > F:=-1.3*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4803844614, -0.4803844612], [-0.4803844612, 0.4803844614], [0.4803844610, -0.4803844612], [-0.4803844612, 0.4803844610], [0.4803844614, -0.4803844612], [-0.4803844612, 0.4803844614], [0.4803844610, -0.4803844612], [-0.4803844612, 0.4803844610], [0.4803844614, -0.4803844612], [-0.4803844612, 0.4803844614], [0.4803844610, -0.4803844612]] > F:=-.1*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0., 0.]} > F:=-.99999999994*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {[0., 0.]} > F:=-.99999999995*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > SFP(A,[z[1],z[2]]) {} > #Lose stability of [0,0] when we reach a k value of -.99999999995. From there > we no longer have a stable fixed point. We reach periodic orbits. Therefore > this value of k is another bifurcation point. > F:=-.99999999995*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.02183846301, -0.02184912152], [-0.02182827553, 0.02183846301], [0.02181764771, -0.02182827553], [-0.02180748912, 0.02181764771], [0.02179689184, -0.02180748912], [-0.02178676199, 0.02179689184], [0.02177619512, -0.02178676199], [-0.02176609389, 0.02177619512], [0.02175555728, -0.02176609389], [-0.02174548453, 0.02175555728], [0.02173497804, -0.02174548453]] > #It is less clear at these values of -.99999999995 & -1, but we do have a > orbit of period 2 at these values of k, it is abundantly more clear when we > reach values of -1.1. Let's see when this behavior ends. > F:=-1.3*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4803844614, -0.4803844612], [-0.4803844612, 0.4803844614], [0.4803844610, -0.4803844612], [-0.4803844612, 0.4803844610], [0.4803844614, -0.4803844612], [-0.4803844612, 0.4803844614], [0.4803844610, -0.4803844612], [-0.4803844612, 0.4803844610], [0.4803844614, -0.4803844612], [-0.4803844612, 0.4803844614], [0.4803844610, -0.4803844612]] > #It seems our period doubles pretty quickly, as at k=-1.3 we now have a period > 4 orbit. Time to check where this second bifurcation value is located. > F:=-1.2*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],100000,100010) [[0.4082482903, -0.4082482903], [-0.4082482903, 0.4082482903], [0.4082482903, -0.4082482903], [-0.4082482903, 0.4082482903], [0.4082482903, -0.4082482903], [-0.4082482903, 0.4082482903], [0.4082482903, -0.4082482903], [-0.4082482903, 0.4082482903], [0.4082482903, -0.4082482903], [-0.4082482903, 0.4082482903], [0.4082482903, -0.4082482903]] > #Somewhere between -1.2 & -1.3 we have another period doubling. > F:=-1.290000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4741373239, -0.4741373238], [-0.4741373238, 0.4741373239], [0.4741373236, -0.4741373238], [-0.4741373238, 0.4741373236], [0.4741373239, -0.4741373238], [-0.4741373238, 0.4741373239], [0.4741373236, -0.4741373238], [-0.4741373238, 0.4741373236], [0.4741373239, -0.4741373238], [-0.4741373238, 0.4741373239], [0.4741373236, -0.4741373238]] > F:=-1.209000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4157765513, -0.4157765513], [-0.4157765513, 0.4157765513], [0.4157765512, -0.4157765513], [-0.4157765513, 0.4157765512], [0.4157765513, -0.4157765513], [-0.4157765513, 0.4157765513], [0.4157765512, -0.4157765513], [-0.4157765513, 0.4157765512], [0.4157765513, -0.4157765513], [-0.4157765513, 0.4157765513], [0.4157765512, -0.4157765513]] > F:=-1.200900*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4090124665, -0.4090124665], [-0.4090124665, 0.4090124665], [0.4090124665, -0.4090124665], [-0.4090124665, 0.4090124665], [0.4090124665, -0.4090124665], [-0.4090124665, 0.4090124665], [0.4090124665, -0.4090124665], [-0.4090124665, 0.4090124665], [0.4090124665, -0.4090124665], [-0.4090124665, 0.4090124665], [0.4090124665, -0.4090124665]] > F:=-1.208999*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4157757284, -0.4157757284], [-0.4157757284, 0.4157757284], [0.4157757284, -0.4157757284], [-0.4157757284, 0.4157757284], [0.4157757284, -0.4157757284], [-0.4157757284, 0.4157757284], [0.4157757284, -0.4157757284], [-0.4157757284, 0.4157757284], [0.4157757284, -0.4157757284], [-0.4157757284, 0.4157757284], [0.4157757284, -0.4157757284]] > F:=-1.209001*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4157773739, -0.4157773741], [-0.4157773741, 0.4157773739], [0.4157773742, -0.4157773741], [-0.4157773741, 0.4157773742], [0.4157773739, -0.4157773741], [-0.4157773741, 0.4157773739], [0.4157773742, -0.4157773741], [-0.4157773741, 0.4157773742], [0.4157773739, -0.4157773741], [-0.4157773741, 0.4157773739], [0.4157773742, -0.4157773741]] > > F:=-1.209005*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4157806645, -0.4157806645], [-0.4157806645, 0.4157806645], [0.4157806645, -0.4157806645], [-0.4157806645, 0.4157806645], [0.4157806645, -0.4157806645], [-0.4157806645, 0.4157806645], [0.4157806645, -0.4157806645], [-0.4157806645, 0.4157806645], [0.4157806645, -0.4157806645], [-0.4157806645, 0.4157806645], [0.4157806645, -0.4157806645]] > F:=-1.209090*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4158505849, -0.4158505849], [-0.4158505849, 0.4158505849], [0.4158505849, -0.4158505849], [-0.4158505849, 0.4158505849], [0.4158505849, -0.4158505849], [-0.4158505849, 0.4158505849], [0.4158505849, -0.4158505849], [-0.4158505849, 0.4158505849], [0.4158505849, -0.4158505849], [-0.4158505849, 0.4158505849], [0.4158505849, -0.4158505849]] > F:=-1.209900*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4165158002, -0.4165158002], [-0.4165158002, 0.4165158002], [0.4165158002, -0.4165158002], [-0.4165158002, 0.4165158002], [0.4165158002, -0.4165158002], [-0.4165158002, 0.4165158002], [0.4165158002, -0.4165158002], [-0.4165158002, 0.4165158002], [0.4165158002, -0.4165158002], [-0.4165158002, 0.4165158002], [0.4165158002, -0.4165158002]] > F:=-1.209999*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4165969705, -0.4165969705], [-0.4165969705, 0.4165969705], [0.4165969704, -0.4165969705], [-0.4165969705, 0.4165969704], [0.4165969705, -0.4165969705], [-0.4165969705, 0.4165969705], [0.4165969704, -0.4165969705], [-0.4165969705, 0.4165969704], [0.4165969705, -0.4165969705], [-0.4165969705, 0.4165969705], [0.4165969704, -0.4165969705]] > F:=-1.210000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4165977902, -0.4165977902], [-0.4165977902, 0.4165977902], [0.4165977902, -0.4165977902], [-0.4165977902, 0.4165977902], [0.4165977902, -0.4165977902], [-0.4165977902, 0.4165977902], [0.4165977902, -0.4165977902], [-0.4165977902, 0.4165977902], [0.4165977902, -0.4165977902], [-0.4165977902, 0.4165977902], [0.4165977902, -0.4165977902]] > F:=-1.250000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4472135958, -0.4472135956], [-0.4472135956, 0.4472135958], [0.4472135955, -0.4472135956], [-0.4472135956, 0.4472135955], [0.4472135958, -0.4472135956], [-0.4472135956, 0.4472135958], [0.4472135955, -0.4472135956], [-0.4472135956, 0.4472135955], [0.4472135958, -0.4472135956], [-0.4472135956, 0.4472135958], [0.4472135955, -0.4472135956]] > F:=-1.260000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4542567627, -0.4542567624], [-0.4542567624, 0.4542567627], [0.4542567621, -0.4542567624], [-0.4542567624, 0.4542567621], [0.4542567627, -0.4542567624], [-0.4542567624, 0.4542567627], [0.4542567621, -0.4542567624], [-0.4542567624, 0.4542567621], [0.4542567627, -0.4542567624], [-0.4542567624, 0.4542567627], [0.4542567621, -0.4542567624]] > F:=-1.270000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4610839677, -0.4610839677], [-0.4610839677, 0.4610839677], [0.4610839677, -0.4610839677], [-0.4610839677, 0.4610839677], [0.4610839677, -0.4610839677], [-0.4610839677, 0.4610839677], [0.4610839677, -0.4610839677], [-0.4610839677, 0.4610839677], [0.4610839677, -0.4610839677], [-0.4610839677, 0.4610839677], [0.4610839677, -0.4610839677]] > F:=-1.299000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4797677143, -0.4797677139], [-0.4797677139, 0.4797677143], [0.4797677135, -0.4797677139], [-0.4797677139, 0.4797677135], [0.4797677143, -0.4797677139], [-0.4797677139, 0.4797677143], [0.4797677135, -0.4797677139], [-0.4797677139, 0.4797677135], [0.4797677143, -0.4797677139], [-0.4797677139, 0.4797677143], [0.4797677135, -0.4797677139]] > F:=-1.298000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4791492207, -0.4791492203], [-0.4791492203, 0.4791492207], [0.4791492199, -0.4791492203], [-0.4791492203, 0.4791492199], [0.4791492207, -0.4791492203], [-0.4791492203, 0.4791492207], [0.4791492199, -0.4791492203], [-0.4791492203, 0.4791492199], [0.4791492207, -0.4791492203], [-0.4791492203, 0.4791492207], [0.4791492199, -0.4791492203]] > F:=-1.297000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4785289719, -0.4785289720], [-0.4785289720, 0.4785289719], [0.4785289719, -0.4785289720], [-0.4785289720, 0.4785289719], [0.4785289719, -0.4785289720], [-0.4785289720, 0.4785289719], [0.4785289719, -0.4785289720], [-0.4785289720, 0.4785289719], [0.4785289719, -0.4785289720], [-0.4785289720, 0.4785289719], [0.4785289719, -0.4785289720]] > F:=-1.297500*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4788393161, -0.4788393161], [-0.4788393161, 0.4788393161], [0.4788393160, -0.4788393161], [-0.4788393161, 0.4788393160], [0.4788393161, -0.4788393161], [-0.4788393161, 0.4788393161], [0.4788393160, -0.4788393161], [-0.4788393161, 0.4788393160], [0.4788393161, -0.4788393161], [-0.4788393161, 0.4788393161], [0.4788393160, -0.4788393161]] > F:=-1.297499*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4788386962, -0.4788386958], [-0.4788386958, 0.4788386962], [0.4788386955, -0.4788386958], [-0.4788386958, 0.4788386955], [0.4788386962, -0.4788386958], [-0.4788386958, 0.4788386962], [0.4788386955, -0.4788386958], [-0.4788386958, 0.4788386955], [0.4788386962, -0.4788386958], [-0.4788386958, 0.4788386962], [0.4788386955, -0.4788386958]] > F:=-1.297050*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4785600264, -0.4785600260], [-0.4785600260, 0.4785600264], [0.4785600257, -0.4785600260], [-0.4785600260, 0.4785600257], [0.4785600264, -0.4785600260], [-0.4785600260, 0.4785600264], [0.4785600257, -0.4785600260], [-0.4785600260, 0.4785600257], [0.4785600264, -0.4785600260], [-0.4785600260, 0.4785600264], [0.4785600257, -0.4785600260]] > F:=-1.297005*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4785320783, -0.4785320776], [-0.4785320776, 0.4785320783], [0.4785320770, -0.4785320776], [-0.4785320776, 0.4785320770], [0.4785320783, -0.4785320776], [-0.4785320776, 0.4785320783], [0.4785320770, -0.4785320776], [-0.4785320776, 0.4785320770], [0.4785320783, -0.4785320776], [-0.4785320776, 0.4785320783], [0.4785320770, -0.4785320776]] > F:=-1.297004*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4785314564, -0.4785314561], [-0.4785314561, 0.4785314564], [0.4785314559, -0.4785314561], [-0.4785314561, 0.4785314559], [0.4785314564, -0.4785314561], [-0.4785314561, 0.4785314564], [0.4785314559, -0.4785314561], [-0.4785314561, 0.4785314559], [0.4785314564, -0.4785314561], [-0.4785314561, 0.4785314564], [0.4785314559, -0.4785314561]] > F:=-1.297003*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4785308353, -0.4785308352], [-0.4785308352, 0.4785308353], [0.4785308350, -0.4785308352], [-0.4785308352, 0.4785308350], [0.4785308353, -0.4785308352], [-0.4785308352, 0.4785308353], [0.4785308350, -0.4785308352], [-0.4785308352, 0.4785308350], [0.4785308353, -0.4785308352], [-0.4785308352, 0.4785308353], [0.4785308350, -0.4785308352]] > F:=-1.297002*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4785302149, -0.4785302141], [-0.4785302141, 0.4785302149], [0.4785302133, -0.4785302141], [-0.4785302141, 0.4785302133], [0.4785302149, -0.4785302141], [-0.4785302141, 0.4785302149], [0.4785302133, -0.4785302141], [-0.4785302141, 0.4785302133], [0.4785302149, -0.4785302141], [-0.4785302141, 0.4785302149], [0.4785302133, -0.4785302141]] > F:=-1.297001*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.4785295935, -0.4785295929], [-0.4785295929, 0.4785295935], [0.4785295923, -0.4785295929], [-0.4785295929, 0.4785295923], [0.4785295935, -0.4785295929], [-0.4785295929, 0.4785295935], [0.4785295923, -0.4785295929], [-0.4785295929, 0.4785295923], [0.4785295935, -0.4785295929], [-0.4785295929, 0.4785295935], [0.4785295923, -0.4785295929]] > #Appears that our orbit becomes a period of 4 at the k value equal to > -1.297001. > F:=-1.5*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],10000,10030) [[0.3174085305, -0.4654678880], [-0.4654678910, 0.3174085305], [0.6825914780, -0.4654678910], [-0.4654678853, 0.6825914780], [0.3174085207, -0.4654678853], [-0.4654678823, 0.3174085207], [0.6825914708, -0.4654678823], [-0.4654678880, 0.6825914708], [0.3174085305, -0.4654678880], [-0.4654678910, 0.3174085305], [0.6825914780, -0.4654678910], [-0.4654678853, 0.6825914780], [0.3174085207, -0.4654678853], [-0.4654678823, 0.3174085207], [0.6825914708, -0.4654678823], [-0.4654678880, 0.6825914708], [0.3174085305, -0.4654678880], [-0.4654678910, 0.3174085305], [0.6825914780, -0.4654678910], [-0.4654678853, 0.6825914780], [0.3174085207, -0.4654678853], [-0.4654678823, 0.3174085207], [0.6825914708, -0.4654678823], [-0.4654678880, 0.6825914708], [0.3174085305, -0.4654678880], [-0.4654678910, 0.3174085305], [0.6825914780, -0.4654678910], [-0.4654678853, 0.6825914780], [0.3174085207, -0.4654678853], [-0.4654678823, 0.3174085207], [0.6825914708, -0.4654678823]] > #We notice a period of 8 at k=-1.5, the bifurcation value should be somewhere > between -1.4 and -1.5. > F:=-1.490000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.2579655150, -0.4453939761], [-0.4122477091, 0.2579655150], [0.6436940388, -0.4122477091], [-0.4826137853, 0.6436940388], [0.3798718514, -0.4826137853], [-0.5203946951, 0.3798718514], [0.7310665611, -0.5203946951], [-0.4453939761, 0.7310665611], [0.2579655150, -0.4453939761], [-0.4122477091, 0.2579655150], [0.6436940388, -0.4122477091]] > F:=-1.409000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3588212866, -0.4796546372], [-0.4796546372, 0.3588212866], [0.6411787135, -0.4796546372], [-0.4796546372, 0.6411787135], [0.3588212866, -0.4796546372], [-0.4796546372, 0.3588212866], [0.6411787135, -0.4796546372], [-0.4796546372, 0.6411787135], [0.3588212866, -0.4796546372], [-0.4796546372, 0.3588212866], [0.6411787135, -0.4796546372]] > F:=-1.480000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3603351325, -0.4773573444], [-0.5039719637, 