> #OK to post #Timothy Nasralla, HW13, 10/18/21 > read "/Users/tan88/OneDrive - Rutgers University/M13.txt" > #Question 2: RUID: 192005950 #x(n) = ((x(n-1)^2)+9*x(n-1) + 2) / ((x(n-1))^2) > #y(n) = (5*(x(n-1)^2)+9*x(n-1) + 2) / (2*(x(n-1)^2)+9) > #Find the fixed points, stable fixed points, and confirm their stability w/ > Orb2 and IC's of x(0) = 9.5, y(0) = 0.5 > evalf(FP2([((x^2+9*x+2)/(x^2)),(5*x^2+9*x+2)/(2*x^2+9)],x,y)) [[3.630633802, 2.844306279]] > #The one fixed point is (3.6306, 2.8443) > evalf(SFP2([((x^2+9*x+2)/(x^2)),(5*x^2+9*x+2)/(2*x^2+9)],x,y)) [[3.630633802, 2.844306279]] > #This fixed point is seemingly stable. > Orb2([((x^2+9*x+2)/(x^2)),(5*x^2+9*x+2)/(2*x^2+9)],x,y,[9.5,0.5],1000,1020) [[3.630633811, 2.844306279], [3.630633795, 2.844306282], [3.630633811, 2.844306279], [3.630633795, 2.844306282], [3.630633811, 2.844306279], [3.630633795, 2.844306282], [3.630633811, 2.844306279], [3.630633795, 2.844306282], [3.630633811, 2.844306279], [3.630633795, 2.844306282], [3.630633811, 2.844306279], [3.630633795, 2.844306282], [3.630633811, 2.844306279], [3.630633795, 2.844306282], [3.630633811, 2.844306279], [3.630633795, 2.844306282], [3.630633811, 2.844306279], [3.630633795, 2.844306282], [3.630633811, 2.844306279], [3.630633795, 2.844306282]] > #Orb 2 shows that over the 1000th to 1020th term of the sequence the point > [3.6306, 2.8443] is a stable fixed point. > #Question 3: Run RT2(x,y,1,100) and generate 200 random transformations and > find the stable equilibria for each. Then confirm #w/ Orb2 #Since using FP2drz > returns 4 imaginary fixed points as well as the real fixed points, I used FP2 > instead as imaginary fixed points are relatively meaningless and did not > appear to be a stable fixed point (it also saved space). > #TRANSFORMATION 1 RT2(x,y,1,100) [96 + 60 y + 99 x 73 + 94 y + 62 x] [----------------, ----------------] [11 + 46 y + 19 x 91 + 50 y + 25 x] > evalf(FP2drz([(47+8*y+46*x)/(44+9*y+77*x),(59+16*y+1*x)/(70+77*y+39*x)],x,y)) [[0.7976878657, 0.4875444232], [-0.8606085011, 0.7463118211], [-0.6336382504, -1.081346978], [0.7815239396, -1.586503204]] > SFP2drz([(47+8*y+46*x)/(44+9*y+77*x),(59+16*y+1*x)/(70+77*y+39*x)],x,y) [[0.7976878657, 0.4875444232]] > > Orb2([(47+8*y+46*x)/(44+9*y+77*x),(59+16*y+1*x)/(70+77*y+39*x)],x,y,[0.5,-4],1 > 010,1020) [[0.7976878659, 0.4875444230], [0.7976878654, 0.4875444230], [0.7976878659, 0.4875444230], [0.7976878654, 0.4875444230], [0.7976878659, 0.4875444230], [0.7976878654, 0.4875444230], [0.7976878659, 0.4875444230], [0.7976878654, 0.4875444230], [0.7976878659, 0.4875444230], [0.7976878654, 0.4875444230]] > #TRANSFORMATION 2 > RT2(x,y,1,100) [52 + 44 y + 52 x 32 + 95 y + 5 x ] [----------------, ----------------] [9 + 52 y + 25 x 13 + 97 y + 82 x] > > evalf(FP2drz([(49+49*y+67*x)/(74+90*y+74*x),(27+98*y+72*x)/(2+73*y+85*x)],x,y) > ) [[-1.248897313, 0.3573639451], [0.6528216044, 1.321846941], [1.459574837 - 0.6192287391 I, -1.531861573 + 0.6401885356 I], [1.459574837 + 0.6192287391 I, -1.531861573 - 0.6401885356 I]] > SFP2drz([(49+49*y+67*x)/(74+90*y+74*x),(27+98*y+72*x)/(2+73*y+85*x)],x,y) [[0.6528216044, 1.321846941]] > > Orb2([(49+49*y+67*x)/(74+90*y+74*x),(27+98*y+72*x)/(2+73*y+85*x)],x,y,[0.5,-4] > ,1010,1020) [[0.6528216034, 1.321846940], [0.6528216034, 1.321846940], [0.6528216034, 1.321846940], [0.6528216034, 1.321846940], [0.6528216034, 1.321846940], [0.6528216034, 1.321846940], [0.6528216034, 1.321846940], [0.6528216034, 1.321846940], [0.6528216034, 1.321846940], [0.6528216034, 1.321846940]] > #TRANSFORMATION 3 > RT2(x,y,1,100) [ 92 + 9 y + 4 x 