0.3603351325], [0.7175634865, -0.5039719637], [-0.4511101812, 0.7175634865], [0.2736311749, -0.4511101812], [-0.4268594262, 0.2736311749], [0.6547642767, -0.4268594262], [-0.4773573444, 0.6547642767], [0.3603351325, -0.4773573444], [-0.5039719637, 0.3603351325], [0.7175634865, -0.5039719637]] > F:=-1.450000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3276545574, -0.4693581238], [-0.4693581238, 0.3276545574], [0.6723454425, -0.4693581238], [-0.4693581238, 0.6723454425], [0.3276545574, -0.4693581238], [-0.4693581238, 0.3276545574], [0.6723454425, -0.4693581238], [-0.4693581238, 0.6723454425], [0.3276545574, -0.4693581238], [-0.4693581238, 0.3276545574], [0.6723454425, -0.4693581238]] > F:=-1.460000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3211507468, -0.4669185631], [-0.4669185637, 0.3211507468], [0.6788492554, -0.4669185637], [-0.4669185624, 0.6788492554], [0.3211507443, -0.4669185624], [-0.4669185615, 0.3211507443], [0.6788492536, -0.4669185615], [-0.4669185631, 0.6788492536], [0.3211507468, -0.4669185631], [-0.4669185637, 0.3211507468], [0.6788492554, -0.4669185637]] > F:=-1.455000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3243591860, -0.4681349209], [-0.4681349228, 0.3243591860], [0.6756408193, -0.4681349228], [-0.4681349193, 0.6756408193], [0.3243591800, -0.4681349193], [-0.4681349174, 0.3243591800], [0.6756408148, -0.4681349174], [-0.4681349209, 0.6756408148], [0.3243591860, -0.4681349209], [-0.4681349228, 0.3243591860], [0.6756408193, -0.4681349228]] > F:=-1.453000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3256666109, -0.4686233768], [-0.4686233774, 0.3256666109], [0.6743333904, -0.4686233774], [-0.4686233765, 0.6743333904], [0.3256666094, -0.4686233765], [-0.4686233760, 0.3256666094], [0.6743333894, -0.4686233760], [-0.4686233768, 0.6743333894], [0.3256666109, -0.4686233768], [-0.4686233774, 0.3256666109], [0.6743333904, -0.4686233774]] > F:=-1.451000*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3269882775, -0.4691129328], [-0.4691129336, 0.3269882775], [0.6730117248, -0.4691129336], [-0.4691129323, 0.6730117248], [0.3269882750, -0.4691129323], [-0.4691129314, 0.3269882750], [0.6730117228, -0.4691129314], [-0.4691129328, 0.6730117228], [0.3269882775, -0.4691129328], [-0.4691129336, 0.3269882775], [0.6730117248, -0.4691129336]] > F:=-1.450900*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3270547400, -0.4691374394], [-0.4691374394, 0.3270547400], [0.6729452601, -0.4691374394], [-0.4691374394, 0.6729452601], [0.3270547400, -0.4691374394], [-0.4691374394, 0.3270547400], [0.6729452601, -0.4691374394], [-0.4691374394, 0.6729452601], [0.3270547400, -0.4691374394], [-0.4691374394, 0.3270547400], [0.6729452601, -0.4691374394]] > F:=-1.450990*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3269949231, -0.4691153837], [-0.4691153850, 0.3269949231], [0.6730050805, -0.4691153850], [-0.4691153825, 0.6730050805], [0.3269949189, -0.4691153825], [-0.4691153812, 0.3269949189], [0.6730050775, -0.4691153812], [-0.4691153837, 0.6730050775], [0.3269949231, -0.4691153837], [-0.4691153850, 0.3269949231], [0.6730050805, -0.4691153850]] > F:=-1.450950*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3270215049, -0.4691251859], [-0.4691251868, 0.3270215049], [0.6729784977, -0.4691251868], [-0.4691251852, 0.6729784977], [0.3270215020, -0.4691251852], [-0.4691251843, 0.3270215020], [0.6729784955, -0.4691251843], [-0.4691251859, 0.6729784955], [0.3270215049, -0.4691251859], [-0.4691251868, 0.3270215049], [0.6729784977, -0.4691251868]] > F:=-1.450910*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3270480927, -0.4691349887], [-0.4691349895, 0.3270480927], [0.6729519095, -0.4691349895], [-0.4691349882, 0.6729519095], [0.3270480902, -0.4691349882], [-0.4691349873, 0.3270480902], [0.6729519076, -0.4691349873], [-0.4691349887, 0.6729519076], [0.3270480927, -0.4691349887], [-0.4691349895, 0.3270480927], [0.6729519095, -0.4691349895]] > F:=-1.450905*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3270514151, -0.4691362137], [-0.4691362137, 0.3270514151], [0.6729485848, -0.4691362137], [-0.4691362137, 0.6729485848], [0.3270514151, -0.4691362137], [-0.4691362137, 0.3270514151], [0.6729485848, -0.4691362137], [-0.4691362137, 0.6729485848], [0.3270514151, -0.4691362137], [-0.4691362137, 0.3270514151], [0.6729485848, -0.4691362137]] > F:=-1.450906*z[1]*(1-z[1])*(1-z[2]): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.3,.4],1000,1010) [[0.3270507520, -0.4691359693], [-0.4691359697, 0.3270507520], [0.6729492500, -0.4691359697], [-0.4691359685, 0.6729492500], [0.3270507497, -0.4691359685], [-0.4691359677, 0.3270507497], [0.6729492484, -0.4691359677], [-0.4691359693, 0.6729492484], [0.3270507520, -0.4691359693], [-0.4691359697, 0.3270507520], [0.6729492500, -0.4691359697]] > #Orbit of period 8 starts to occur at bifurcation value of k=-1.450906. > #Now that we've done work on the second order version of this we will now add > 3 more parameters (a,b,c) and look at the behavior at different values of k, > a, b, c for the equation x(n)=k*x(n-1)^a*((1-x(n-1)^b)*((1-x(n-2)^c) > F:=1*(z[1]^.5)*((1-z[1])^.3)*((1-z[2])^.4): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459], [0.4419351459, 0.4419351459]] > FP(A,[z[1],z[2]]) {[0., 0.], [0.4419351459, 0.4419351459], [0.4386283803 - 0.2913789343 I, 0.4386283803 - 0.2913789343 I], [0.4386283803 + 0.2913789343 I, 0.4386283803 + 0.2913789343 I], [0.5000000000 - 0.8660254038 I, 0.5000000000 - 0.8660254038 I], [0.5000000000 + 0.8660254038 I, 0.5000000000 + 0.8660254038 I], [2.340404047 - 1.637875945 I, 2.340404047 - 1.637875945 I], [2.340404047 + 1.637875945 I, 2.340404047 + 1.637875945 I]} > SFP(A,[z[1],z[2]]) Error, (in sqrfree) argument must be a polynomial or a rational function in lam > F:=.4*(z[1]^.5)*((1-z[1])^.3)*((1-z[2])^.4): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370], [0.1313681370, 0.1313681370]] > FP(A,[z[1],z[2]]) {[0., 0.], [0.1313681370, 0.1313681370], [-0.1347156998 - 0.1381284451 I, -0.1347156998 - 0.1381284451 I], [-0.1347156998 + 0.1381284451 I, -0.1347156998 + 0.1381284451 I], [0.07086115858 - 0.1279552054 I, 0.07086115858 - 0.1279552054 I], [0.07086115858 + 0.1279552054 I, 0.07086115858 + 0.1279552054 I], [3.498170472 - 97.70095549 I, 3.498170472 - 97.70095549 I], [3.498170472 + 97.70095549 I, 3.498170472 + 97.70095549 I]} > SFP(A,[z[1],z[2]]) Error, (in sqrfree) argument must be a polynomial or a rational function in lam > F:=.1*(z[1]^.5)*((1-z[1])^.3)*((1-z[2])^.4): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046], [0.009862202046, 0.009862202046]] > FP(A,[z[1],z[2]]) / { [0., 0.], [0.009862202043, 0.009862202043], \ [-0.00813270815 - 0.006013153504 I, -0.00813270815 - 0.006013153504 I], [-0.00813270815 + 0.006013153504 I, -0.00813270815 + 0.006013153504 I], [0.003201608882 - 0.009426995323 I, 0.003201608882 - 0.009426995323 I], [0.003201608882 + 0.009426995323 I, 0.003201608882 + 0.009426995323 I], [ 5 5 ] [3.499999998 - 1.000000000 10 I, 3.499999998 - 1.000000000 10 I], [ 5 5 ]\ [3.499999998 + 1.000000000 10 I, 3.499999998 + 1.000000000 10 I] } / > SFP(A,[z[1],z[2]]) Error, (in sqrfree) argument must be a polynomial or a rational function in lam > F:=1.1*(z[1]^.5)*((1-z[1])^.3)*((1-z[2])^.4): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020], [0.4819036020, 0.4819036020]] > F:=1*(z[1]^.5)*((1-z[1])^2)*((1-z[2])^3): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573], [0.1649209573, 0.1649209573]] > FP(A,[z[1],z[2]]) {[0., 0.], [0.1649209573, 0.1649209573], [0.5938152686 - 0.9241055305 I, 0.5938152686 - 0.9241055305 I], [0.5938152686 + 0.9241055305 I, 0.5938152686 + 0.9241055305 I], [1.843146650 - 0.6578940462 I, 1.843146650 - 0.6578940462 I], [1.843146650 + 0.6578940462 I, 1.843146650 + 0.6578940462 I]} > SFP(A,[z[1],z[2]]) Error, (in sqrfree) argument must be a polynomial or a rational function in lam > F:=.4*(z[1]^.5)*((1-z[1])^2)*((1-z[2])^3): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160], [0.07409456160, 0.07409456160]] > F:=1*(z[1]^.5)*((1-z[1])^10)*((1-z[2])^.4): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1033672968, 0.1033672969], [0.1033672969, 0.1033672968], [0.1033672968, 0.1033672969], [0.1033672969, 0.1033672968], [0.1033672968, 0.1033672969], [0.1033672969, 0.1033672968], [0.1033672968, 0.1033672969], [0.1033672969, 0.1033672968], [0.1033672968, 0.1033672969], [0.1033672969, 0.1033672968], [0.1033672968, 0.1033672969]] > > F:=1*(z[1]^.5)*((1-z[1])^(-10))*((1-z[2])^.4): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[-0.2082947259 - 0.2227373119 I, -0.01620044006 - 0.1525716901 I], [-0.06954347223 - 0.01532620806 I, -0.2082947259 - 0.2227373119 I], [0.005627389154 - 0.1476739004 I, -0.06954347223 - 0.01532620806 I], [-0.2309389894 - 0.2950276412 I, 0.005627389154 - 0.1476739004 I], [-0.05599378182 + 0.01545457837 I, -0.2309389894 - 0.2950276412 I], [-0.01620043955 + 0.1525716893 I, -0.05599378182 + 0.01545457837 I], [-0.2082947237 + 0.2227373135 I, -0.01620043955 + 0.1525716893 I], [-0.06954347332 + 0.01532620745 I, -0.2082947237 + 0.2227373135 I], [0.005627389231 + 0.1476738999 I, -0.06954347332 + 0.01532620745 I], [-0.2309389875 + 0.2950276420 I, 0.005627389231 + 0.1476738999 I], [-0.05599378204 - 0.01545457877 I, -0.2309389875 + 0.2950276420 I]] > > F:=.1*(z[1]^.5)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353], [0.008974643353, 0.008974643353]] > #We are seeing some things that are new to us in regards to the behavior of > the equation. With an a value of .5 we seem to have stability to a point that > is not [0,0] even when k is between 0 & 1. This is not something we had seen > prior in the other two cases. We even see in some instances we reach an orbit > of complex numbers, when b=-10 a=.5 and c=.4. This is interesting behavior in > it of itself to have a complex orbit. FP & SFP were not working as well as I > would have hoped for this equation, but we'll see if that continues. We will > now try values of a>1 to see how stability is affected then. > F:=.1*(z[1]^2)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.1*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.5*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.5*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.1*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.2*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.3*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.4*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.5*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2.0*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > FP(A,[z[1],z[2]]) > SFP(A,[z[1],z[2]]) {[0., 0.]} > F:=2.0*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.9,.9],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=3.0*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=4.0*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=5.0*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=10*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(-4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.1*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=3.0*(z[1]^4)*((1-z[1])^(10))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.1*(z[1]^4)*((1-z[1])^(.10))*((1-z[2])^(.4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.1*(z[1]^4)*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=5*(z[1]^4)*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.3*(z[1]^4)*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.1*(z[1]^2)*((1-z[1])^(1.5))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.4*(z[1]^(-4))*((1-z[1])^(2))*((1-z[2])^(2)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[ 7814 -1563] [[1.540250251 10 , 1.007640788 10 ], [ -15629 7814] [1.686077042 10 , 1.540250251 10 ], [ 78143 -15629] [1.174175213 10 , 1.686077042 10 ], [ -156287 78143] [2.901310269 10 , 1.174175213 10 ], [ 781431 -156287] [7.783037212 10 , 2.901310269 10 ], [ -1562865 781431] [6.603311388 10 , 7.783037212 10 ], [ 7814320 -1562865] [1.274415647 10 , 6.603311388 10 ], [ -15628641 7814320] [2.462849089 10 , 1.274415647 10 ], [ 78143202 -15628641] [1.765757161 10 , 2.462849089 10 ], [ -156286405 78143202] [1.282915460 10 , 1.765757161 10 ], [ 781432023 -156286405] [4.603941420 10 , 1.282915460 10 ], [ -1562864048 781432023] [1.887123893 10 , 4.603941420 10 ], [ 7814320237 -1562864048] [6.685254060 10 , 1.887123893 10 ], [ -15628640477 7814320237] [8.950023148 10 , 6.685254060 10 ], [ 78143202379 -15628640477] [2.786115972 10 , 8.950023148 10 ], [ -156286404760 78143202379] [5.153017428 10 , 2.786115972 10 ], [ 781432023795 -156286404760] [4.403641196 10 , 5.153017428 10 ], [ -1562864047592 781432023795] [2.062700336 10 , 4.403641196 10 ], [ 7814320237957 -1562864047592] [4.284886136 10 , 2.062700336 10 ], [ -15628640475916 7814320237957] [2.178619667 10 , 4.284886136 10 ], [ 78143202379577 -15628640475916]] [3.259964596 10 , 2.178619667 10 ]] > #For this case where I gave a negative value to a (-4) I could only go only a > few timesteps into the future as apposed to the typical 1000,1010 or > 10000,10010 that I have been doing. The float becomes undefined, we see that > we bounce back from extraordinarly small numbers to extraordinarly big > numbers. What we care about though, is we have no stability, nor a periodic > orbit. > F:=1.1*(z[1]^(-2))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[ 1996 755] [[3.535470499 10 , -2.111431277 10 ], [ 5017 1996] [-7.729434212 10 , 3.535470499 10 ], [ 13004 5017] [1.328401205 10 , -7.729434212 10 ], [ 33075 13004] [-5.215701957 10 , 1.328401205 10 ], [ 85092 33075] [1.786580112 10 , -5.215701957 10 ], [ 217395 85092] [-1.454340994 10 , 1.786580112 10 ], [ 557764 217395] [1.629854785 10 , -1.454340994 10 ], [ 1427344 557764] [-8.020594878 10 , 1.629854785 10 ], [ 3658401 1427344] [6.225796387 10 , -8.020594878 10 ], [ 9367781 3658401] [-2.834091846 10 , 6.225796387 10 ], [ 24001388 9367781] [4.683669675 10 , -2.834091846 10 ], [ 61472514 24001388] [-3.323798026 10 , 4.683669675 10 ], [ 157478069 61472514] [1.759431418 10 , -3.323798026 10 ], [ 403368127 157478069] [-2.362128558 10 , 1.759431418 10 ], [ 1033280404 403368127] [2.489921108 10 , -2.362128558 10 ], [ 2646752913 1033280404] [-8.526924017 10 , 2.489921108 10 ], [ 6779874531 2646752913] [3.605183922 10 , -8.526924017 10 ], [ 17366886187 6779874531] [-2.096474975 10 , 3.605183922 10 ], [ 44486384313 17366886187] [3.895758888 10 , -2.096474975 10 ], [ 113953929062 44486384313] [-8.278344324 10 , 3.895758888 10 ], [ 291899466317 113953929062]] [2.097511989 10 , -8.278344324 10 ]] > #Similar situation in this case, but we jump back and forth from negative to > positive numbers. Getting increasingly more negative and increasingly more > positive each timestep. > F:=1.1*(z[1]^(-.5))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [ [[16143.37242 + 1283.031044 I, -15.46078115 - 7.180160930 I], [ 9 10 ] [2.837626405 10 + 1.281771334 10 I, 16143.37242 + 1283.031044 I], [ 27 27 9 10 ] [4.507550756 10 - 5.389280999 10 I, 2.837626405 10 + 1.281771334 10 I], [ 72 71 27 27 [1.353277113 10 - 5.629190379 10 I, 4.507550756 10 - 5.389280999 10 I ] [ 191 190 ], [6.751602582 10 - 4.901276301 10 I, 72 71 ] [ 1.353277113 10 - 5.629190379 10 I], [ 503 504 -5.321896900 10 + 1.854162357 10 I, 191 190 ] [ 6.751602582 10 - 4.901276301 10 I], [ 1331 1331 7.629249287 10 - 5.037067927 10 I, 503 504 ] [ -5.321896900 10 + 1.854162357 10 I], [ 3509 3510 2.537275173 10 + 6.897326043 10 I, 1331 1331 ] [ 7.629249287 10 - 5.037067927 10 I], [ 9262 9261 -1.298908584 10 - 7.971503283 10 I, 3509 3510 ] [ 2.537275173 10 + 6.897326043 10 I], [ 24425 24425 -5.136507938 10 - 4.462956260 10 I, 9262 9261 ] [ -1.298908584 10 - 7.971503283 10 I], [ 64424 64425 -8.856946211 10 - 1.998012939 10 I, 24425 24425 ] [ -5.136507938 10 - 4.462956260 10 I], [ 169915 169915 2.373884327 10 - 2.627369870 10 I, 64424 64425 ] [ -8.856946211 10 - 1.998012939 10 I], [ 448149 448149 -1.436721431 10 + 1.946846070 10 I, 169915 169915 ] [ 2.373884327 10 - 2.627369870 10 I], [ 1181970 1181970 -4.042476384 10 - 4.176610517 10 I, 448149 448149 ] [ -1.436721431 10 + 1.946846070 10 I], [ 3117404 3117402 2.182451809 10 - 6.929073059 10 I, 1181970 1181970 ] [ -4.042476384 10 - 4.176610517 10 I], [ 8222018 8222018 -4.935219964 10 + 4.921936445 10 I, 3117404 3117402 ] [ 2.182451809 10 - 6.929073059 10 I], [ 21685241 21685240 2.013544910 10 + 6.217713417 10 I, 8222018 8222018 ] [ -4.935219964 10 + 4.921936445 10 I], [ 57193918 57193918 -1.174801518 10 - 3.406506191 10 I, 21685241 21685240 ] [ 2.013544910 10 + 6.217713417 10 I], [ 150846601 150846601 2.644747448 10 + 6.526137677 10 I, 57193918 57193918 ] [ -1.174801518 10 - 3.406506191 10 I], [ 397851658 397851658 2.143893758 10 - 2.156957967 10 I, 150846601 150846601 ] [ 2.644747448 10 + 6.526137677 10 I], [ 1049317293 1049317293 1.466685746 10 - 1.413520085 10 I, 397851658 397851658 ]] 2.143893758 10 - 2.156957967 10 I]] > #With an a value of -.5 we have an orbit of complex numbers. > F:=1.1*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [ [[0.1362461486 + 0.2302106146 I, 0.6284359866 - 0.2041912299 I], [0.04852575319 + 0.05904824823 I, 0.1362461486 + 0.2302106146 I], [0.5038519375 - 0.7738618492 I, 0.04852575319 + 0.05904824823 I], [-0.2737551189 + 0.7641859004 I, 0.5038519375 - 0.7738618492 I], [-0.3048050581 + 1.900783694 I, -0.2737551189 + 0.7641859004 I], [-14.90447333 + 10.00023027 I, -0.3048050581 + 1.900783694 I], [-1476.510847 + 3244.571459 I, -14.90447333 + 10.00023027 I], [ 10 10 ] [-2.109886261 10 + 3.506027422 10 I, -1476.510847 + 3244.571459 I], [ 30 30 10 10 [6.050222983 10 + 3.993143801 10 I, -2.109886261 10 + 3.506027422 10 I ] [ 90 90 ], [-1.288993624 10 - 2.982144889 10 I, 30 30 ] [ 6.050222983 10 + 3.993143801 10 I], [ 264 264 1.670093738 10 + 3.558857272 10 I, 90 90 ] [ -1.288993624 10 - 2.982144889 10 I], [ 774 774 1.618558662 10 - 1.214925022 10 I, 264 264 ] [ 1.670093738 10 + 3.558857272 10 I], [ 2264 2264 5.749954253 10 - 8.370931467 10 I, 774 774 ] [ 1.618558662 10 - 1.214925022 10 I], [ 6626 6626 2.397305698 10 - 1.751773011 10 I, 2264 2264 ] [ 5.749954253 10 - 8.370931467 10 I], [ 19385 19385 1.304520045 10 - 1.879931490 10 I, 6626 6626 ] [ 2.397305698 10 - 1.751773011 10 I], [ 56711 56711 3.662539021 10 - 2.417789601 10 I, 19385 19385 ] [ 1.304520045 10 - 1.879931490 10 I], [ 165908 165908 1.135267192 10 - 1.317321156 10 I, 56711 56711 ] [ 3.662539021 10 - 2.417789601 10 I], [ 485360 485360 4.093753536 10 - 1.009855994 10 I, 165908 165908 ] [ 1.135267192 10 - 1.317321156 10 I], [ 1419909 1419908 8.858698749 10 + 9.210237029 10 I, 485360 485360 ] [ 4.093753536 10 - 1.009855994 10 I], [ 4153910 4153910 -5.713675352 10 + 3.337985150 10 I, 1419909 1419908 ] [ 8.858698749 10 + 9.210237029 10 I], [ 12152160 12152160 -1.503159549 10 + 2.382505441 10 I, 4153910 4153910 ]] -5.713675352 10 + 3.337985150 10 I]] > #Seems to be the case for values of -1 F:=.3*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.1696750060, 0.1685971151], [0.1419352513, 0.1696750060], [0.1537080743, 0.1419352513], [0.1637013130, 0.1537080743], [0.1524050860, 0.1637013130], [0.1521536018, 0.1524050860], [0.1585230488, 0.1521536018], [0.1556495972, 0.1585230488], [0.1534894601, 0.1556495972], [0.1560796502, 0.1534894601], [0.1560582221, 0.1560796502], [0.1546400353, 0.1560582221], [0.1553137381, 0.1546400353], [0.1557815992, 0.1553137381], [0.1551906526, 0.1557815992], [0.1552088788, 0.1551906526], [0.1555265046, 0.1552088788], [0.1553677522, 0.1555265046], [0.1552667827, 0.1553677522], [0.1554016120, 0.1552667827], [0.1553942396, 0.1554016120]] > F:=.9*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [ [[0.2561219602 - 0.7567391138 I, 1.780285680 - 0.5784498822 I], [-0.9305594379 + 0.1899981519 I, 0.2561219602 - 0.7567391138 I], [-1.282991721 + 3.860041597 I, -0.9305594379 + 0.1899981519 I], [-95.75975718 - 63.38916232 I, -1.282991721 + 3.860041597 I], [ 5 ] [-83375.07298 - 6.707531556 10 I, -95.75975718 - 63.38916232 I], [ 16 17 5 ] [1.514244233 10 - 1.655555897 10 I, -83375.07298 - 6.707531556 10 I], [ 48 50 16 17 [5.915751554 10 + 1.455429568 10 I, 1.514244233 10 - 1.655555897 10 I ] [ 145 146 ], [-3.325731151 10 + 8.443624083 10 I, 48 50 ] [ 5.915751554 10 + 1.455429568 10 I], [ 428 429 -8.170924748 10 - 3.946589490 10 I, 145 146 ] [ -3.325731151 10 + 8.443624083 10 I], [ 1256 1256 -1.086476448 10 - 9.598709601 10 I, 428 429 ] [ -8.170924748 10 - 3.946589490 10 I], [ 3676 3676 6.831730470 10 + 8.631497651 10 I, 1256 1256 ] [ -1.086476448 10 - 9.598709601 10 I], [ 10757 10756 1.905359422 10 - 3.793362088 10 I, 3676 3676 ] [ 6.831730470 10 + 8.631497651 10 I], [ 31469 31469 5.825752726 10 - 6.134155299 10 I, 10757 10756 ] [ 1.905359422 10 - 3.793362088 10 I], [ 92064 92064 2.551069288 10 + 4.069001172 10 I, 31469 31469 ] [ 5.825752726 10 - 6.134155299 10 I], [ 269332 269332 -3.726182125 10 + 2.099106207 10 I, 92064 92064 ] [ 2.551069288 10 + 4.069001172 10 I], [ 787925 787925 1.709445718 10 - 9.801130392 10 I, 269332 269332 ] [ -3.726182125 10 + 2.099106207 10 I], [ 2305056 2305057 -8.654691726 10 + 1.522572381 10 I, 787925 787925 ] [ 1.709445718 10 - 9.801130392 10 I], [ 6743386 6743386 -4.996214482 10 + 1.158118588 10 I, 2305056 2305057 ] [ -8.654691726 10 + 1.522572381 10 I], [ 19727606 19727606 -1.952862116 10 + 1.882643569 10 I, 6743386 6743386 ] [ -4.996214482 10 + 1.158118588 10 I], [ 57712612 57712612 -1.570654361 10 - 1.287437593 10 I, 19727606 19727606 ] [ -1.952862116 10 + 1.882643569 10 I], [ 168836782 168836782 3.379347312 10 - 2.746896540 10 I, 57712612 57712612 ]] -1.570654361 10 - 1.287437593 10 I]] > F:=.4*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.1965634587, 0.2272606417], [0.1401883654, 0.1965634587], [0.1866595306, 0.1401883654], [0.1989343097, 0.1866595306], [0.1623089360, 0.1989343097], [0.1730604814, 0.1623089360], [0.1916193411, 0.1730604814], [0.1743689394, 0.1916193411], [0.1715271482, 0.1743689394], [0.1843067906, 0.1715271482], [0.1792232696, 0.1843067906], [0.1736805410, 0.1792232696], [0.1799102642, 0.1736805410], [0.1801860190, 0.1799102642], [0.1759959412, 0.1801860190], [0.1780386896, 0.1759959412], [0.1796798420, 0.1780386896], [0.1774724525, 0.1796798420], [0.1775815946, 0.1774724525], [0.1789605040, 0.1775815946], [0.1781520540, 0.1789605040]] > F:=.5*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.1934714162, 0.3275286198], [0.1165652953, 0.1934714162], [0.2538173354, 0.1165652953], [0.2201565216, 0.2538173354], [0.1469762640, 0.2201565216], [0.2090209656, 0.1469762640], [0.2270723798, 0.2090209656], [0.1714455210, 0.2270723798], [0.1890671202, 0.1714455210], [0.2209232536, 0.1890671202], [0.1882179650, 0.2209232536], [0.1841295512, 0.1882179650], [0.2108717339, 0.1841295512], [0.1975727188, 0.2108717339], [0.1860593406, 0.1975727188], [0.2024917812, 0.1860593406], [0.2011767696, 0.2024917812], [0.1899849048, 0.2011767696], [0.1974405616, 0.1899849048], [0.2013075747, 0.1974405616], [0.1935417857, 0.2013075747]] > F:=.6*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.1314122372, 0.5083469332], [0.06590089914, 0.1314122372], [0.4502867253, 0.06590089914], [0.1600492514, 0.4502867253], [0.08445827412, 0.1600492514], [0.3815968480, 0.08445827412], [0.1938958831, 0.3815968480], [0.1086404995, 0.1938958831], [0.3117678213, 0.1086404995], [0.2261500894, 0.3117678213], [0.1359032901, 0.2261500894], [0.2534680167, 0.1359032901], [0.2474815492, 0.2534680167], [0.1625460908, 0.2474815492], [0.2150437793, 0.1625460908], [0.2532049407, 0.2150437793], [0.1856706302, 0.2532049407], [0.1961017341, 0.1856706302], [0.2464368142, 0.1961017341], [0.2036213509, 0.2464368142], [0.1909269151, 0.2036213509]] > F:=.7*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.01974211807, 0.8281036820], [0.005058701739, 0.01974211807], [1.107414174, 0.005058701739], [0.007873767615, 1.107414174], [-0.001386112327, 0.007873767615], [1.259007883 - 0.4090764587 I, -0.001386112327], [-0.06394068960 - 0.1469140718 I, 1.259007883 - 0.4090764587 I], [0.09115085294 + 0.06170468137 I, -0.06394068960 - 0.1469140718 I], [0.8764010115 + 0.1928208524 I, 0.09115085294 + 0.06170468137 I], [-0.01710953866 - 0.02223899449 I, 0.8764010115 + 0.1928208524 I], [-0.01143338921 - 0.004910410794 I, -0.01710953866 - 0.02223899449 I], [1.098798432 + 0.3997077934 I, -0.01143338921 - 0.004910410794 I], [-0.1057982082 + 0.05848152234 I, 1.098798432 + 0.3997077934 I], [0.06583269085 + 0.03381434398 I, -0.1057982082 + 0.05848152234 I], [1.036076240 - 0.2960492352 I, 0.06583269085 + 0.03381434398 I], [-0.04981965268 - 0.008132416761 I, 1.036076240 - 0.2960492352 I], [0.01728883007 - 0.02150004683 I, -0.04981965268 - 0.008132416761 I], [1.106974595 + 0.1744915792 I, 0.01728883007 - 0.02150004683 I], [-0.01370101069 + 0.02388245361 I, 1.106974595 + 0.1744915792 I], [0.008345525405 - 0.002908900804 I, -0.01370101069 + 0.02388245361 I], [1.150975904 - 0.03599145591 I, 0.008345525405 - 0.002908900804 I]] > F:=.65*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.07583324014, 0.6463152208], [0.03178902402, 0.07583324014], [0.6790107844, 0.03178902402], [0.06318537644, 0.6790107844], [0.02486739806, 0.06318537644], [0.7352561392, 0.02486739806], [0.04356245192, 0.7352561392], [0.01509360757, 0.04356245192], [0.8390615422, 0.01509360757], [0.01636961686, 0.8390615422], [0.003955058225, 0.01636961686], [1.067214562, 0.003955058225], [0.002883047058, 1.067214562], [-0.0003522171531, 0.002883047058], [1.358259685 - 0.4413253241 I, -0.0003522171531], [-0.03544692661 - 0.1997672080 I, 1.358259685 - 0.4413253241 I], [0.08008053840 + 0.1335560034 I, -0.03544692661 - 0.1997672080 I], [0.7804153617 + 0.1422890511 I, 0.08008053840 + 0.1335560034 I], [-0.001085862493 - 0.03656593158 I, 0.7804153617 + 0.1422890511 I], [0.001283373604 - 0.01621592939 I, -0.001085862493 - 0.03656593158 I], [0.9427306284 + 0.2825374690 I, 0.001283373604 - 0.01621592939 I]] > F:=.61*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.1213040398, 0.5329226536], [0.05926332045, 0.1213040398], [0.4858512019, 0.05926332045], [0.1442983581, 0.4858512019], [0.07367411054, 0.1442983581], [0.4256896050, 0.07367411054], [0.1741828094, 0.4256896050], [0.09385062628, 0.1741828094], [0.3573859409, 0.09385062628], [0.2077379866, 0.3573859409], [0.1188958178, 0.2077379866], [0.2913897040, 0.1188958178], [0.2370167940, 0.2913897040], [0.1459164501, 0.2370167940], [0.2395879989, 0.1459164501], [0.2535025461, 0.2395879989], [0.1714490770, 0.2535025461], [0.2077972152, 0.1714490770], [0.2547981137, 0.2077972152], [0.1930932977, 0.2547981137], [0.1937410269, 0.1930932977]] > F:=.63*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.09936883039, 0.5864555188], [0.04552774779, 0.09936883039], [0.5710583250, 0.04552774779], [0.1066004189, 0.5710583250], [0.04964218372, 0.1066004189], [0.5478567251, 0.04964218372], [0.1174052327, 0.5478567251], [0.05619824168, 0.1174052327], [0.5145337726, 0.05619824168], [0.1334011435, 0.5145337726], [0.06621217974, 0.1334011435], [0.4690266345, 0.06621217974], [0.1559950750, 0.4690266345], [0.08089797164, 0.1559950750], [0.4114425264, 0.08089797164], [0.1851733839, 0.4114425264], [0.1009438217, 0.1851733839], [0.3464988955, 0.1009438217], [0.2173819626, 0.3464988955], [0.1254460188, 0.2173819626], [0.2842612507, 0.1254460188]] > F:=.64*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.08773525101, 0.6155565454], [0.03860128499, 0.08773525101], [0.6218556048, 0.03860128499], [0.08527799213, 0.6218556048], [0.03703800082, 0.08527799213], [0.6315400584, 0.03703800082], [0.08123588000, 0.6315400584], [0.03473634143, 0.08123588000], [0.6471508973, 0.03473634143], [0.07485060589, 0.6471508973], [0.03118539372, 0.07485060589], [0.6728245235, 0.03118539372], [0.06481466982, 0.6728245235], [0.02577212592, 0.06481466982], [0.7162738714, 0.02577212592], [0.04925523337, 0.7162738714], [0.01785503145, 0.04925523337], [0.7935000070, 0.01785503145], [0.02646014084, 0.7935000070], [0.007680479910, 0.02646014084], [0.9462489696, 0.007680479910]] > F:=.645*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.08180785879, 0.6307237127], [0.03517236868, 0.08180785879], [0.6495906004, 0.03517236868], [0.07426709458, 0.6495906004], [0.03084456969, 0.07426709458], [0.6805903911, 0.03084456969], [0.06225115135, 0.6805903911], [0.02439843590, 0.06225115135], [0.7338851680, 0.02439843590], [0.04374742838, 0.7338851680], [0.01519906723, 0.04374742838], [0.8313669811, 0.01519906723], [0.01784477272, 0.8313669811], [0.004462690024, 0.01784477272], [1.040532192, 0.004462690024], [0.001041374568, 1.040532192], [-0.00008517149961, 0.001041374568], [1.561180397 - 0.5072582604 I, -0.00008517149961], [0.04633438413 - 0.3482650899 I, 1.561180397 - 0.5072582604 I], [-0.06661143033 + 0.3125077645 I, 0.04633438413 - 0.3482650899 I], [0.8922249693 + 0.2782545215 