27 + 12 y + 48 x] [----------------, ----------------] [18 + 22 y + 60 x 14 + 20 y + 4 x ] > > evalf(FP2drz([(41+4*y+44*x)/(13+19*y+10*x),(15+64*y+9*x)/(12+52*y+25*x)],x,y)) [[3.169979666, 0.6897016763], [-3.047286318, 2.363959656], [-0.9652962545, -0.07868388747], [22.80704716, -10.37326804]] > SFP2drz([(41+4*y+44*x)/(13+19*y+10*x),(15+64*y+9*x)/(12+52*y+25*x)],x,y) [[3.169979666, 0.6897016763]] > > Orb2([(41+4*y+44*x)/(13+19*y+10*x),(15+64*y+9*x)/(12+52*y+25*x)],x,y,[0.5,-4], > 1010,1020) [[3.169979666, 0.6897016765], [3.169979666, 0.6897016765], [3.169979666, 0.6897016765], [3.169979666, 0.6897016765], [3.169979666, 0.6897016765], [3.169979666, 0.6897016765], [3.169979666, 0.6897016765], [3.169979666, 0.6897016765], [3.169979666, 0.6897016765], [3.169979666, 0.6897016765]] > #TRANSFORMATION 4 > RT2(x,y,1,100) [94 + 28 y + 84 x 55 + 41 y + 82 x] [----------------, ----------------] [89 + 66 y + 37 x 55 + 97 y + 96 x] > > evalf(FP2drz([(72+90*y+18*x)/(43+55*y+40*x),(17+70*y+52*x)/(81+87*y+34*x)],x,y > )) [[1.208133500, 0.7042432306], [-1.334418894 - 2.438930620 I, -0.3962214209 + 1.545602888 I], [0.4024744760, -0.8172048294], [-1.334418894 + 2.438930620 I, -0.3962214209 - 1.545602888 I]] > SFP2drz([(72+90*y+18*x)/(43+55*y+40*x),(17+70*y+52*x)/(81+87*y+34*x)],x,y) [[1.208133500, 0.7042432306]] > > Orb2([(72+90*y+18*x)/(43+55*y+40*x),(17+70*y+52*x)/(81+87*y+34*x)],x,y,[0.5,-4 > ],1010,1020) [[1.208133499, 0.7042432307], [1.208133501, 0.7042432303], [1.208133499, 0.7042432312], [1.208133500, 0.7042432303], [1.208133499, 0.7042432307], [1.208133501, 0.7042432303], [1.208133499, 0.7042432312], [1.208133500, 0.7042432303], [1.208133499, 0.7042432307], [1.208133501, 0.7042432303]] > #TRANSFORMATION 5 > RT2(x,y,1,100) [79 + 20 y + 70 x 53 + 6 y + 96 x] [----------------, ---------------] [49 + 40 y + 72 x 3 + 19 y + 94 x] (20)> evalf(FP2drz(,x,y)) [[-1.130382004, 0.5633187831], [1.012808383, 1.287652350], [0.09334645223 - 0.7135850849 I, -1.620587661 + 3.175334548 I], [0.09334645223 + 0.7135850849 I, -1.620587661 - 3.175334548 I]] (20)> SFP2drz(,x,y) [[1.012808383, 1.287652350]] (20)> Orb2(,x,y,[0.5,-4],1010,1020) [[1.012808384, 1.287652350], [1.012808384, 1.287652350], [1.012808384, 1.287652350], [1.012808384, 1.287652350], [1.012808384, 1.287652350], [1.012808384, 1.287652350], [1.012808384, 1.287652350], [1.012808384, 1.287652350], [1.012808384, 1.287652350], [1.012808384, 1.287652350]] > #TRANSFORMATION 6 > RT2(x,y,1,100) [70 + 85 y + 14 x 73 + 44 y + 49 x] [----------------, ----------------] [63 + 39 y + 11 x 67 + 53 y + 79 x] > evalf(FP2drz([(87 + 16*y + 98*x)/(43 + 53*y + 61*x), (47 + 28*y + 75*x)/(3 + > 5*y + 11*x)],x,y)) [[0.5868505538, 6.506538819], [-6.182864281, 7.521139391], [-0.7578075050 - 0.6409471594 I, 0.2843623337 + 1.510749439 I], [-0.7578075050 + 0.6409471594 I, 0.2843623337 - 1.510749439 I]] (24)> SFP2drz(,x,y) [[1.437649375, 0.8011984353]] (24)> Orb2(,x,y,[0.5,-4],1010,1020) [[1.437649374, 0.8011984354], [1.437649374, 0.8011984354], [1.437649374, 0.8011984354], [1.437649374, 0.8011984354], [1.437649374, 0.8011984354], [1.437649374, 0.8011984354], [1.437649374, 0.8011984354], [1.437649374, 0.8011984354], [1.437649374, 0.8011984354], [1.437649374, 0.8011984354]] > #Transformation 7 > RT2(x,y,1,100) [11 + 48 y + 29 x 33 + 89 y + 43 x] [----------------, ----------------] [50 + 21 y + 90 x 92 + 96 y + 24 x] (28)> evalf(FP2drz(,x,y)) [[-0.6868108449, 0.2728287168], [0.5287015733, 0.6846039554], [-0.2154819916 - 0.4492114448 I, -0.5177557904 + 0.2451062158 I], [-0.2154819916 + 0.4492114448 I, -0.5177557904 - 0.2451062158 I]] (28)> SFP2drz(,x,y) [[0.5287015733, 0.6846039554]] (28)> Orb2(,x,y,[0.5,-4],1010,1020) [[0.5287015735, 0.6846039554], [0.5287015735, 0.6846039554], [0.5287015735, 0.6846039554], [0.5287015735, 0.6846039554], [0.5287015735, 0.6846039554], [0.5287015735, 0.6846039554], [0.5287015735, 0.6846039554], [0.5287015735, 0.6846039554], [0.5287015735, 0.6846039554], [0.5287015735, 0.6846039554]] > #TRANSFORMATION 8 > RT2(x,y,1,100) [72 + 64 y + 98 x 58 + 92 y + 43 x] [----------------, ----------------] [36 + 79 y + 19 x 20 + 91 y + 9 x ] (32)> evalf(FP2drz(,x,y)) [[-1.849525887, 0.5124148063], [1.797919051, 1.564071119], [-0.5147839664, -0.3348527563], [17.09317190, -3.435721988]] (32)> SFP2drz(,x,y) [[1.797919051, 1.564071119]] (32)> Orb2(,x,y,[0.5,-4],1010,1020) [[1.797919049, 1.564071119], [1.797919049, 1.564071119], [1.797919049, 1.564071119], [1.797919049, 1.564071119], [1.797919049, 1.564071119], [1.797919049, 1.564071119], [1.797919049, 1.564071119], [1.797919049, 1.564071119], [1.797919049, 1.564071119], [1.797919049, 1.564071119]] > #TRANSFORMATION 9 > RT2(x,y,1,100) [84 + 2 y + 74 x 35 + 80 y + 45 x] [----------------, ----------------] [16 + 13 y + 82 x 83 + 36 y + 38 x] (36)> evalf(FP2drz(,x,y)) [[-0.7680205180, 0.7437536664], [1.330924020, 1.041835212], [-0.7116186848, -0.1067653726], [1.728181925, -2.964263340]] (36)> SFP2drz(,x,y) [[1.330924020, 1.041835212]] (36)> Orb2(,x,y,[0.5,-4],1010,1020) [[1.330924024, 1.041835212], [1.330924023, 1.041835213], [1.330924024, 1.041835212], [1.330924023, 1.041835213], [1.330924024, 1.041835212], [1.330924023, 1.041835213], [1.330924024, 1.041835212], [1.330924023, 1.041835213], [1.330924024, 1.041835212], [1.330924023, 1.041835213]] > #TRANSFORMATION 10 > RT2(x,y,1,100) [14 + 39 y + 85 x 61 + 2 y + 58 x] [----------------, ---------------] [50 + 10 y + 37 x 3 + 77 y + 22 x] (40)> evalf(FP2drz(,x,y)) [[-0.6975046513, 0.6180229555], [1.630301850, 1.201972506], [0.3876616224 - 0.7463257751 I, -1.133656917 + 0.3750310121 I], [0.3876616224 + 0.7463257751 I, -1.133656917 - 0.3750310121 I]] (40)> SFP2drz(,x,y) [[1.630301850, 1.201972506], [0.3876616224 - 0.7463257751 I, -1.133656917 + 0.3750310121 I], [0.3876616224 + 0.7463257751 I, -1.133656917 - 0.3750310121 I]] (40)> Orb2(,x,y,[0.5,-4],1010,1020) [[1.630301850, 1.201972505], [1.630301849, 1.201972507], [1.630301850, 1.201972505], [1.630301849, 1.201972507], [1.630301850, 1.201972505], [1.630301849, 1.201972507], [1.630301850, 1.201972505], [1.630301849, 1.201972507], [1.630301850, 1.201972505], [1.630301849, 1.201972507]] > #TRANSFORMATION 11 > RT2(x,y,1,100) [57 + 61 y + 94 x 11 + 74 y + 30 x] [----------------, ----------------] [97 + 52 y + 63 x 83 + 4 y + 39 x ] (44)> evalf(FP2drz(,x,y)) [[-1.639303284, 0.7342736810], [0.9894625693, 0.8009750334], [-0.2661608063, -0.7126564053], [1.226140056, -15.00122407]] (44)> SFP2drz(,x,y) [[0.9894625693, 0.8009750334]] (44)> Orb2(,x,y,[0.5,-4],1010,1020) [[0.9894625685, 0.8009750334], [0.9894625685, 0.8009750334], [0.9894625685, 0.8009750334], [0.9894625685, 0.8009750334], [0.9894625685, 0.8009750334], [0.9894625685, 0.8009750334], [0.9894625685, 0.8009750334], [0.9894625685, 0.8009750334], [0.9894625685, 0.8009750334], [0.9894625685, 0.8009750334]] > #TRANSFORMATION 12 > RT2(x,y,1,100) [55 + 97 y + 98 x 25 + 73 y + 62 x] [----------------, ----------------] [91 + 48 y + 23 x 86 + 23 y + 61 x] (48)> evalf(FP2drz(,x,y)) [[1.825495687, 0.9457474792], [-3.799190246, 1.086668009], [-0.2899112830, -0.4601466273], [2.118273701, -7.135782644]] (48)> SFP2drz(,x,y) [[1.825495687, 0.9457474792]] (48)> Orb2(,x,y,[0.5,-4],1010,1020) [[1.825495690, 0.9457474794], [1.825495690, 0.9457474794], [1.825495690, 0.9457474794], [1.825495690, 0.9457474794], [1.825495690, 0.9457474794], [1.825495690, 0.9457474794], [1.825495690, 0.9457474794], [1.825495690, 0.9457474794], [1.825495690, 0.9457474794], [1.825495690, 0.9457474794]] > #TRANSFORMATION 13 > RT2(x,y,1,100) [16 + 99 y + 51 x 20 + 83 y + 10 x] [----------------, ----------------] [47 + 33 y + 77 x 40 + 16 y + 19 x] (52)> evalf(FP2drz(,x,y)) [[1.332686951, 2.097830922], [-6.203594804, 9.785793243], [-0.09606908705 - 0.5587881086 I, -0.3512129087 + 0.1661793403 I], [-0.09606908705 + 0.5587881086 I, -0.3512129087 - 0.1661793403 I]] (52)> SFP2drz(,x,y) [[1.332686951, 2.097830922]] (52)> Orb2(,x,y,[0.5,-4],1010,1020) [[1.332686948, 2.097830922], [1.332686949, 2.097830922], [1.332686948, 2.097830922], [1.332686949, 2.097830922], [1.332686948, 2.097830922], [1.332686949, 2.097830922], [1.332686948, 2.097830922], [1.332686949, 2.097830922], [1.332686948, 2.097830922], [1.332686949, 2.097830922]] > #TRANSFORMATION 14 > RT2(x,y,1,100) [40 + 40 y + 78 x 5 + 78 y + 29 x ] [----------------, ----------------] [91 + 71 y + 75 x 23 + 81 y + 21 x] (56)> evalf(FP2drz(,x,y)) [[0.6181237365, 0.8512484601], [-1.152203890 - 0.3704522058 I, 0.3693120525 + 0.4063487329 I], [0.6733317472, -0.3530989241], [-1.152203890 + 0.3704522058 I, 0.3693120525 - 0.4063487329 I]] (56)> SFP2drz(,x,y) [[0.6181237365, 0.8512484601]] (56)> Orb2(,x,y,[0.5,-4],1010,1020) [[0.6181237358, 0.8512484603], [0.6181237358, 0.8512484603], [0.6181237358, 0.8512484603], [0.6181237358, 0.8512484603], [0.6181237358, 0.8512484603], [0.6181237358, 0.8512484603], [0.6181237358, 0.8512484603], [0.6181237358, 0.8512484603], [0.6181237358, 0.8512484603], [0.6181237358, 0.8512484603]] > #TRANSFORMATION 15 > RT2(x,y,1,100) [6 + 36 y + 90 x 91 + 44 y + 44 x] [----------------, ----------------] [42 + 69 y + 73 x 25 + 37 y + 37 x] (60)> evalf(FP2drz(,x,y)) [[0.5945181001, 1.739411860], [-2.723288820, 2.974932100], [0.4041499918, -1.660853245], [15.91353221, -16.68186909]] (60)> SFP2drz(,x,y) [[0.5945181001, 1.739411860]] (60)> Orb2(,x,y,[0.5,-4],1010,1020) [[0.5945180994, 1.739411860], [0.5945180994, 1.739411860], [0.5945180994, 1.739411860], [0.5945180994, 1.739411860], [0.5945180994, 1.739411860], [0.5945180994, 1.739411860], [0.5945180994, 1.739411860], [0.5945180994, 1.739411860], [0.5945180994, 1.739411860], [0.5945180994, 1.739411860]] > #TRANSFORMATION 16 > RT2(x,y,1,100) [66 + 100 y + 61 x 99 + 87 y + 81 x] [-----------------, ----------------] [44 + 45 y + 75 x 52 + 91 y + 41 x] (64)> evalf(FP2drz(,x,y)) [[-1.480555423, 0.7416220466], [1.379967985, 1.407965414], [-0.03125246657 - 0.6886163714 I, -0.8694084531 + 0.4129486768 I], [-0.03125246657 + 0.6886163714 I, -0.8694084531 - 0.4129486768 I]] (64)> SFP2drz(,x,y) [[1.379967985, 1.407965414]] (64)> Orb2(,x,y,[0.5,-4],1010,1020) [[1.379967987, 1.407965414], [1.379967987, 1.407965414], [1.379967987, 1.407965414], [1.379967987, 1.407965414], [1.379967987, 1.407965414], [1.379967987, 1.407965414], [1.379967987, 1.407965414], [1.379967987, 1.407965414], [1.379967987, 1.407965414], [1.379967987, 1.407965414]] > #TRANSFORMATION 17 > RT2(x,y,1,100) [85 + 48 y + 75 x 7 + 75 y + 64 x ] [----------------, ----------------] [74 + 93 y + 92 x 25 + 22 y + 33 x] (68)> evalf(FP2drz(,x,y)) [[-2.266086345, 1.506108672], [0.7057124533, 2.262285861], [0.8790687844 - 1.217795585 I, -1.221996310 + 1.699438577 I], [0.8790687844 + 1.217795585 I, -1.221996310 - 1.699438577 