I, -0.06661143033 + 0.3125077645 I]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(2))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.08061602244, 0.6338077415], [0.03449147127, 0.08061602244], [0.6553365153, 0.03449147127], [0.07205182305, 0.6553365153], [0.02962792866, 0.07205182305], [0.6910510555, 0.02962792866], [0.05846147942, 0.6910510555], [0.02243271666, 0.05846147942], [0.7532721754, 0.02243271666], [0.03779319080, 0.7532721754], [0.01246471643, 0.03779319080], [0.8701112787, 0.01246471643], [0.01064328730, 0.8701112787], [0.002182453871, 0.01064328730], [1.149473591, 0.002182453871], [0.01414050646, 1.149473591], [-0.003210032815, 0.01414050646], [1.052002322 - 0.3418162751 I, -0.003210032815], [-0.07293838486 - 0.02525826257 I, 1.052002322 - 0.3418162751 I], [0.02796682785 - 0.02822359074 I, -0.07293838486 - 0.02525826257 I], [1.020230969 + 0.2149623918 I, 0.02796682785 - 0.02822359074 I]] > #When k=.646 we first begin to see the orbit start with real numbers, and then > it eventually moves into complex points. This is while keeping the a value > between -1 F:=.646*(z[1]^(-.1))*((1-z[1])^(5))*((1-z[2])^(7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[ 559 560 [[-6.406315033 10 - 2.179487659 10 I, 91 92 ] [ -6.377885425 10 - 2.195282561 10 I], [ 3392 3390 -1.170868506 10 + 9.548116526 10 I, 559 560 ] [ -6.406315033 10 - 2.179487659 10 I], [ 20543 20543 -1.161667142 10 + 2.549494216 10 I, 3392 3390 ] [ -1.170868506 10 + 9.548116526 10 I], [ 124407 124406 1.524916042 10 - 3.130933240 10 I, 20543 20543 ] [ -1.161667142 10 + 2.549494216 10 I], [ 753399 753398 1.388685501 10 + 6.363905799 10 I, 124407 124406 ] [ 1.524916042 10 - 3.130933240 10 I], [ 4562506 4562505 1.109693541 10 + 9.122249812 10 I, 753399 753398 ] [ 1.388685501 10 + 6.363905799 10 I], [ 27630074 27630073 1.849388435 10 + 1.784341715 10 I, 4562506 4562505 ] [ 1.109693541 10 + 9.122249812 10 I], [ 167324906 167324906 3.675472904 10 - 5.668912270 10 I, 27630074 27630073 ] [ 1.849388435 10 + 1.784341715 10 I], [ 1013302562 1013302563 -7.007440807 10 + 1.260581718 10 I, 167324906 167324906 ] [ 3.675472904 10 - 5.668912270 10 I], [ 6136456907 6136456905 -1.248257969 10 - 9.040403628 10 I, 1013302562 1013302563 ] [ -7.007440807 10 + 1.260581718 10 I], [ 37161756786 37161756786 4.426295444 10 - 2.375135418 10 I, 6136456907 6136456905 ] [ -1.248257969 10 - 9.040403628 10 I], [ 225047806603 225047806604 7.009952203 10 + 2.004986962 10 I, 37161756786 37161756786 ] [ 4.426295444 10 - 2.375135418 10 I], [ 1362866549867 1362866549867 -7.136898542 10 + 4.280256669 10 I, 225047806603 225047806604 ] [ 7.009952203 10 + 2.004986962 10 I], [ 8253380740582 8253380740582 -6.684315832 10 + 4.605102039 10 I, 1362866549867 1362866549867 ] [ -7.136898542 10 + 4.280256669 10 I], [ 49981631477931 49981631477931 2.400944134 10 - 2.321390610 10 I, 8253380740582 8253380740582 ] [ -6.684315832 10 + 4.605102039 10 I], [ 302683659425943 302683659425944 4.401896367 10 + 3.245966754 10 I, 49981631477931 49981631477931 ] [ 2.400944134 10 - 2.321390610 10 I], [ 1833021351532647 1833021351532648 -2.410402964 10 + 2.803583480 10 I, 302683659425943 302683659425944 ] [ 4.401896367 10 + 3.245966754 10 I], [ 11100590238491587 11100590238491587 4.054642830 10 - 3.281212315 10 I, 1833021351532647 1833021351532648 ] [ -2.410402964 10 + 2.803583480 10 I], [ 67224041629337316 67224041629337316 -1.324854521 10 + 3.063223703 10 I, 11100590238491587 11100590238491587 ] 4.054642830 10 - 3.281212315 10 I], [ 407101935653194105 407101935653194106 [7.300260514 10 - 3.230391486 10 I, 67224041629337316 67224041629337316 ] -1.324854521 10 + 3.063223703 10 I], [ 2465367776106012161 2465367776106012161 [2.896927177 10 + 4.155436096 10 I, 407101935653194105 407101935653194106 ]] 7.300260514 10 - 3.230391486 10 I]] > #Greater values of b,c get us into the complex orbit much quicker and we see > the numbers get much greater. > F:=.646*(z[1]^(-.1))*((1-z[1])^(-2))*((1-z[2])^(-3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[-0.08129824457 + 0.08194453404 I, 0.04423823772 + 0.006695238633 I], [0.7792570764 - 0.04910403747 I, -0.08129824457 + 0.08194453404 I], [9.942863347 - 2.062767771 I, 0.7792570764 - 0.04910403747 I], [0.5172245383 - 0.09561995274 I, 9.942863347 - 2.062767771 I], [-0.003506686417 - 0.001112871517 I, 0.5172245383 - 0.09561995274 I], [8.989622278 - 2.833957695 I, -0.003506686417 - 0.001112871517 I], [0.005395037155 + 0.004627177360 I, 8.989622278 - 2.833957695 I], [-0.001006707253 - 0.001440976607 I, 0.005395037155 + 0.004627177360 I], [1.203351025 + 0.2806624601 I, -0.001006707253 - 0.001440976607 I], [-1.770447048 - 4.941765408 I, 1.203351025 + 0.2806624601 I], [-0.01931377797 - 0.4091122502 I, -1.770447048 - 4.941765408 I], [-0.002583616038 + 0.001921522148 I, -0.01931377797 - 0.4091122502 I], [0.1536141858 - 0.8471466679 I, -0.002583616038 + 0.001921522148 I], [0.06511051093 - 0.4490501438 I, 0.1536141858 - 0.8471466679 I], [-0.3782925563 - 0.01179280997 I, 0.06511051093 - 0.4490501438 I], [0.1672989442 - 0.2912628988 I, -0.3782925563 - 0.01179280997 I], [0.2930418820 - 0.1977756488 I, 0.1672989442 - 0.2912628988 I], [0.1454807425 - 1.931928369 I, 0.2930418820 - 0.1977756488 I], [-0.3378974382 - 0.05595375831 I, 0.1454807425 - 1.931928369 I], [-0.04229430460 + 0.004556149835 I, -0.3378974382 - 0.05595375831 I], [0.3100670388 - 0.1384968359 I, -0.04229430460 + 0.004556149835 I]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(-.5))*((1-z[2])^(-.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[0.5225109959 - 0.2670025748 I, 0.2291421599 - 0.3189360632 I], [0.9265372159 - 0.4850757226 I, 0.5225109959 - 0.2670025748 I], [0.7342708205 - 1.192602282 I, 0.9265372159 - 0.4850757226 I], [0.002383421331 - 0.9302078434 I, 0.7342708205 - 1.192602282 I], [0.1913760455 - 0.4448098803 I, 0.002383421331 - 0.9302078434 I], [0.4595106751 - 0.3568613819 I, 0.1913760455 - 0.4448098803 I], [0.7508423206 - 0.4895478457 I, 0.4595106751 - 0.3568613819 I], [0.7417492001 - 0.9360921134 I, 0.7508423206 - 0.4895478457 I], [0.2328936887 - 0.9514603610 I, 0.7417492001 - 0.9360921134 I], [0.2030495103 - 0.5621615743 I, 0.2328936887 - 0.9514603610 I], [0.4129175913 - 0.4331716722 I, 0.2030495103 - 0.5621615743 I], [0.6366047647 - 0.5011867273 I, 0.4129175913 - 0.4331716722 I], [0.6900515901 - 0.7855048983 I, 0.6366047647 - 0.5011867273 I], [0.3793602133 - 0.9025434813 I, 0.6900515901 - 0.7855048983 I], [0.2500984020 - 0.6498026086 I, 0.3793602133 - 0.9025434813 I], [0.3849389422 - 0.4998463805 I, 0.2500984020 - 0.6498026086 I], [0.5592236249 - 0.5205123015 I, 0.3849389422 - 0.4998463805 I], [0.6333633806 - 0.7020932078 I, 0.5592236249 - 0.5205123015 I], [0.4591039454 - 0.8364082972 I, 0.6333633806 - 0.7020932078 I], [0.3098138187 - 0.7001052345 I, 0.4591039454 - 0.8364082972 I], [0.3745623049 - 0.5567403941 I, 0.3098138187 - 0.7001052345 I]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(.5))*((1-z[2])^(7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [ [ [ 69 70 6 5 [-5.889205118 10 + 4.209005368 10 I, -1.609314966 10 - 7.573274003 10 I ] [ 71 71 ], [-3.776618740 10 + 5.276224466 10 I, 69 70 ] [ -5.889205118 10 + 4.209005368 10 I], [ 522 522 -2.365797105 10 + 8.254204257 10 I, 71 71 ] [ -3.776618740 10 + 5.276224466 10 I], [ 711 711 1.713887998 10 - 4.338134106 10 I, 522 522 ] [ -2.365797105 10 + 8.254204257 10 I], [ 3943 3945 -9.773810858 10 - 1.029281526 10 I, 711 711 ] [ 1.713887998 10 - 4.338134106 10 I], [ 6559 6559 -1.215384447 10 + 2.900581371 10 I, 3943 3945 ] [ -9.773810858 10 - 1.029281526 10 I], [ 30238 30237 -5.096776673 10 - 6.369069717 10 I, 6559 6559 ] [ -1.215384447 10 + 2.900581371 10 I], [ 58008 58011 -1.071580711 10 - 5.994671104 10 I, 30238 30237 ] [ -5.096776673 10 - 6.369069717 10 I], [ 234874 234875 -7.503146562 10 + 3.041830758 10 I, 58008 58011 ] [ -1.071580711 10 - 5.994671104 10 I], [ 500032 500032 -2.126307736 10 - 1.879268612 10 I, 234874 234875 ] [ -7.503146562 10 + 3.041830758 10 I], [ 1844141 1844141 -1.371042424 10 - 1.216513016 10 I, 500032 500032 ] [ -2.126307736 10 - 1.879268612 10 I], [ 4237883 4237883 2.507829258 10 - 1.760656733 10 I, 1844141 1844141 ] [ -1.371042424 10 - 1.216513016 10 I], [ 14604142 14604141 1.105344642 10 + 1.366567573 10 I, 4237883 4237883 ] [ 2.507829258 10 - 1.760656733 10 I], [ 35506840 35506840 -9.588990568 10 - 4.953477820 10 I, 14604142 14604141 ] [ 1.105344642 10 + 1.366567573 10 I], [ 116431729 116431730 -7.234681023 10 - 3.482189589 10 I, 35506840 35506840 ] [ -9.588990568 10 - 4.953477820 10 I], [ 295120578 295120579 -9.001901769 10 - 1.593996111 10 I, 116431729 116431730 ] [ -7.234681023 10 - 3.482189589 10 I], [ 933070345 933070345 -1.513576338 10 - 1.807614511 10 I, 295120578 295120579 ] [ -9.001901769 10 - 1.593996111 10 I], [ 2439072192 2439072192 -1.286703786 10 + 6.138603588 10 I, 933070345 933070345 ] [ -1.513576338 10 - 1.807614511 10 I], [ 7507121294 7507121294 1.779335799 10 - 2.943672563 10 I, 2439072192 2439072192 ] [ -1.286703786 10 + 6.138603588 10 I], [ 20076353866 20076353867 -8.210096630 10 - 1.384311826 10 I, 7507121294 7507121294 ] [ 1.779335799 10 - 2.943672563 10 I], [ 60580390608 60580390607 -2.764397964 10 + 4.978532902 10 I, 20076353866 20076353867 ]] -8.210096630 10 - 1.384311826 10 I]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.5345554632, 0.01486686426], [0.01486686422, 0.5345554632], [0.5345554630, 0.01486686422], [0.01486686426, 0.5345554630], [0.5345554632, 0.01486686426], [0.01486686422, 0.5345554632], [0.5345554630, 0.01486686422], [0.01486686426, 0.5345554630], [0.5345554632, 0.01486686426], [0.01486686422, 0.5345554632], [0.5345554630, 0.01486686422]] > #We've found an orbit! > F:=.646*(z[1]^(2))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.646*(z[1]^(.1))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753]] > F:=.646*(z[1]^(.2))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476]] > F:=.646*(z[1]^(.2))*((1-z[1])^(.5))*((1-z[2])^(7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.09588206350, 0.01766431366], [0.3392438511, 0.09588206350], [0.2088964779, 0.3392438511], [0.02310063370, 0.2088964779], [0.05827944899, 0.02310063370], [0.3014709189, 0.05827944899], [0.2790154068, 0.3014709189], [0.03448359903, 0.2790154068], [0.03278058904, 0.03448359903], [0.2508626228, 0.03278058904], [0.3357959617, 0.2508626228]] > FP(A,[z[1],z[2]]) {[0., 0.], [0.1403300612, 0.1403300612], [0.2230714196 - 0.6609666023 I, 0.2230714196 - 0.6609666023 I], [0.2230714196 + 0.6609666023 I, 0.2230714196 + 0.6609666023 I], [1.018258471 - 1.107013482 I, 1.018258471 - 1.107013482 I], [1.018258471 + 1.107013482 I, 1.018258471 + 1.107013482 I], [1.898423679 - 0.7064086193 I, 1.898423679 - 0.7064086193 I], [1.898423679 + 0.7064086193 I, 1.898423679 + 0.7064086193 I]} > F:=.646*(z[1]^(2))*((1-z[1])^(5))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.646*(z[1]^(.1))*((1-z[1])^(5))*((1-z[2])^(.6)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1795806394, 0.1795806388], [0.1795806388, 0.1795806394], [0.1795806394, 0.1795806388], [0.1795806388, 0.1795806394], [0.1795806394, 0.1795806388], [0.1795806388, 0.1795806394], [0.1795806394, 0.1795806388], [0.1795806388, 0.1795806394], [0.1795806394, 0.1795806388], [0.1795806388, 0.1795806394], [0.1795806394, 0.1795806388]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(5))*((1-z[2])^(.6)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.5906878220, 0.007786336502], [0.007786336502, 0.5906878220], [0.5906878220, 0.007786336502], [0.007786336502, 0.5906878220], [0.5906878220, 0.007786336502], [0.007786336502, 0.5906878220], [0.5906878220, 0.007786336502], [0.007786336502, 0.5906878220], [0.5906878220, 0.007786336502], [0.007786336502, 0.5906878220], [0.5906878220, 0.007786336502]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(5))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.4858310759, 0.02446291144], [0.02446291132, 0.4858310759], [0.4858310754, 0.02446291132], [0.02446291144, 0.4858310754], [0.4858310759, 0.02446291144], [0.02446291132, 0.4858310759], [0.4858310754, 0.02446291132], [0.02446291144, 0.4858310754], [0.4858310759, 0.02446291144], [0.02446291132, 0.4858310759], [0.4858310754, 0.02446291132]] > F:=.646*(z[1]^(-.2))*((1-z[1])^(5))*((1-z[2])^(.6)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(.1))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753], [0.1779277753, 0.1779277759], [0.1779277759, 0.1779277753]] > F:=.646*(z[1]^(.2))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476], [0.1630034476, 0.1630034476]] > F:=.646*(z[1]^(3))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.646*(z[1]^(.9))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351], [0.008005055351, 0.008005055351]] > F:=.646*(z[1]^(.999999))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[ -1891 -1891] [[3.499601310 10 , 5.393811741 10 ], [ -1891 -1891] [2.270604764 10 , 3.499601310 10 ], [ -1891 -1891] [1.473210163 10 , 2.270604764 10 ], [ -1892 -1891] [9.558462830 10 , 1.473210163 10 ], [ -1892 -1892] [6.201711991 10 , 9.558462830 10 ], [ -1892 -1892] [4.023790113 10 , 6.201711991 10 ], [ -1892 -1892] [2.610713607 10 , 4.023790113 10 ], [ -1892 -1892] [1.693882705 10 , 2.610713607 10 ], [ -1892 -1892] [1.099025130 10 , 1.693882705 10 ], [ -1893 -1892] [7.130698931 10 , 1.099025130 10 ], [ -1893 -1893]] [4.626544728 10 , 7.130698931 10 ]] > F:=.646*(z[1]^(1))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[ -1900 -1900] [[1.163194324 10 , 1.800610409 10 ], [ -1901 -1900] [7.514235333 10 , 1.163194324 10 ], [ -1901 -1901] [4.854196025 10 , 7.514235333 10 ], [ -1901 -1901] [3.135810632 10 , 4.854196025 10 ], [ -1901 -1901] [2.025733668 10 , 3.135810632 10 ], [ -1901 -1901] [1.308623950 10 , 2.025733668 10 ], [ -1902 -1901] [8.453710717 10 , 1.308623950 10 ], [ -1902 -1902] [5.461097123 10 , 8.453710717 10 ], [ -1902 -1902] [3.527868741 10 , 5.461097123 10 ], [ -1902 -1902] [2.279003207 10 , 3.527868741 10 ], [ -1902 -1902]] [1.472236072 10 , 2.279003207 10 ]] > F:=4*(z[1]^(2))*((1-z[1])^(5))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(6))*((1-z[2])^(.