I]] (68)> SFP2drz(,x,y) [[0.7057124533, 2.262285861]] (68)> Orb2(,x,y,[0.5,-4],1010,1020) [[0.7057124540, 2.262285860], [0.7057124542, 2.262285861], [0.7057124540, 2.262285862], [0.7057124538, 2.262285862], [0.7057124538, 2.262285860], [0.7057124540, 2.262285860], [0.7057124542, 2.262285861], [0.7057124540, 2.262285862], [0.7057124538, 2.262285862], [0.7057124538, 2.262285860]] > #TRANSFORMATION 18 > RT2(x,y,1,100) [60 + 85 y + 95 x 58 + 27 y + 41 x] [----------------, ----------------] [64 + 99 y + 3 x 82 + 27 y + 98 x] (72)> evalf(FP2drz(,x,y)) [[-6.396054757, 0.3634124520], [3.605031680, 0.4882976530], [-1.287892503, -0.07105522177], [0.7027608654, -5.205470198]] (72)> SFP2drz(,x,y) [[3.605031680, 0.4882976530]] (72)> Orb2(,x,y,[0.5,-4],1010,1020) [[3.605031676, 0.4882976531], [3.605031676, 0.4882976531], [3.605031676, 0.4882976531], [3.605031676, 0.4882976531], [3.605031676, 0.4882976531], [3.605031676, 0.4882976531], [3.605031676, 0.4882976531], [3.605031676, 0.4882976531], [3.605031676, 0.4882976531], [3.605031676, 0.4882976531]] > #TRANSFORMATION 19 > RT2(x,y,1,100) [51 + 58 y + 77 x 45 + 99 y + 17 x] [----------------, ----------------] [ 3 + 47 y + 8 x 74 + 7 y + 6 x ] (76)> evalf(FP2drz(,x,y)) [[2.051107240, 4.404078698], [-4.680796469 - 6.745042264 I, 1.957656129 + 1.380147996 I], [0.2369237471, -1.452766430], [-4.680796469 + 6.745042264 I, 1.957656129 - 1.380147996 I]] (76)> SFP2drz(,x,y) [[2.051107240, 4.404078698]] (76)> Orb2(,x,y,[0.5,-4],1010,1020) [[2.051107246, 4.404078693], [2.051107246, 4.404078696], [2.051107246, 4.404078695], [2.051107245, 4.404078694], [2.051107246, 4.404078693], [2.051107246, 4.404078696], [2.051107246, 4.404078695], [2.051107245, 4.404078694], [2.051107246, 4.404078693], [2.051107246, 4.404078696]] > #TRANSFORMATION 20 > RT2(x,y,1,100) [86 + 8 y + 10 x 74 + 54 y + 15 x] [----------------, ----------------] [68 + 43 y + 65 x 72 + 10 y + 73 x] (80)> evalf(FP2drz(,x,y)) [[0.6441054109, 1.100458684], [-0.07708340766 - 0.6200877468 I, -2.497492652 + 3.256088063 I], [-1.396988075, -0.5901660285], [-0.07708340766 + 0.6200877468 I, -2.497492652 - 3.256088063 I]] (80)> SFP2drz(,x,y) [[0.6441054109, 1.100458684]] (80)> Orb2(,x,y,[0.5,-4],1010,1020) [[0.6441054113, 1.100458684], [0.6441054113, 1.100458684], [0.6441054113, 1.100458684], [0.6441054113, 1.100458684], [0.6441054113, 1.100458684], [0.6441054113, 1.100458684], [0.6441054113, 1.100458684], [0.6441054113, 1.100458684], [0.6441054113, 1.100458684], [0.6441054113, 1.100458684]] > > #Question 4, generate new analog versions of the 2d functions RT, Orb, FP, and > SFP for a 3d function. > RT3 := proc(x,y,z,d,K) local ra,i,j,k,f,g,h: ra:=rand(1..K): #random integer > from -K to K > f:=add(add(add(ra()*x^i*y^j*z^k,k=0..d-i),j=0..d-i),i=0..d)/add(add(add(ra()*x > ^i*y^j*z^k,k=0..d-i),j=0..d-i),i=0..d): > g:=add(add(add(ra()*x^i*y^j*z^k,k=0..d-i),j=0..d-i),i=0..d)/add(add(add(ra()*x > ^i*y^j*z^k,k=0..d-i),j=0..d-i),i=0..d): > h:=add(add(add(ra()*x^i*y^j*z^k,k=0..d-i),j=0..d-i),i=0..d)/add(add(add(ra()*x > ^i*y^j*z^k,k=0..d-i),j=0..d-i),i=0..d): [f,g,h]: end: > > Orb3:=proc(F,x,y,z,pt0,K1,K2) local pt,L,i: pt:=pt0: for i from 1 to K1 do > pt:=subs({x=pt[1],y=pt[2],z=pt[3]},F): od: L:=[]: for i from K1+1 to K2 do > L:=[op(L),pt]: pt:=subs({x=pt[1],y=pt[2],z=pt[3]},F): od: L: end: > FP3:=proc(F,x,y,z) local L,i: L:=[solve({F[1]=x,F[2]=y,F[3]=z},{x,y,z})]: > [seq(subs(L[i],[x,y,z]),i=1..nops(L))]: end: > SFP3:=proc(F,x,y,z) local L,J,S,J0,i,pt,EV: L:=evalf(FP3(F,x,y,z)): > J:=Matrix(normal([[diff(F[1],x),diff(F[2],x),diff(F[3],x)],[diff(F[1],y),diff( > F[2],y),diff(F[3],y)],[diff(F[1],z),diff(F[2],z),diff(F[3],z)]])): S:=[]: for i > from 1 to nops(L) do pt:=L[i]: J0:=subs({x=pt[1],y=pt[2],z=pt[3]},J): > EV:=Eigenvalues(J0): if abs(EV[1])<1 and abs(EV[2])<1 and abs(EV[3])<1 then > S:=[op(S),pt]: fi: od: S: end: > #Transformation 1. RT3(x,y,z,1,100) [18 y z + 7 x + 20 y + 14 z + 33 73 y z + 56 x + 66 y + 68 z + 54 [-------------------------------, --------------------------------, [96 y z + 42 x + 31 y + 8 z + 5 54 y z + 44 x + 58 y + 69 z + 82 90 y z + 18 x + 37 y + 49 z + 32 ] ---------------------------------] 29 y z + 49 x + 100 y + 38 z + 74] (84)> evalf(FP3(,x,y,z)) [[0.592309683, 0.9997237413, 0.712674977]] (84)> SFP3(,x,y,z) [[0.592309683, 0.9997237413, 0.712674977]] (84)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1010) [[0.5923096768, 0.9997237412, 0.7126749748], [0.5923096768, 0.9997237412, 0.7126749748], [0.5923096768, 0.9997237412, 0.7126749748], [0.5923096768, 0.9997237412, 0.7126749748], [0.5923096768, 0.9997237412, 0.7126749748], [0.5923096768, 0.9997237412, 0.7126749748], [0.5923096768, 0.9997237412, 0.7126749748], [0.5923096768, 0.9997237412, 0.7126749748], [0.5923096768, 0.9997237412, 0.7126749748], [0.5923096768, 0.9997237412, 0.7126749748]] > #Transformation 2. RT3(x,y,z,1,100) [6 y z + 54 x + 16 y + 79 z + 70 60 y z + 95 x + 16 y + 38 z + 13 [--------------------------------, --------------------------------, [11 y z + 30 x + 23 y + 30 z + 51 40 y z + 2 x + 77 y + 16 z + 80 75 y z + 96 x + 100 y + 85 z + 46] ---------------------------------] 64 y z + 35 x + 11 y + 10 z + 6 ] (88)> evalf(FP3(,x,y,z)) [[0.330997942, 0.06325486146, -0.985805181]] (88)> SFP3(,x,y,z) [] (88)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1010) [[1.636339381, 1.347612870, 2.516654275], [1.636339382, 1.347612871, 2.516654274], [1.636339381, 1.347612870, 2.516654275], [1.636339382, 1.347612871, 2.516654274], [1.636339381, 1.347612870, 2.516654275], [1.636339382, 1.347612871, 2.516654274], [1.636339381, 1.347612870, 2.516654275], [1.636339382, 1.347612871, 2.516654274], [1.636339381, 1.347612870, 2.516654275], [1.636339382, 1.347612871, 2.516654274]] > #Transformation 3. RT3(x,y,z,1,100) [45 y z + 78 x + 76 y + 94 z + 88 25 y z + 20 x + 80 y + 67 z + 47 [--------------------------------, --------------------------------, [34 y z + 20 x + 39 y + 77 z + 1 58 y z + 97 x + 82 y + 83 z + 32 73 y z + 11 x + 59 y + 46 z + 77] --------------------------------] 8 y z + 83 x + 76 y + 24 z + 1 ] (92)> evalf(FP3(,x,y,z)) [[3.248376884, 0.429665326, 0.5715650679]] (92)> SFP3(,x,y,z) [[3.248376884, 0.429665326, 0.5715650679]] (92)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1010) [[3.248376882, 0.4296653258, 0.5715650680], [3.248376882, 0.4296653258, 0.5715650680], [3.248376882, 0.4296653258, 0.5715650680], [3.248376882, 0.4296653258, 0.5715650680], [3.248376882, 0.4296653258, 0.5715650680], [3.248376882, 0.4296653258, 0.5715650680], [3.248376882, 0.4296653258, 0.5715650680], [3.248376882, 0.4296653258, 0.5715650680], [3.248376882, 0.4296653258, 0.5715650680], [3.248376882, 0.4296653258, 0.5715650680]] > > #Transformation 4. RT3(x,y,z,1,100) [68 y z + 62 x + 19 y + 92 z + 38 9 y z + 61 x + 96 y + 26 z + 76 [--------------------------------, -------------------------------, [21 y z + 6 x + 48 y + 98 z + 53 96 y z + 44 x + 72 y + 5 z + 34 27 y z + 6 x + 48 y + 19 z + 29 ] --------------------------------] 56 y z + 36 x + 12 y + 88 z + 52] (96)> evalf(FP3(,x,y,z)) [[1.271361259, 1.174237246, 0.603840190]] (96)> SFP3(,x,y,z) [[1.271361259, 1.174237246, 