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.6026187680, 0.002670910313], [0.002670910310, 0.6026187680], [0.6026187679, 0.002670910310], [0.002670910313, 0.6026187679], [0.6026187680, 0.002670910313], [0.002670910310, 0.6026187680], [0.6026187679, 0.002670910310], [0.002670910313, 0.6026187679], [0.6026187680, 0.002670910313], [0.002670910310, 0.6026187680], [0.6026187679, 0.002670910310]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(6))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.5529210283, 0.005449705932], [0.005449705924, 0.5529210283], [0.5529210282, 0.005449705924], [0.005449705932, 0.5529210282], [0.5529210283, 0.005449705932], [0.005449705924, 0.5529210283], [0.5529210282, 0.005449705924], [0.005449705932, 0.5529210282], [0.5529210283, 0.005449705932], [0.005449705924, 0.5529210283], [0.5529210282, 0.005449705924]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(7))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.6074409229, 0.0009747168274], [0.0009747168293, 0.6074409229], [0.6074409230, 0.0009747168293], [0.0009747168274, 0.6074409230], [0.6074409229, 0.0009747168274], [0.0009747168293, 0.6074409229], [0.6074409230, 0.0009747168293], [0.0009747168274, 0.6074409230], [0.6074409229, 0.0009747168274], [0.0009747168293, 0.6074409229], [0.6074409230, 0.0009747168293]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(10))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(8))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.6716899652, 0.00009073134826], [0.00009073134826, 0.6716899652], [0.6716899652, 0.00009073134826], [0.00009073134826, 0.6716899652], [0.6716899652, 0.00009073134826], [0.00009073134826, 0.6716899652], [0.6716899652, 0.00009073134826], [0.00009073134826, 0.6716899652], [0.6716899652, 0.00009073134826], [0.00009073134826, 0.6716899652], [0.6716899652, 0.00009073134826]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[ -7] [ -7 ] [[0.7851213959, 6.463640281 10 ], [6.463640281 10 , 0.7851213959], [ -7] [ -7 ] [0.7851213959, 6.463640281 10 ], [6.463640281 10 , 0.7851213959], [ -7] [ -7 ] [0.7851213959, 6.463640281 10 ], [6.463640281 10 , 0.7851213959], [ -7] [ -7 ] [0.7851213959, 6.463640281 10 ], [6.463640281 10 , 0.7851213959], [ -7] [ -7 ] [0.7851213959, 6.463640281 10 ], [6.463640281 10 , 0.7851213959], [ -7]] [0.7851213959, 6.463640281 10 ]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.5))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.3))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.1))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[ -7] [ -7 ] [[0.8069731728, 2.082894882 10 ], [2.082894882 10 , 0.8069731728], [ -7] [ -7 ] [0.8069731728, 2.082894882 10 ], [2.082894882 10 , 0.8069731728], [ -7] [ -7 ] [0.8069731728, 2.082894882 10 ], [2.082894882 10 , 0.8069731728], [ -7] [ -7 ] [0.8069731728, 2.082894882 10 ], [2.082894882 10 , 0.8069731728], [ -7] [ -7 ] [0.8069731728, 2.082894882 10 ], [2.082894882 10 , 0.8069731728], [ -7]] [0.8069731728, 2.082894882 10 ]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[ -8] [ -8 ] [[0.8382918968, 3.453229023 10 ], [3.453229023 10 , 0.8382918968], [ -8] [ -8 ] [0.8382918968, 3.453229023 10 ], [3.453229023 10 , 0.8382918968], [ -8] [ -8 ] [0.8382918968, 3.453229023 10 ], [3.453229023 10 , 0.8382918968], [ -8] [ -8 ] [0.8382918968, 3.453229023 10 ], [3.453229023 10 , 0.8382918968], [ -8] [ -8 ] [0.8382918968, 3.453229023 10 ], [3.453229023 10 , 0.8382918968], [ -8]] [0.8382918968, 3.453229023 10 ]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.25))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[ -9] [ -9 ] [[0.8680354429, 4.793015544 10 ], [4.793015544 10 , 0.8680354429], [ -9] [ -9 ] [0.8680354429, 4.793015544 10 ], [4.793015544 10 , 0.8680354429], [ -9] [ -9 ] [0.8680354429, 4.793015544 10 ], [4.793015544 10 , 0.8680354429], [ -9] [ -9 ] [0.8680354429, 4.793015544 10 ], [4.793015544 10 , 0.8680354429], [ -9] [ -9 ] [0.8680354429, 4.793015544 10 ], [4.793015544 10 , 0.8680354429], [ -9]] [0.8680354429, 4.793015544 10 ]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.29))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.28))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.26))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.259))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[ -9] [ -9 ] [[0.8844477028, 1.373511211 10 ], [1.373511211 10 , 0.8844477028], [ -9] [ -9 ] [0.8844477028, 1.373511211 10 ], [1.373511211 10 , 0.8844477028], [ -9] [ -9 ] [0.8844477028, 1.373511211 10 ], [1.373511211 10 , 0.8844477028], [ -9] [ -9 ] [0.8844477028, 1.373511211 10 ], [1.373511211 10 , 0.8844477028], [ -9] [ -9 ] [0.8844477028, 1.373511211 10 ], [1.373511211 10 , 0.8844477028], [ -9]] [0.8844477028, 1.373511211 10 ]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2599))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2595))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2593))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[ -9] [ -9 ] [[0.8853080062, 1.281285845 10 ], [1.281285845 10 , 0.8853080062], [ -9] [ -9 ] [0.8853080062, 1.281285845 10 ], [1.281285845 10 , 0.8853080062], [ -9] [ -9 ] [0.8853080062, 1.281285845 10 ], [1.281285845 10 , 0.8853080062], [ -9] [ -9 ] [0.8853080062, 1.281285845 10 ], [1.281285845 10 , 0.8853080062], [ -9] [ -9 ] [0.8853080062, 1.281285845 10 ], [1.281285845 10 , 0.8853080062], [ -9]] [0.8853080062, 1.281285845 10 ]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2592))*((1-z[2])^(.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(.9)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.6932129882, 0.00001187718137], [0.00001187718137, 0.6932129882], [0.6932129882, 0.00001187718137], [0.00001187718137, 0.6932129882], [0.6932129882, 0.00001187718137], [0.00001187718137, 0.6932129882], [0.6932129882, 0.00001187718137], [0.00001187718137, 0.6932129882], [0.6932129882, 0.00001187718137], [0.00001187718137, 0.6932129882], [0.6932129882, 0.00001187718137]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(.99999)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.6242509640, 0.00007845117231], [0.00007845117250, 0.6242509640], [0.6242509642, 0.00007845117250], [0.00007845117205, 0.6242509642], [0.6242509641, 0.00007845117205], [0.00007845117231, 0.6242509641], [0.6242509640, 0.00007845117231], [0.00007845117250, 0.6242509640], [0.6242509642, 0.00007845117250], [0.00007845117205, 0.6242509642], [0.6242509641, 0.00007845117205]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.6242454189, 0.00007846196115], [0.00007846196161, 0.6242454189], [0.6242454191, 0.00007846196161], [0.00007846196115, 0.6242454191], [0.6242454189, 0.00007846196115], [0.00007846196161, 0.6242454189], [0.6242454191, 0.00007846196161], [0.00007846196115, 0.6242454191], [0.6242454189, 0.00007846196115], [0.00007846196161, 0.6242454189], [0.6242454191, 0.00007846196161]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(1.1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.5766170141, 0.0002387354688], [0.0002387354688, 0.5766170141], [0.5766170141, 0.0002387354688], [0.0002387354688, 0.5766170141], [0.5766170141, 0.0002387354688], [0.0002387354688, 0.5766170141], [0.5766170141, 0.0002387354688], [0.0002387354688, 0.5766170141], [0.5766170141, 0.0002387354688], [0.0002387354688, 0.5766170141], [0.5766170141, 0.0002387354688]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(1.6)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.4396712651, 0.003269278089], [0.003269278110, 0.4396712651], [0.4396712655, 0.003269278110], [0.003269278089, 0.4396712655], [0.4396712651, 0.003269278089], [0.003269278110, 0.4396712651], [0.4396712655, 0.003269278110], [0.003269278089, 0.4396712655], [0.4396712651, 0.003269278089], [0.003269278110, 0.4396712651], [0.4396712655, 0.003269278110]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(2.5)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.3029428389, 0.02422569638], [0.02422569645, 0.3029428389], [0.3029428391, 0.02422569645], [0.02422569638, 0.3029428391], [0.3029428389, 0.02422569638], [0.02422569645, 0.3029428389], [0.3029428391, 0.02422569645], [0.02422569638, 0.3029428391], [0.3029428389, 0.02422569638], [0.02422569645, 0.3029428389], [0.3029428391, 0.02422569645]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(4.0)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852], [0.1283487852, 0.1283487852]] > #We eventually achieved stability with a higher value of c. With values > ranging from 0 to 2.5 we did not see that, let's see if stability is reached > earlier than 4. > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(3.0)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.2324915272, 0.05451643148], [0.05451643148, 0.2324915272], [0.2324915272, 0.05451643148], [0.05451643148, 0.2324915272], [0.2324915272, 0.05451643148], [0.05451643148, 0.2324915272], [0.2324915272, 0.05451643148], [0.05451643148, 0.2324915272], [0.2324915272, 0.05451643148], [0.05451643148, 0.2324915272], [0.2324915272, 0.05451643148]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(3.5)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1313174954, 0.1313174970], [0.1313174970, 0.1313174954], [0.1313174954, 0.1313174970], [0.1313174970, 0.1313174954], [0.1313174954, 0.1313174970], [0.1313174970, 0.1313174954], [0.1313174954, 0.1313174970], [0.1313174970, 0.1313174954], [0.1313174954, 0.1313174970], [0.1313174970, 0.1313174954], [0.1313174954, 0.1313174970]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(3.6)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1307105363, 0.1307105364], [0.1307105364, 0.1307105363], [0.1307105363, 0.1307105364], [0.1307105364, 0.1307105363], [0.1307105363, 0.1307105364], [0.1307105364, 0.1307105363], [0.1307105363, 0.1307105364], [0.1307105364, 0.1307105363], [0.1307105363, 0.1307105364], [0.1307105364, 0.1307105363], [0.1307105363, 0.1307105364]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(3.7)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1301103039, 0.1301103036], [0.1301103037, 0.1301103039], [0.1301103038, 0.1301103037], [0.1301103037, 0.1301103038], [0.1301103039, 0.1301103037], [0.1301103036, 0.1301103039], [0.1301103039, 0.1301103036], [0.1301103037, 0.1301103039], [0.1301103038, 0.1301103037], [0.1301103037, 0.1301103038], [0.1301103039, 0.1301103037]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(3.8)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1295166785, 0.1295166786], [0.1295166786, 0.1295166785], [0.1295166785, 0.1295166786], [0.1295166786, 0.1295166785], [0.1295166785, 0.1295166786], [0.1295166786, 0.1295166785], [0.1295166785, 0.1295166786], [0.1295166786, 0.1295166785], [0.1295166785, 0.1295166786], [0.1295166786, 0.1295166785], [0.1295166785, 0.1295166786]] > F:=.646*(z[1]^(-.1))*((1-z[1])^(9.2591))*((1-z[2])^(3.9)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1289295438, 0.1289295436], [0.1289295437, 0.1289295438], [0.1289295437, 0.1289295437], [0.1289295438, 0.1289295437], [0.1289295436, 0.1289295438], [0.1289295438, 0.1289295436], [0.1289295437, 0.1289295438], [0.1289295437, 0.1289295437], [0.1289295438, 0.1289295437], [0.1289295436, 0.1289295438], [0.1289295438, 0.1289295436]] > #Looks like stability is probably achieved earlier, probably somewhere between > 3.5 and 3.6, but it is much more noticeable at 4.0 for C. > F:=.646*(z[1]^(-.1))*((1-z[1])^(4))*((1-z[2])^(9.2591)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=.646*(z[1]^(-.1))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=2*(z[1]^(-.1))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=2*(z[1]^(.1))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I]] > F:=2*(z[1]^(.3))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1302372680, 0.5144456540], [0.03968298622, 0.1302372680], [0.3849978896, 0.03968298622], [0.2971363688, 0.3849978896], [0.06902976248, 0.2971363688], [0.1766170345, 0.06902976248], [0.4985268964, 0.1766170345], [0.09407777112, 0.4985268964], [0.04627425430, 0.09407777112], [0.4647998816, 0.04627425430], [0.2015823144, 0.4647998816]] > FP(A,[z[1],z[2]]) {[0., 0.], [0.2211359613, 0.2211359613], [0.3852266943 - 0.6274239887 I, 0.3852266943 - 0.6274239887 I], [0.3852266943 + 0.6274239887 I, 0.3852266943 + 0.6274239887 I], [1.146532224 - 0.9302401911 I, 1.146532224 - 0.9302401911 I], [1.146532224 + 0.9302401911 I, 1.146532224 + 0.9302401911 I], [1.860710229 - 0.4393651227 I, 1.860710229 - 0.4393651227 I], [1.860710229 + 0.4393651227 I, 1.860710229 + 0.4393651227 I]} > F:=-2*(z[1]^(-.1))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=-2*(z[1]^(2))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.]] > F:=-.3*(z[1]^(2))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.], [-0., -0.]] > F:=-.3*(z[1]^(-.1))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=-.3*(z[1]^(-.1))*((1-z[1])^(4))*((1-z[2])^(3)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [ [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I ], [Float(undefined) + Float(undefined) I, Float(undefined) + Float(undefined) I] ] > F:=-.3*(z[1]^(.1))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I, -Float(infinity) - Float(infinity) I]] > F:=1*(z[1]^(4))*((1-z[1])^(3))*((1-z[2])^(4)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(4))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(1.1))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714], [0.2407045714, 0.2407045714]] > F:=2*(z[1]^(1.2))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666], [0.1380537666, 0.1380537666]] > F:=2*(z[1]^(1.5))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(1.4))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(1.3))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(1.25))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(1.22))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(1.21))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(1.200001))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947], [0.1380514947, 0.1380514947]] > F:=2*(z[1]^(1.200090))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367], [0.1378488367, 0.1378488367]] > F:=2*(z[1]^(1.200900))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815], [0.1359599815, 0.1359599815]] > F:=2*(z[1]^(1.209))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889], [0.1059474889, 0.1059474889]] > F:=2*(z[1]^(1.2099))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(1.2094))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753], [0.1017793753, 0.1017793753]] > F:=2*(z[1]^(1.2095))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833], [0.1002652833, 0.1002652833]] > F:=2*(z[1]^(1.2096))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160], [0.09811684160, 0.09811684160]] > F:=2*(z[1]^(1.2097))*((1-z[1])^(1))*((1-z[2])^(1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > #At a=1.2097 we start to get [0,0] as our stable point. > F:=2*(z[1]^(1.2097))*((1-z[1])^(.1))*((1-z[2])^(.1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I, Float(infinity) + Float(infinity) I]] > F:=2*(z[1]^(1.2097))*((1-z[1])^(-.1))*((1-z[2])^(-.2)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[-2065.907552 - 605.7855352 I, -2065.907552 + 605.7855354 I], [-2065.907552 + 605.7855354 I, -2065.907552 - 605.7855352 I], [-2065.907552 - 605.7855352 I, -2065.907552 + 605.7855354 I], [-2065.907552 + 605.7855354 I, -2065.907552 - 605.7855352 I], [-2065.907552 - 605.7855352 I, -2065.907552 + 605.7855354 I], [-2065.907552 + 605.7855354 I, -2065.907552 - 605.7855352 I], [-2065.907552 - 605.7855352 I, -2065.907552 + 605.7855354 I], [-2065.907552 + 605.7855354 I, -2065.907552 - 605.7855352 I], [-2065.907552 - 605.7855352 I, -2065.907552 + 605.7855354 I], [-2065.907552 + 605.7855354 I, -2065.907552 - 605.7855352 I], [-2065.907552 - 605.7855352 I, -2065.907552 + 605.7855354 I]] > F:=2*(z[1]^(1.2097))*((1-z[1])^(5))*((1-z[2])^(5)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=2*(z[1]^(1.2097))*((1-z[1])^(.2))*((1-z[2])^(2)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.1*(z[1]^(1.2097))*((1-z[1])^(1.3))*((1-z[2])^(1.1)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=.646*(z[1]^(1.2097))*((1-z[1])^(2))*((1-z[2])^(.9)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > #Not much new stuff is being seen with these manipulation of parameters, so > we'll just do a modified version of the first order case so now we will only > have 3 parameters k, a, and b. The equation is x(n)=k*x(n-1)^a*(1-x(n-1))^b > F:=[1*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.4817498545], [0.4817498542], [0.4817498545], [0.4817498542], [0.4817498545], [0.4817498542], [0.4817498545], [0.4817498542], [0.4817498545], [0.4817498542], [0.4817498545]] > F:=[.1*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.07131594698], [0.07131594698], [0.07131594698], [0.07131594698], [0.07131594698], [0.07131594698], [0.07131594698], [0.07131594698], [0.07131594698], [0.07131594698], [0.07131594698]] > F:=[.1*(x^(.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.05256388079], [0.05256388079], [0.05256388079], [0.05256388079], [0.05256388079], [0.05256388079], [0.05256388079], [0.05256388079], [0.05256388079], [0.05256388079], [0.05256388079]] > F:=[.1*(x^(.9))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[ -10] [ -10] [ -10] [[1.000000004 10 ], [1.000000004 10 ], [1.000000004 10 ], [ -10] [ -10] [ -10] [1.000000004 10 ], [1.000000004 10 ], [1.000000004 10 ], [ -10] [ -10] [ -10] [1.000000004 10 ], [1.000000004 10 ], [1.000000004 10 ], [ -10] [ -10]] [1.000000004 10 ], [1.000000004 10 ]] > F:=[.1*(x^(.9999999))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[ -9996] [ -9997] [ -9998] [[2.422271425 10 ], [2.427852905 10 ], [2.433447805 10 ], [ -9999] [ -10000] [ -10001] [2.439056159 10 ], [2.444678001 10 ], [2.450313365 10 ], [ -10002] [ -10003] [ -10004] [2.455962284 10 ], [2.461624792 10 ], [2.467300923 10 ], [ -10005] [ -10006]] [2.472990711 10 ], [2.478694191 10 ]] > F:=[.2*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.1412216285], [0.1412216285], [0.1412216285], [0.1412216285], [0.1412216285], [0.1412216285], [0.1412216285], [0.1412216285], [0.1412216285], [0.1412216285], [0.1412216285]] > F:=[1*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.4817498545], [0.4817498542], [0.4817498545], [0.4817498542], [0.4817498545], [0.4817498542], [0.4817498545], [0.4817498542], [0.4817498545], [0.4817498542], [0.4817498545]] > F:=[2*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > F:=[1.8*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > F:=[1.7*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > F:=[1.5*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > F:=[1.2*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.2746913380], [0.7648717248], [0.2746913380], [0.7648717248], [0.2746913380], [0.7648717248], [0.2746913380], [0.7648717248], [0.2746913380], [0.7648717248], [0.2746913380]] > F:=[1.3*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.09053931780], [0.9298459470], [0.09053931806], [0.9298459472], [0.09053931780], [0.9298459470], [0.09053931806], [0.9298459472], [0.09053931780], [0.9298459470], [0.09053931806]] > F:=[1.4*(x^(.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > F:=[1.4*(x^(.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.3226870804], [0.7562644937], [0.3226870804], [0.7562644937], [0.3226870804], [0.7562644937], [0.3226870804], [0.7562644937], [0.3226870804], [0.7562644937], [0.3226870804]] > F:=[1.1*(x^(.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.4879530752], [0.4879530751], [0.4879530752], [0.4879530751], [0.4879530752], [0.4879530751], [0.4879530752], [0.4879530751], [0.4879530752], [0.4879530751], [0.4879530752]] > F:=[1.4*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806]] > F:=[1.4*(x^(.4))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.5184110492], [0.5184110497], [0.5184110492], [0.5184110497], [0.5184110492], [0.5184110497], [0.5184110492], [0.5184110497], [0.5184110492], [0.5184110497], [0.5184110492]] > F:=[1.4*(x^(.5))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456]] > F:=[1.4*(x^(.99))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.2943956046], [0.2943956046], [0.2943956046], [0.2943956046], [0.2943956046], [0.2943956046], [0.2943956046], [0.2943956046], [0.2943956046], [0.2943956046], [0.2943956046]] > F:=[1.4*(x^(2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[.1*(x^(2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.4*(x^(1.2097))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.4*(x^(1.2096))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.4*(x^(1.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024]] > F:=[1.4*(x^(1.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.4*(x^(1.19))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.4*(x^(1.109))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.5*(x^(1.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[3*(x^(1.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.6349704315], [0.6349704318], [0.6349704315], [0.6349704318], [0.6349704315], [0.6349704318], [0.6349704315], [0.6349704318], [0.6349704315], [0.6349704318], [0.6349704315]] > F:=[1.4*(x^(.2))*((1-x)^(.2))]: > Orb(F,[x],[.5],10000,10010) [[1.169470952 + 0.3989402320 I], [1.169470953 - 0.3989402313 I], [1.169470952 + 0.3989402320 I], [1.169470953 - 0.3989402313 I], [1.169470952 + 0.3989402320 I], [1.169470953 - 0.3989402313 I], [1.169470952 + 0.3989402320 I], [1.169470953 - 0.3989402313 I], [1.169470952 + 0.3989402320 I], [1.169470953 - 0.3989402313 I], [1.169470952 + 0.3989402320 I]] > F:=[1.4*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806]] > F:=[1.4*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806], [0.5374832816], [0.5374832806]] > F:=[1.4*(x^(.4))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.5184110492], [0.5184110497], [0.5184110492], [0.5184110497], [0.5184110492], [0.5184110497], [0.5184110492], [0.5184110497], [0.5184110492], [0.5184110497], [0.5184110492]] > F:=[1.4*(x^(.5))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456], [0.4966291456]] > F:=[1.4*(x^(.6))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.4713154366], [0.4713154366], [0.4713154366], [0.4713154366], [0.4713154366], [0.4713154366], [0.4713154366], [0.4713154366], [0.4713154366], [0.4713154366], [0.4713154366]] > F:=[1.4*(x^(.7))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.4411942023], [0.4411942023], [0.4411942023], [0.4411942023], [0.4411942023], [0.4411942023], [0.4411942023], [0.4411942023], [0.4411942023], [0.4411942023], [0.4411942023]] > F:=[1.4*(x^(1.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024], [0.1059492024]] > F:=[1.4*(x^(-.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[-1.932689261 + 29.66748954 I], [-1.932689247 - 29.66748954 I], [-1.932689261 + 29.66748954 I], [-1.932689247 - 29.66748954 I], [-1.932689261 + 29.66748954 I], [-1.932689247 - 29.66748954 I], [-1.932689261 + 29.66748954 I], [-1.932689247 - 29.66748954 I], [-1.932689261 + 29.66748954 I], [-1.932689247 - 29.66748954 I], [-1.932689261 + 29.66748954 I]] > F:=[1.4*(x^(.21))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.3775887247], [0.7101980586], [0.3775887247], [0.7101980586], [0.3775887247], [0.7101980586], [0.3775887247], [0.7101980586], [0.3775887247], [0.7101980586], [0.3775887247]] > F:=[1.5*(x^(.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.1866467070], [0.8721166036], [0.1866467070], [0.8721166036], [0.1866467070], [0.8721166036], [0.1866467070], [0.8721166036], [0.1866467070], [0.8721166036], [0.1866467070]] > F:=[1.6*(x^(.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.1221624345], [0.9224050675], [0.1221624344], [0.9224050674], [0.1221624345], [0.9224050675], [0.1221624344], [0.9224050674], [0.1221624345], [0.9224050675], [0.1221624344]] > F:=[1.7*(x^(.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10030) [[0.1163962620], [0.9769993449], [0.03891956537], [0.8535761146], [0.2411622344], [0.9706434266], [0.04960965728], [0.8860621423], [0.1890644202], [0.9880006683], [0.02034967267], [0.7642430446], [0.3798041752], [0.8687397759], [0.2169501747], [0.9806416150], [0.03278084226], [0.8300377281], [0.2783690939], [0.9499226639], [0.08426122903], [0.9491745415], [0.08550656158], [0.9506691560], [0.08301820672], [0.9476420068], [0.08805636922], [0.9536062103], [0.07812366068], [0.9411917398], [0.09876950428]] > F:=[1.8*(x^(.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > F:=[1.5*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.5572596772], [0.5572596812], [0.5572596772], [0.5572596812], [0.5572596772], [0.5572596812], [0.5572596772], [0.5572596812], [0.5572596772], [0.5572596812], [0.5572596772]] > F:=[1.6*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.3679863986], [0.7491968408], [0.3679863986], [0.7491968408], [0.3679863986], [0.7491968408], [0.3679863986], [0.7491968408], [0.3679863986], [0.7491968408], [0.3679863986]] > F:=[1.7*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.2579522208], [0.8401224155], [0.2579522208], [0.8401224155], [0.2579522208], [0.8401224155], [0.2579522208], [0.8401224155], [0.2579522208], [0.8401224155], [0.2579522208]] > F:=[1.8*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.1954762299], [0.8874411678], [0.1954762303], [0.8874411680], [0.1954762299], [0.8874411678], [0.1954762303], [0.8874411680], [0.1954762299], [0.8874411678], [0.1954762303]] > F:=[1.9*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.2345434396], [0.9413283478], [0.1094723045], [0.8713493463], [0.2345434404], [0.9413283476], [0.1094723048], [0.8713493467], [0.2345434396], [0.9413283478], [0.1094723045]] > F:=[2.0*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.1772461107], [0.9792034928], [0.04133160618], [0.7371962034], [0.4796633736], [0.8348248342], [0.3129344350], [0.9697598174], [0.05992577514], [0.8081207144], [0.3599987230]] > F:=[2.1*(x^(.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > F:=[.1*(x^(1))*((1-x)^(2))]: > Orb(F,[x],[.5],10000,10010) [[ -10001] [ -10002] [ -10003] [[1.215646925 10 ], [1.215646925 10 ], [1.215646925 10 ], [ -10004] [ -10005] [ -10006] [1.215646925 10 ], [1.215646925 10 ], [1.215646925 10 ], [ -10007] [ -10008] [ -10009] [1.215646925 10 ], [1.215646925 10 ], [1.215646925 10 ], [ -10010] [ -10011]] [1.215646925 10 ], [1.215646925 10 ]] > SFP(F,[x]) {[0.]} > F:=[.2*(x^(1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[ -6991] [ -6992] [ -6993] [[1.189951298 10 ], [2.379902596 10 ], [4.759805192 10 ], [ -6994] [ -6994] [ -6995] [9.519610384 10 ], [1.903922077 10 ], [3.807844154 10 ], [ -6996] [ -6996] [ -6997] [7.615688308 10 ], [1.523137662 10 ], [3.046275324 10 ], [ -6998] [ -6998]] [6.092550648 10 ], [1.218510130 10 ]] > SFP(F,[x]) {[0.]} > F:=[.2*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > SFP(F,[x]) {[0.]} > F:=[.3*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[.4*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[.5*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[.99*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.1*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[2*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.1555015804], [0.1555015803], [0.1555015804], [0.1555015803], [0.1555015804], [0.1555015803], [0.1555015804], [0.1555015803], [0.1555015804], [0.1555015803], [0.1555015804]] > F:=[1.2*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.3*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.7*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.09312277323], [0.09312277323], [0.09312277323], [0.09312277323], [0.09312277323], [0.09312277323], [0.09312277323], [0.09312277323], [0.09312277323], [0.09312277323], [0.09312277323]] > F:=[1.4*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.5*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.6*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.06195301962], [0.06195301962], [0.06195301962], [0.06195301962], [0.06195301962], [0.06195301962], [0.06195301962], [0.06195301962], [0.06195301962], [0.06195301962], [0.06195301962]] > F:=[3.1*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.2848588327], [0.2848588327], [0.2848588327], [0.2848588327], [0.2848588327], [0.2848588327], [0.2848588327], [0.2848588327], [0.2848588327], [0.2848588327], [0.2848588327]] > F:=[4.5*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.3741132987], [0.3741132987], [0.3741132987], [0.3741132987], [0.3741132987], [0.3741132987], [0.3741132987], [0.3741132987], [0.3741132987], [0.3741132987], [0.3741132987]] > F:=[6*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.5520275981], [0.2805833512], [0.5520275982], [0.2805833511], [0.5520275981], [0.2805833512], [0.5520275982], [0.2805833511], [0.5520275981], [0.2805833512], [0.5520275982]] > F:=[4.9*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.3926076056], [0.3926076050], [0.3926076056], [0.3926076050], [0.3926076056], [0.3926076050], [0.3926076056], [0.3926076050], [0.3926076056], [0.3926076050], [0.3926076056]] > F:=[5.3*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.4091003789], [0.4091003715], [0.4091003789], [0.4091003715], [0.4091003789], [0.4091003715], [0.4091003789], [0.4091003715], [0.4091003789], [0.4091003715], [0.4091003789]] > F:=[5.7*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.5147035163], [0.3137683532], [0.5147035163], [0.3137683532], [0.5147035163], [0.3137683532], [0.5147035163], [0.3137683532], [0.5147035163], [0.3137683532], [0.5147035163]] > F:=[5.4*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.4435279487], [0.3804844580], [0.4435279487], [0.3804844580], [0.4435279487], [0.3804844580], [0.4435279487], [0.3804844580], [0.4435279487], [0.3804844580], [0.4435279487]] > F:=[5.4*(x^(1.1))*((1-x)^(2))]: > Orb(F,[x],[.5],10000,10010) [[0.3119443414], [0.7097832144], [0.3119443414], [0.7097832144], [0.3119443414], [0.7097832144], [0.3119443414], [0.7097832144], [0.3119443414], [0.7097832144], [0.3119443414]] > F:=[5.4*(x^(1.