0.603840190]] (96)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1010) [[1.271361280, 1.174237246, 0.6038401885], [1.271361279, 1.174237246, 0.6038401882], [1.271361280, 1.174237246, 0.6038401885], [1.271361279, 1.174237246, 0.6038401882], [1.271361280, 1.174237246, 0.6038401885], [1.271361279, 1.174237246, 0.6038401882], [1.271361280, 1.174237246, 0.6038401885], [1.271361279, 1.174237246, 0.6038401882], [1.271361280, 1.174237246, 0.6038401885], [1.271361279, 1.174237246, 0.6038401882]] > #Transformation 5. RT3(x,y,z,1,100) [40 y z + 55 x + 52 y + 84 z + 93 84 y z + 80 x + 84 y + 84 z + 53 [--------------------------------, --------------------------------, [52 y z + 18 x + 30 y + 16 z + 95 41 y z + 51 x + 56 y + 88 z + 33 14 y z + 84 x + 95 y + 29 z + 13] --------------------------------] 82 y z + 4 x + 57 y + 92 z + 65 ] (100)> evalf(FP3(,x,y,z)) [[1.527395154, 1.485947452, 0.927862900]] (100)> SFP3(,x,y,z) [[1.527395154, 1.485947452, 0.927862900]] (100)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1020) [[1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976], [1.527395171, 1.485947452, 0.9278628976]] > #Transformation 6. RT3(x,y,z,1,100) [99 y z + 60 x + 37 y + 45 z + 81 86 y z + 21 x + 56 y + 15 z + 49 [--------------------------------, --------------------------------, [97 y z + 14 x + 64 y + 43 z + 18 5 y z + 21 x + 83 y + 25 z + 46 16 y z + 29 x + 72 y + 23 z + 91] --------------------------------] 38 y z + 26 x + 10 y + 10 z + 99] (104)> evalf(FP3(,x,y,z)) [[-6.487859, 0.9604266, 0.3385077677]] (104)> SFP3(,x,y,z) [] (104)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1020) [[1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756], [1.247063373, 1.424673353, 1.267912756]] > #Transformation 7. RT3(x,y,z,1,100) [98 y z + 5 x + 77 y + 31 z + 66 y z + 42 x + 7 y + 24 z + 99 [--------------------------------, --------------------------------, [59 y z + 76 x + 66 y + 71 z + 61 91 y z + 69 x + 45 y + 25 z + 89 39 y z + 74 x + 36 y + 6 z + 92 ] --------------------------------] 41 y z + 75 x + 47 y + 97 z + 41] (108)> evalf(FP3(,x,y,z)) [[0.781372985, 0.646600986, 0.8496870289]] (108)> SFP3(,x,y,z) [[0.781372985, 0.646600986, 0.8496870289]] (108)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1020) [[0.7813729848, 0.6466009781, 0.8496870289], [0.7813729854, 0.6466009787, 0.8496870289], [0.7813729852, 0.6466009781, 0.8496870289], [0.7813729851, 0.6466009784, 0.8496870289], [0.7813729848, 0.6466009781, 0.8496870289], [0.7813729854, 0.6466009787, 0.8496870289], [0.7813729852, 0.6466009781, 0.8496870289], [0.7813729851, 0.6466009784, 0.8496870289], [0.7813729848, 0.6466009781, 0.8496870289], [0.7813729854, 0.6466009787, 0.8496870289], [0.7813729852, 0.6466009781, 0.8496870289], [0.7813729851, 0.6466009784, 0.8496870289], [0.7813729848, 0.6466009781, 0.8496870289], [0.7813729854, 0.6466009787, 0.8496870289], [0.7813729852, 0.6466009781, 0.8496870289], [0.7813729851, 0.6466009784, 0.8496870289], [0.7813729848, 0.6466009781, 0.8496870289], [0.7813729854, 0.6466009787, 0.8496870289], [0.7813729852, 0.6466009781, 0.8496870289], [0.7813729851, 0.6466009784, 0.8496870289]] > #Transformation 8. RT3(x,y,z,1,100) [89 y z + 82 x + 46 y + 42 z + 69 55 y z + 23 x + 87 y + 57 z + 34 [--------------------------------, --------------------------------, [14 y z + 81 x + 31 y + 33 z + 52 58 y z + 87 x + 85 y + 18 z + 74 68 y z + 14 x + 93 y + 71 z + 40] --------------------------------] 49 y z + 20 x + 18 y + 99 z + 54] (112)> evalf(FP3(,x,y,z)) [[1.050403739, 