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > SFP(F,[x]) {[0.]} > F:=[5.4*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > F:=[5*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I], [Float(infinity) + Float(infinity) I]] > F:=[4.3*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10200) [[0.2164527095], [0.6107163111], [0.8854922296], [0.3468088248], [0.8396764039], [0.4736840810], [0.9329906269], [0.2037429429], [0.5815913142], [0.9084665044], [0.2788343793], [0.7365257309], [0.7082510078], [0.7589002185], [0.6638777288], [0.8258713180], [0.5093653774], [0.9354806967], [0.1960042124], [0.5632997416], [0.9193560884], [0.2457702613], [0.6734490391], [0.8127364348], [0.5421404026], [0.9285321835], [0.2175538022], [0.6131852993], [0.8832275990], [0.3533642267], [0.8476937758], [0.4523980937], [0.9265156190], [0.2237798711], [0.6269818480], [0.8696647810], [0.3920337810], [0.8879564977], [0.3396445556], [0.8305219554], [0.4974844086], [0.9358157979], [0.1949614861], [0.5608032072], [0.9206298860], [0.2418715213], [0.6654704222], [0.8237360321], [0.5147723620], [0.9349433184], [0.1976757357], [0.5672860118], [0.9172163044], [0.2523054835], [0.6865693775], [0.7935547497], [0.5878796480], [0.9040910958], [0.2919756868], [0.7592814711], [0.6630892852], [0.8269209067], [0.5066964920], [0.9356572070], [0.1954550065], [0.5619857523], [0.9200328887], [0.2436995303], [0.6692254122], [0.8186221144], [0.5275978826], [0.9327061127], [0.2046260585], [0.5836520062], [0.9070682390], [0.2830433760], [0.7439588606], [0.6939257358], [0.7822079853], [0.6137182994], [0.8827322308], [0.3547944912], [0.8493973150], [0.4478230197], [0.9246377879], [0.2295653086], [0.6395503688], [0.8559723353], [0.4299962499], [0.9156828380], [0.2569778055], [0.6957538185], [0.7793225877], [0.6201419589], [0.8765811771], [0.3724426006], [0.8690655627], [0.3937182315], [0.8894355696], [0.3353292013], [0.8248105866], [0.5120551628], [0.9352434597], [0.1967422370], [0.5650621869], [0.9184260836], [0.2486128263], [0.6791950224], [0.8045028881], [0.5620865645], [0.9199814645], [0.2438569284], [0.6695475717], [0.8181781574], [0.5287030123], [0.9324499914], [0.2054208440], [0.5855018894], [0.9057834833], [0.2869029678], [0.7506551908], [0.6806649091], [0.8023548136], [0.5672133320], [0.9172564832], [0.2521829367], [0.6863262830], [0.7939224680], [0.5870269863], [0.9047032889], [0.2901422304], [0.7561865865], [0.6694585247], [0.8183009521], [0.5283974762], [0.9325218065], [0.2051980095], [0.5849836857], [0.9061461960], [0.2858140817], [0.7487776126], [0.6844169618], [0.7967944754], [0.5803345988], [0.9093022124], [0.2763146369], [0.7320108246], [0.7167483251], [0.7443231836], [0.6932129089], [0.7833260163], [0.6112131744], [0.8850404639], [0.3481187362], [0.8413059151], [0.4693908350], [0.9319846157], [0.2068644893], [0.5888506103], [0.9033867429], [0.2940830108], [0.7628064958], [0.6557485621], [0.8364555680], [0.4821197900], [0.9345255016], [0.1989748227], [0.5703706611], [0.9154711778], [0.2576219709], [0.6970072014], [0.7773292045], [0.6245446975], [0.8721725221], [0.3849618898], [0.8814956974], [0.3583589066], [0.8535714213], [0.4365366785], [0.9192708422], [0.2460309555], [0.6739785136], [0.8119886480], [0.5439711771], [0.9278835066], [0.2195580261], [0.6176570254], [0.8790002647], [0.3655267999], [0.8616565883], [0.4143720173], [0.9056948943], [0.2871688267], [0.7511122257], [0.6797476111], [0.8036973390], [0.5640129095], [0.9189828308], [0.2469115262], [0.6757632294], [0.8094517374] ] > F:=[4.2*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.3341081163], [0.8040241847], [0.5501334367], [0.9039745903], [0.2855261711], [0.7308777853], [0.7021388951], [0.7511554126], [0.6638547854], [0.8066949085], [0.5438725753]] > F:=[3.4*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.6096769527], [0.7008987542], [0.6096769527], [0.7008987542], [0.6096769527], [0.7008987542], [0.6096769527], [0.7008987542], [0.6096769527], [0.7008987542], [0.6096769527]] > F:=[3.7*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.4839725127], [0.8043491546], [0.4839725127], [0.8043491546], [0.4839725127], [0.8043491546], [0.4839725127], [0.8043491546], [0.4839725127], [0.8043491546], [0.4839725127]] > F:=[4.1*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10020) [[0.6418291930], [0.8136628248], [0.5147569709], [0.8914593666], [0.3140849676], [0.7575879808], [0.6355793879], [0.8204155215], [0.4987889108], [0.8923085688], [0.3117092500], [0.7541498381], [0.6422806108], [0.8131635030], [0.5159259948], [0.8913185513], [0.3144785713], [0.7581535254], [0.6344690890], [0.8215838460], [0.4959964045] ] > F:=[3.9*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.4701924182], [0.8454691725], [0.4157084045], [0.8222897857], [0.4701924182], [0.8454691725], [0.4157084045], [0.8222897857], [0.4701924182], [0.8454691725], [0.4157084045]] > F:=[4*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.5112671220], [0.8700643112], [0.3636367605], [0.7995951328], [0.5337233936], [0.8661953416], [0.3737281204], [0.8096746620], [0.5112671220], [0.8700643112], [0.3636367605]] > #Appears that at some point with a, b = 1.1 we lose the periodic orbit we once > had entirely. We had orbits of period 2, then 4, then 8, but once we got to > 4.3 as our k value. We lose it, we have no stability and no orbit, completely > erratic behavior. > F:=[2.2*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.4777609778], [0.4777609778], [0.4777609778], [0.4777609778], [0.4777609778], [0.4777609778], [0.4777609778], [0.4777609778], [0.4777609778], [0.4777609778], [0.4777609778]] > F:=[2.9*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.6022029498], [0.6022029498], [0.6022029498], [0.6022029498], [0.6022029498], [0.6022029498], [0.6022029498], [0.6022029498], [0.6022029498], [0.6022029498], [0.6022029498]] > F:=[3*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.6150133035], [0.6150133041], [0.6150133035], [0.6150133041], [0.6150133035], [0.6150133041], [0.6150133035], [0.6150133041], [0.6150133035], [0.6150133041], [0.6150133035]] > F:=[3.1*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.6269737194], [0.6269737203], [0.6269737194], [0.6269737203], [0.6269737194], [0.6269737203], [0.6269737194], [0.6269737203], [0.6269737194], [0.6269737203], [0.6269737194]] > F:=[3.2*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.6381689181], [0.6381689197], [0.6381689181], [0.6381689197], [0.6381689181], [0.6381689197], [0.6381689181], [0.6381689197], [0.6381689181], [0.6381689197], [0.6381689181]] > F:=[3.6*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.5123887495], [0.7829664156], [0.5123887495], [0.7829664156], [0.5123887495], [0.7829664156], [0.5123887495], [0.7829664156], [0.5123887495], [0.7829664156], [0.5123887495]] > F:=[3.3*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.6486723807], [0.6486723847], [0.6486723807], [0.6486723847], [0.6486723807], [0.6486723847], [0.6486723807], [0.6486723847], [0.6486723807], [0.6486723847], [0.6486723807]] > F:=[4.1*(x^(1.1))*((1-x)^(1.2))]: > Orb(F,[x],[.5],10000,10010) [[0.4405156358], [0.8288708420], [0.4009666218], [0.8112302316], [0.4405156358], [0.8288708420], [0.4009666218], [0.8112302316], [0.4405156358], [0.8288708420], [0.4009666218]] > F:=[4.1*(x^(1.1))*((1-x)^(1.3))]: > Orb(F,[x],[.5],10000,10010) [[0.4337597202], [0.7810143017], [0.4337597202], [0.7810143017], [0.4337597202], [0.7810143017], [0.4337597202], [0.7810143017], [0.4337597202], [0.7810143017], [0.4337597202]] > #Changing the value of b took us from a period 4 orbit to a period 2 orbit. > F:=[4.1*(x^(1.1))*((1-x)^(1.4))]: > Orb(F,[x],[.5],10000,10010) [[0.4508519601], [0.7375414416], [0.4508519601], [0.7375414416], [0.4508519601], [0.7375414416], [0.4508519601], [0.7375414416], [0.4508519601], [0.7375414416], [0.4508519601]] > F:=[4.1*(x^(1.1))*((1-x)^(1.5))]: > Orb(F,[x],[.5],10000,10010) [[0.4728330937], [0.6884695183], [0.4728330937], [0.6884695183], [0.4728330937], [0.6884695183], [0.4728330937], [0.6884695183], [0.4728330937], [0.6884695183], [0.4728330937]] > F:=[4.1*(x^(1.1))*((1-x)^(1.9))]: > Orb(F,[x],[.5],10000,10010) [[0.5068071471], [0.5068071484], [0.5068071471], [0.5068071484], [0.5068071471], [0.5068071484], [0.5068071471], [0.5068071484], [0.5068071471], [0.5068071484], [0.5068071471]] > F:=[4.1*(x^(1.1))*((1-x)^(2.0))]: > Orb(F,[x],[.5],10000,10010) [[0.4881031087], [0.4881031066], [0.4881031087], [0.4881031066], [0.4881031087], [0.4881031066], [0.4881031087], [0.4881031066], [0.4881031087], [0.4881031066], [0.4881031087]] > F:=[4.1*(x^(1.1))*((1-x)^(2.1))]: > Orb(F,[x],[.5],10000,10010) [[0.4705971279], [0.4705971271], [0.4705971279], [0.4705971271], [0.4705971279], [0.4705971271], [0.4705971279], [0.4705971271], [0.4705971279], [0.4705971271], [0.4705971279]] > F:=[4.1*(x^(1.1))*((1-x)^(2.4))]: > Orb(F,[x],[.5],10000,10010) [[0.4243129442], [0.4243129438], [0.4243129442], [0.4243129438], [0.4243129442], [0.4243129438], [0.4243129442], [0.4243129438], [0.4243129442], [0.4243129438], [0.4243129442]] > F:=[4.1*(x^(1.1))*((1-x)^(2.7))]: > Orb(F,[x],[.5],10000,10010) [[0.3857201595], [0.3857201595], [0.3857201595], [0.3857201595], [0.3857201595], [0.3857201595], [0.3857201595], [0.3857201595], [0.3857201595], [0.3857201595], [0.3857201595]] > F:=[4.1*(x^(1.1))*((1-x)^(2.5))]: > Orb(F,[x],[.5],10000,10010) [[0.4106870038], [0.4106870034], [0.4106870038], [0.4106870034], [0.4106870038], [0.4106870034], [0.4106870038], [0.4106870034], [0.4106870038], [0.4106870034], [0.4106870038]] > F:=[4.1*(x^(1.1))*((1-x)^(2.6))]: > Orb(F,[x],[.5],10000,10010) [[0.3978431988], [0.3978431988], [0.3978431988], [0.3978431988], [0.3978431988], [0.3978431988], [0.3978431988], [0.3978431988], [0.3978431988], [0.3978431988], [0.3978431988]] > #If we brought b high enough to 2.6 we actually lose a periodic orbit > altogether and just reach a stable fixed point. > F:=[4.1*(x^(1.1))*((1-x)^(3))]: > Orb(F,[x],[.5],10000,10010) [[0.3531449901], [0.3531449901], [0.3531449901], [0.3531449901], [0.3531449901], [0.3531449901], [0.3531449901], [0.3531449901], [0.3531449901], [0.3531449901], [0.3531449901]] > F:=[4.1*(x^(1.1))*((1-x)^(4))]: > Orb(F,[x],[.5],10000,10010) [[0.2741366687], [0.2741366687], [0.2741366687], [0.2741366687], [0.2741366687], [0.2741366687], [0.2741366687], [0.2741366687], [0.2741366687], [0.2741366687], [0.2741366687]] > F:=[4.1*(x^(1.1))*((1-x)^(5))]: > Orb(F,[x],[.5],10000,10010) [[0.2228920785], [0.2228920785], [0.2228920785], [0.2228920785], [0.2228920785], [0.2228920785], [0.2228920785], [0.2228920785], [0.2228920785], [0.2228920785], [0.2228920785]] > F:=[4.1*(x^(1.1))*((1-x)^(9))]: > Orb(F,[x],[.5],10000,10010) [[0.1251328445], [0.1251328445], [0.1251328445], [0.1251328445], [0.1251328445], [0.1251328445], [0.1251328445], [0.1251328445], [0.1251328445], [0.1251328445], [0.1251328445]] > F:=[4.1*(x^(1.1))*((1-x)^(12))]: > Orb(F,[x],[.5],10000,10010) [[0.09317416899], [0.09317416899], [0.09317416899], [0.09317416899], [0.09317416899], [0.09317416899], [0.09317416899], [0.09317416899], [0.09317416899], [0.09317416899], [0.09317416899]] > F:=[4.1*(x^(1.1))*((1-x)^(30))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > #Eventually as b keeps getting larger, we maintain stability to a point that > gets closer and closer to 0. > F:=[4.1*(x^(1.2))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.4756163672], [0.8262396785], [0.4756163615], [0.8262396760], [0.4756163672], [0.8262396785], [0.4756163615], [0.8262396760], [0.4756163672], [0.8262396785], [0.4756163615]] > F:=[4.1*(x^(2.6))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[4.1*(x^(2.))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[4.1*(x^(1.7))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.6266867003], [0.6266867003], [0.6266867003], [0.6266867003], [0.6266867003], [0.6266867003], [0.6266867003], [0.6266867003], [0.6266867003], [0.6266867003], [0.6266867003]] > F:=[4.1*(x^(1.9))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.5430353925], [0.5430353925], [0.5430353925], [0.5430353925], [0.5430353925], [0.5430353925], [0.5430353925], [0.5430353925], [0.5430353925], [0.5430353925], [0.5430353925]] > F:=[4.1*(x^(1.1))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10030) [[0.1665412727], [0.4757097168], [0.9493621004], [0.1960805965], [0.5491290876], [0.9560441947], [0.1715243820], [0.4884521642], [0.9536178032], [0.1804874281], [0.5110120358], [0.9579796165], [0.1643371548], [0.4700286059], [0.9470502088], [0.2044835674], [0.5690589604], [0.9503305802], [0.1925462738], [0.5406175376], [0.9574969764], [0.1661325898], [0.4746584059], [0.9489535141], [0.1975691535], [0.5526911684], [0.9552611312], [0.1744227676], [0.4957983249], [0.9554846218], [0.1735961127]] > F:=[4.1*(x^(1.2))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10030) [[0.4975473840], [0.8914157536], [0.3878350402], [0.8054318082], [0.6153046989], [0.8806615115], [0.4200830992], [0.8397535991], [0.5327879447], [0.8998355862], [0.3618204486], [0.7725355444], [0.6842259846], [0.8211069389], [0.5789712771], [0.8959504750], [0.3739080054], [0.7883874887], [0.6522481654], [0.8537854038], [0.4958987995], [0.8907857415], [0.3897544715], [0.8076771384], [0.6102392214], [0.8834503183], [0.4118257579], [0.8316604824], [0.5532293520], [0.9002326216], [0.3605770761]] > F:=[4.1*(x^(1.3))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10030) [[0.5201792586], [0.8411274665], [0.5201792656], [0.8411274693], [0.5201792586], [0.8411274665], [0.5201792656], [0.8411274693], [0.5201792586], [0.8411274665], [0.5201792656], [0.8411274693], [0.5201792586], [0.8411274665], [0.5201792656], [0.8411274693], [0.5201792586], [0.8411274665], [0.5201792656], [0.8411274693], [0.5201792586], [0.8411274665], [0.5201792656], [0.8411274693], [0.5201792586], [0.8411274665], [0.5201792656], [0.8411274693], [0.5201792586], [0.8411274665], [0.5201792656]] > F:=[4.1*(x^(1.4))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.5945711153], [0.8027565783], [0.5945711153], [0.8027565783], [0.5945711153], [0.8027565783], [0.5945711153], [0.8027565783], [0.5945711153], [0.8027565783], [0.5945711153]] > F:=[4.1*(x^(1.5))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.7106794614], [0.7106794680], [0.7106794614], [0.7106794680], [0.7106794614], [0.7106794680], [0.7106794614], [0.7106794680], [0.7106794614], [0.7106794680], [0.7106794614]] > F:=[4.1*(x^(1.6))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.6971567664], [0.6971567656], [0.6971567664], [0.6971567656], [0.6971567664], [0.6971567656], [0.6971567664], [0.6971567656], [0.6971567664], [0.6971567656], [0.6971567664]] > F:=[4.1*(x^(1.7))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.6807566950], [0.6807566946], [0.6807566950], [0.6807566946], [0.6807566950], [0.6807566946], [0.6807566950], [0.6807566946], [0.6807566950], [0.6807566946], [0.6807566950]] > F:=[4.1*(x^(1.8))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.6598640069], [0.6598640069], [0.6598640069], [0.6598640069], [0.6598640069], [0.6598640069], [0.6598640069], [0.6598640069], [0.6598640069], [0.6598640069], [0.6598640069]] > F:=[3.2*(x^(1.8))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.3*(x^(1.8))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[.3*(x^(1.8))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[-1*(x^(1.8))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I]] > F:=[-4.1*(x^(1.8))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I] ] > F:=[-3.2*(x^(1.8))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I] ] > F:=[-5.1*(x^(1.8))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I], [-Float(infinity) - Float(infinity) I] ] > F:=[-2.1*(x^(1.8))*((1-x)^(1))]: > Orb(F,[x],[.5],10000,10010) [[-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I], [-0. - 0. I]] > F:=[1*(x^(1.1))*((1-x)^(1.1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(1.2))*((1-x)^(1.2))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(1.3))*((1-x)^(1.3))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(2.0))*((1-x)^(2.0))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(3.5))*((1-x)^(3.5))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(4.0))*((1-x)^(4.0))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(4.0))*((1-x)^(1.2))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(1.2))*((1-x)^(4.0))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(.9))*((1-x)^(1.2))]: > Orb(F,[x],[.5],10000,10010) [[0.1474492855], [0.1474492855], [0.1474492855], [0.1474492855], [0.1474492855], [0.1474492855], [0.1474492855], [0.1474492855], [0.1474492855], [0.1474492855], [0.1474492855]] > F:=[1.*(x^(1.2))*((1-x)^(.9))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(1.2))*((1-x)^(.1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > F:=[1.*(x^(1.2))*((1-x)^(-.1))]: > Orb(F,[x],[.5],10000,10010) [[0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.], [0.]] > #Seems as long as k=1 and a>1 our stable fixed point will always be 0. > F:=x(n)=k*(x(n-1)^a)*(1-x(n-1))^b*(1-x(n-2))^c; > F:=1*(z[1]^.99)*((1-z[1])^1)*((1-z[2])^1): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454], [0.01949535454, 0.01949535454]] > F:=1*(z[1]^1.2)*((1-z[1])^1)*((1-z[2])^1): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1*(z[1]^1.1)*((1-z[1])^1)*((1-z[2])^1): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1*(z[1]^1.001)*((1-z[1])^1)*((1-z[2])^1): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[ -54344 -54290] [[1.357801392 10 , 2.644928008 10 ], [ -54399 -54344] [6.151331012 10 , 1.357801392 10 ], [ -54453 -54399] [2.458997397 10 , 6.151331012 10 ], [ -54508 -54453] [8.672590364 10 , 2.458997397 10 ], [ -54562 -54508] [2.698279752 10 , 8.672590364 10 ], [ -54617 -54562] [7.404880840 10 , 2.698279752 10 ], [ -54671 -54617] [1.792204592 10 , 7.404880840 10 ], [ -54726 -54671] [3.825083864 10 , 1.792204592 10 ], [ -54781 -54726] [7.198195048 10 , 3.825083864 10 ], [ -54835 -54781] [1.194210417 10 , 7.198195048 10 ], [ -54890 -54835]] [1.746457138 10 , 1.194210417 10 ]] > F:=1.1*(z[1]^1.1)*((1-z[1])^1)*((1-z[2])^1): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1.1*(z[1]^1.2)*((1-z[1])^1)*((1-z[2])^1): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=4.0*(z[1]^1.1)*((1-z[1])^1)*((1-z[2])^1): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0.4813846180, 0.4813846196], [0.4813846160, 0.4813846180], [0.4813846172, 0.4813846160], [0.4813846192, 0.4813846172], [0.4813846184, 0.4813846192], [0.4813846164, 0.4813846184], [0.4813846168, 0.4813846164], [0.4813846188, 0.4813846168], [0.4813846188, 0.4813846188], [0.4813846168, 0.4813846188], [0.4813846164, 0.4813846168]] > F:=1*(z[1]^2)*((1-z[1])^10)*((1-z[2])^9): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=1*(z[1]^2)*((1-z[1])^(-10))*((1-z[2])^(-9)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10000,10010) [[0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.], [0., 0.]] > F:=3*(z[1]^1.1)*((1-z[1])^(-.2))*((1-z[2])^(2)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [ [[0.1718707902, 0.08605804368], [0.3750257991, 0.1718707902], [0.7684441401, 0.3750257991], [1.175135139, 0.7684441401], [0.2201975016 - 0.1599828494 I, 1.175135139], [0.01712567540 - 0.01537611671 I, 0.2201975016 - 0.1599828494 I], [0.02769519375 - 0.01181428579 I, 0.01712567540 - 0.01537611671 I], [0.05659444977 - 0.02491297394 I, 0.02769519375 - 0.01181428579 I], [0.1217011644 - 0.05686742484 I, 0.05659444977 - 0.02491297394 I], [0.2724206393 - 0.1285714027 I, 0.1217011644 - 0.05686742484 I], [0.6099286140 - 0.2512332776 I, 0.2724206393 - 0.1285714027 I], [1.185542626 - 0.2334526387 I, 0.6099286140 - 0.2512332776 I], [0.8975568273 + 0.4696190727 I, 1.185542626 - 0.2334526387 I], [0.1700522746 - 0.2631234265 I, 0.8975568273 + 0.4696190727 I], [-0.1483135697 + 0.1323527449 I, 0.1700522746 - 0.2631234265 I], [-0.3694560864 - 0.05571801690 I, -0.1483135697 + 0.1323527449 I], [-1.180021093 + 0.4811196234 I, -0.3694560864 - 0.05571801690 I], [-6.265454613 - 0.08189541390 I, -1.180021093 + 0.4811196234 I], [-56.08631799 + 50.84917320 I, -6.265454613 - 0.08189541390 I], [-7350.101031 + 2500.123367 I, -56.08631799 + 50.84917320 I], [ 6 7 ]] [-7.453530150 10 + 5.507781579 10 I, -7350.101031 + 2500.123367 I]] > F:=5*(z[1]^1.1)*((1-z[1])^(2))*((1-z[2])^(-.9)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [[ 1241 1241 [[2.658728188 10 + 1.622017131 10 I, 446 446 ] [ 3.543567018 10 - 6.890164225 10 I], [ 3447 3446 1.329598577 10 - 1.915396733 10 I, 1241 1241 ] [ 2.658728188 10 + 1.622017131 10 I], [ 9568 9569 -8.915049655 10 + 2.689355424 10 I, 3447 3446 ] [ 1.329598577 10 - 1.915396733 10 I], [ 26563 26561 -3.851226976 10 - 8.281821570 10 I, 9568 9569 ] [ -8.915049655 10 + 2.689355424 10 I], [ 73735 73735 -1.296595658 10 - 1.562473294 10 I, 26563 26561 ] [ -3.851226976 10 - 8.281821570 10 I], [ 204672 204672 6.144427035 10 - 5.757643195 10 I, 73735 73735 ] [ -1.296595658 10 - 1.562473294 10 I], [ 568124 568123 -9.786506535 10 - 1.688421625 10 I, 204672 204672 ] [ 6.144427035 10 - 5.757643195 10 I], [ 1576982 1576982 2.565331711 10 + 2.300841396 10 I, 568124 568123 ] [ -9.786506535 10 - 1.688421625 10 I], [ 4377333 4377333 -7.439956660 10 + 9.198754720 10 I, 1576982 1576982 ] [ 2.565331711 10 + 2.300841396 10 I], [ 12150452 12150451 -1.058739320 10 + 3.017007428 10 I, 4377333 4377333 ] [ -7.439956660 10 + 9.198754720 10 I], [ 33726801 33726800 -2.230260313 10 - 5.819136360 10 I, 12150452 12150451 ] [ -1.058739320 10 + 3.017007428 10 I], [ 93617677 93617677 -9.101267320 10 - 8.094029940 10 I, 33726801 33726800 ] [ -2.230260313 10 - 5.819136360 10 I], [ 259860680 259860681 4.772290624 10 - 3.418763015 10 I, 93617677 93617677 ] [ -9.101267320 10 - 8.094029940 10 I], [ 721312202 721312203 5.762318020 10 + 1.437165808 10 I, 259860680 259860681 ] [ 4.772290624 10 - 3.418763015 10 I], [ 2002193216 2002193217 -8.726378765 10 + 1.337693392 10 I, 721312202 721312203 ] [ 5.762318020 10 + 1.437165808 10 I], [ 5557617990 5557617991 -7.702721580 10 + 1.217038956 10 I, 2002193216 2002193217 ] [ -8.726378765 10 + 1.337693392 10 I], [ 15426641877 15426641877 2.150750792 10 + 6.045545695 10 I, 5557617990 5557617991 ] [ -7.702721580 10 + 1.217038956 10 I], [ 42820733627 42820733629 2.454523176 10 - 7.234001135 10 I, 15426641877 15426641877 ] [ 2.150750792 10 + 6.045545695 10 I], [ 118860296563 118860296560 -1.721629366 10 - 6.843049695 10 I, 42820733627 42820733629 ] [ 2.454523176 10 - 7.234001135 10 I], [ 329928259079 329928259079 1.065166072 10 + 7.153679200 10 I, 118860296563 118860296560 ] [ -1.721629366 10 - 6.843049695 10 I], [ 915803336240 915803336241 -6.757486795 10 - 2.148584666 10 I, 329928259079 329928259079 ]] 1.065166072 10 + 7.153679200 10 I]] > F:=1*(z[1]^(1.1))*((1-z[1])^(.1))*((1-z[2])^(-.999)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],100,130) [[ -9490 -9491 [[-2.212693554 10 - 7.186882523 10 I, -8627 -8631 ] [ -1.149837170 10 + 1.108737734 10 I], [ -10439 -10441 -2.530213780 10 - 7.921966810 10 I, -9490 -9491 ] [ -2.212693554 10 - 7.186882523 10 I], [ -11483 -11484 -3.361159383 10 + 9.655337661 10 I, -10439 -10441 ] [ -2.530213780 10 - 7.921966810 10 I], [ -12631 -12633 -1.986408763 10 - 1.282476644 10 I, -11483 -11484 ] [ -3.361159383 10 + 9.655337661 10 I], [ -13894 -13895 -1.610949089 10 + 5.108095651 10 I, -12631 -12633 ] [ -1.986408763 10 - 1.282476644 10 I], [ -15284 -15285 -7.088477919 10 + 1.673472544 10 I, -13894 -13895 ] [ -1.610949089 10 + 5.108095651 10 I], [ -16812 -16813 -3.291930094 10 - 9.758861066 10 I, -15284 -15285 ] [ -7.088477919 10 + 1.673472544 10 I], [ -18493 -18496 -2.450833184 10 - 6.997468058 10 I, -16812 -16813 ] [ -3.291930094 10 - 9.758861066 10 I], [ -20342 -20343 -1.279069704 10 + 4.111582536 10 I, -18493 -18496 ] [ -2.450833184 10 - 6.997468058 10 I], [ -22377 -22378 -8.727743771 10 + 2.441019692 10 I, -20342 -20343 ] [ -1.279069704 10 + 4.111582536 10 I], [ -24614 -24615 -2.077302935 10 - 6.049961336 10 I, -22377 -22378 ] [ -8.727743771 10 + 2.441019692 10 I], [ -27076 -27078 -9.304554575 10 + 2.249052764 10 I, -24614 -24615 ] [ -2.077302935 10 - 6.049961336 10 I], [ -29783 -29784 -2.780650241 10 - 8.953211623 10 I, -27076 -27078 ] [ -9.304554575 10 + 2.249052764 10 I], [ -32761 -32763 -1.629096848 10 - 4.642745080 10 I, -29783 -29784 ] [ -2.780650241 10 - 8.953211623 10 I], [ -36037 -36038 -1.305359231 10 + 3.793489755 10 I, -32761 -32763 ] [ -1.629096848 10 - 4.642745080 10 I], [ -39641 -39644 -2.796836223 10 - 8.553866064 10 I, -36037 -36038 ] [ -1.305359231 10 + 3.793489755 10 I], [ -43605 -43606 -2.344306074 10 + 7.530012693 10 I, -39641 -39644 ] [ -2.796836223 10 - 8.553866064 10 I], [ -47966 -47967 -8.517315809 10 + 2.361201208 10 I, -43605 -43606 ] [ -2.344306074 10 + 7.530012693 10 I], [ -52762 -52763 -2.545676210 10 - 7.421529226 10 I, -47966 -47967 ] [ -8.517315809 10 + 2.361201208 10 I], [ -58038 -58041 -1.844451350 10 + 3.909400700 10 I, -52762 -52763 ] [ -2.545676210 10 - 7.421529226 10 I], [ -63842 -63843 -2.957937669 10 - 9.534734665 10 I, -58038 -58041 ] [ -1.844451350 10 + 3.909400700 10 I], [ -70226 -70228 -2.195430630 10 - 6.335856230 10 I, -63842 -63843 ] [ -2.957937669 10 - 9.534734665 10 I], [ -77249 -77249 -5.732116824 10 + 1.663341982 10 I, -70226 -70228 ] [ -2.195430630 10 - 6.335856230 10 I], [ -84974 -84976 -8.983694047 10 - 3.139024169 10 I, -77249 -77249 ] [ -5.732116824 10 + 1.663341982 10 I], [ -93471 -93471 -4.241783624 10 + 1.360236804 10 I, -84974 -84976 ] [ -8.983694047 10 - 3.139024169 10 I], [ -102818 -102819 -4.106985488 10 + 1.116884067 10 I, -93471 -93471 ] [ -4.241783624 10 + 1.360236804 10 I], [ -113100 -113100 -7.198884430 10 - 2.103255113 10 I, -102818 -102819 ] [ -4.106985488 10 + 1.116884067 10 I], [ -124410 -124412 -9.173959134 10 + 1.359216515 10 I, -113100 -113100 ] [ -7.198884430 10 - 2.103255113 10 I], [ -136850 -136851 -1.089553783 10 - 3.520553411 10 I, -124410 -124412 ] [ -9.173959134 10 + 1.359216515 10 I], [ -150535 -150537 -1.160122169 10 - 3.437657363 10 I, -136850 -136851 ] [ -1.089553783 10 - 3.520553411 10 I], [ -165589 -165589 -3.578615685 10 + 1.035146747 10 I, -150535 -150537 ]] -1.160122169 10 - 3.437657363 10 I]] > F:=1*(z[1]^(.2))*((1-z[1])^(-.9))*((1-z[2])^(30)): > A:=ToSys(2,z,F)[1]: > Orb(A,[z[1],z[2]],[.5,.5],10,30) [ [[0.3478652976 - 0.001603519724 I, 0.004979799115 - 0.0001122642680 I], [1.024083546 + 0.0002559024817 I, 0.3478652976 - 0.001603519724 I], [-0.00007498084298 + 0.00001913822803 I, 1.024083546 + 0.0002559024817 I], [ -50 -50 [2.661610703 10 + 3.334222882 10 I, ] [ -0.00007498084298 + 0.00001913822803 I], [ -10 -11 1.318268142 10 + 2.383161719 10 I, -50 -50 ] 2.661610703 10 + 3.334222882 10 I], [ -10 -11 ] [0.01059545208 + 0.0003791570245 I, 1.318268142 10 + 2.383161719 10 I], [0.4066593141 + 0.003049521795 I, 0.01059545208 + 0.0003791570245 I], [0.9706920430 - 0.005213808059 I, 0.4066593141 + 0.003049521795 I], [0.000003536771953 - 0.000001147411424 I, 0.9706920430 - 0.005213808059 I], [ -48 -47 [6.493525288 10 - 1.170390167 10 I, ] [ 0.000003536771953 - 0.000001147411424 I], [ -10 -11 4.124438596 10 - 8.912510666 10 I, -48 -47 ] 6.493525288 10 - 1.170390167 10 I], [ -10 -11 ] [0.01332482358 - 0.0005674952839 I, 4.124438596 10 - 8.912510666 10 I], [0.4268118189 - 0.003854369879 I, 0.01332482358 - 0.0005674952839 I], [0.9306438859 + 0.008745327958 I, 0.4268118189 - 0.003854369879 I], [ -7 -7 ] [ [5.764847913 10 + 1.888027034 10 I, 0.9306438859 + 0.008745327958 I], [ -36 -37 -1.048172056 10 + 6.542952650 10 I, -7 -7 ] [ 5.764847913 10 + 1.888027034 10 I], [ -8 -8 5.722889265 10 + 3.251712454 10 I, -36 -37 ] -1.048172056 10 + 6.542952650 10 I], [ -8 -8 ] [0.03642091437 + 0.003777226783 I, 5.722889265 10 + 3.251712454 10 I], [0.5334642076 + 0.01290978660 I, 0.03642091437 + 0.003777226783 I], [0.5732485863 - 0.05049688040 I, 0.5334642076 + 0.01290978660 I], [ -10 -10 ]] [1.306805434 10 - 1.841041650 10 I, 0.5732485863 - 0.05049688040 I]] > F:=x(n)=k*(x(n-1)^a)*((1-x(n-1))^b) >