0.1205863223, -0.7831152000]] (112)> SFP3(,x,y,z) [] (112)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1020) [[1.411961648, 0.7220601864, 1.068745167], [1.411961648, 0.7220601862, 1.068745167], [1.411961647, 0.7220601864, 1.068745167], [1.411961648, 0.7220601864, 1.068745167], [1.411961648, 0.7220601862, 1.068745167], [1.411961647, 0.7220601864, 1.068745167], [1.411961648, 0.7220601864, 1.068745167], [1.411961648, 0.7220601862, 1.068745167], [1.411961647, 0.7220601864, 1.068745167], [1.411961648, 0.7220601864, 1.068745167], [1.411961648, 0.7220601862, 1.068745167], [1.411961647, 0.7220601864, 1.068745167], [1.411961648, 0.7220601864, 1.068745167], [1.411961648, 0.7220601862, 1.068745167], [1.411961647, 0.7220601864, 1.068745167], [1.411961648, 0.7220601864, 1.068745167], [1.411961648, 0.7220601862, 1.068745167], [1.411961647, 0.7220601864, 1.068745167], [1.411961648, 0.7220601864, 1.068745167], [1.411961648, 0.7220601862, 1.068745167]] > #Transformation 9. RT3(x,y,z,1,100) [89 y z + 42 x + 76 y + 64 z + 11 6 y z + 57 x + 37 y + 18 z + 39 [--------------------------------, --------------------------------, [93 y z + 26 x + 84 y + 66 z + 81 47 y z + 93 x + 15 y + 82 z + 81 78 y z + 13 x + 48 y + 49 z + 97] --------------------------------] 42 y z + 3 x + 3 y + 75 z + 39 ] (116)> evalf(FP3(,x,y,z)) [[0.76654758, 0.4216402075, 1.402481495]] (116)> SFP3(,x,y,z) [[0.76654758, 0.4216402075, 1.402481495]] (116)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1020) [[0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495], [0.7665475869, 0.4216402075, 1.402481495]] > #Transformation 10. RT3(x,y,z,1,100) [75 y z + 80 x + 79 y + 81 z + 53 86 y z + 26 x + 27 y + 52 z + 74 [--------------------------------, --------------------------------, [27 y z + 87 x + 86 y + 33 z + 4 34 y z + 84 x + 6 y + 63 z + 96 89 y z + 49 x + 80 y + 5 z + 62 ] --------------------------------] 13 y z + 59 x + 29 y + 41 z + 26] (120)> evalf(FP3(,x,y,z)) [[1.558663435, 0.8923224478, 1.510456690]] (120)> SFP3(,x,y,z) [[1.558663435, 0.8923224478, 1.510456690]] (120)> Orb3(,x,y,z,[0.5,-4.0,1],1000,1020) [[1.558663446, 0.8923224479, 1.510456690], [1.558663445, 0.8923224476, 1.510456690], [1.558663445, 0.8923224479, 1.510456690], [1.558663446, 0.8923224479, 1.510456690], [1.558663445, 0.8923224476, 1.510456690], [1.558663445, 0.8923224479, 1.510456690], [1.558663446, 0.8923224479, 1.510456690], [1.558663445, 0.8923224476, 1.510456690], [1.558663445, 0.8923224479, 1.510456690], [1.558663446, 0.8923224479, 1.510456690], [1.558663445, 0.8923224476, 1.510456690], [1.558663445, 0.8923224479, 1.510456690], [1.558663446, 0.8923224479, 1.510456690], [1.558663445, 0.8923224476, 1.510456690], [1.558663445, 0.8923224479, 1.510456690], [1.558663446, 0.8923224479, 1.510456690], [1.558663445, 0.8923224476, 1.510456690], [1.558663445, 0.8923224479, 1.510456690], [1.558663446, 0.8923224479, 1.510456690], [1.558663445, 0.8923224476, 1.510456690]] > > #Question 6, Brownie Points: check file 3 and page 81, NB(lambda, a, c) > #F(N,P) = lambda**N*e^(-a*P) & G(N,P) = c*N*(1-e^(-a*P)) > solve(x=lambda*x*exp(1)^(-a*y),y) ln(lambda) ---------- a > #Solving F(N,P) for P when setting F(N,P) = N shows that P is equal to > ln(lambda) / a. > solve(ln(lambda)/a = c*x*(1 - exp(-ln(lambda))), x) ln(lambda) lambda ----------------- a c (lambda - 1) > #Then plugging in P to the second equation where P = G(N,P) and solving for N > gives the resultant in Line125. #This gives us the only potential steady > state. NB := proc(y, a, c) local Nl, N2, L: Nl := evalf(y*ln(y)/(a*c*(y-1))): > N2 := evalf(ln(y) / a): L := [Nl, N2]: > end: