Help9:=proc(): print(` Orb(f,x,x0,K1,K2), Orb2D(f,x,x0,K) , FP(f,x) , SFP(f,x) , Comp(f,x) `):end:
#Orb(f,x,x0,K1,K2): Inputs an expression f in x (desccribing) a function of x, an initial point, x0, and a positive integer K, outputs
#the values of x[n] from n=K1 to n=K2. Try: where x[n]=f(x[n-1]), . Try:
#Orb(2*x*(1-x),x,0.4,1000,2000);
Orb:=proc(f,x,x0,K1,K2) local x1,i,L:
x1:=x0:
for i from 1 to K1 do
x1:=subs(x=x1,f): #we don't record the first values of K1, since we are interested in the long-time behavior of the orbit
od:
L:=[x1]:
for i from K1 to K2 do
x1:=subs(x=x1,f): #we compute the next member of the orbit
L:=[op(L),x1]: #we append it to the list
od:
L: #that's the output
end:
#Orb2D(f,x,x0,K): 2D version of Orb(f,x,x0,0,K), just for illustration
Orb2D:=proc(f,x,x0,K) local L,L1,i:
L:=Orb(f,x,x0,0,K):
L1:=[[L[1],0],[L[1],L[2]], [L[2],L[2]] ] :
for i from 3 to nops(L) do
L1:=[op(L1),[L[i-1],L[i]],[L[i],L[i]]]:
od:
L1:
end:
#FP(f,x): The list of fixed points of the map x->f where f is an expression in x. Try:
#FP(2*x*(1-x),x);
FP:=proc(f,x)
evalf([solve(f=x)]):
end:
#SFP(f,x): The list of stable fixed points of the map x->f where f is an expression in x. Try:
#SFP(2*x*(1-x),x);
SFP:=proc(f,x) local L,i,f1,pt,Ls:
L:=FP(f,x): #The list of fixed points (including complex ones)
Ls:=[]: #Ls is the list of stable fixed points, that starts out as the empty list
f1:=diff(f,x): #The derivative of the function f w.r.t. x
for i from 1 to nops(L) do
pt:=L[i]:
if abs(subs(x=pt,f1))<1 then
Ls:=[op(Ls),pt]: # if pt, is stable we add it to the list of stable points
fi:
od:
Ls: #The last line is the output
end:
#Comp(f,x): f(f(x))
Comp:=proc(f,x): normal(subs(x=f,f)):end:
#USING
#(i): when f(x):= 2*x*(1-x)
#Using Orb() to find the stable fixed points
print("orb");
Orb(2*x*(1-x),x,0.5,1000,2000);
#It appears that Orb()
#Using SFP() to find the stable fixed points
print("SFP");
SFP(2*x*(1-x),x);
#OK so SFP gives a unique#(ii): when f(x) := 2.5*x*(1-x)
#Using Orb() to find the stable fixed points
print("Orb");
Orb((2.5)*x*(1-x),x,0.5,1000,2000);
#Using SFP() to find the stable fixed points
print("SFP");
SFP((2.5)*x*(1-x),x);#(iii): when f(x) := (3.1)*x*(1-x)
#Using Orb() to find the stable fixed points
print("orb");
Orb((3.1)*x*(1-x),x,0.5,1000,2000);
#with the constant of the Malthus function being 3.1 , now there are orbits between 2 fixed points
#Using SFP() to find the stable fixed points
print("SFP");
SFP((3.1)*x*(1-x),x);
#Notice how we saw no stable fixed points. A function that has only ONE fixed point has a stable fixed point. A function may have a fixed orbit that cycles through the same points OVER AND OVER and OVER again, but none of the points in the orbit are stable fixed points
#(iv): when f(x) := (4+x) / (3+x)
#Using Orb() to find the stable fixed points
#Maybe see if maple sees 2 stable fixed points with 0nly 10 or 9 digits
Digits:=9;
#Huh it appeared to oscillate between 7 and 9 instead of staying at 8 something happened glitchy. I think i saw a
print(Orb((4+x)/ (3+x),x,1.5,1000,2000));
#interesting, only the last digit alternates between 7 and 8 change the digits to see how the digits after that change
Digits:=20;
#maybe see if maple sees only one stable fixed point
Orb((4+x)/ (3+x),x,1.5,1000,2000);
#Using SFP() to find the stable fixed points
print(SFP((4+x)/ (3+x),x));#(v): when f(x)
#Using Orb() to find the stable fixed points
#Using SFP() to find the stable fixed points#(vi) when f(x) := (3 + x + x^2) / (4 + x + 2*x^2)
#Using Orb() to find the stable fixed points
Orb(
#Using SFP() to find the stable fixed points#2. Consider the discrete dynamical system: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#To find the orbits using the ORB and SFP procedure, we can create lists of known values of a and b and #We are creating lists to iterate through, because Dr. Z's spf command does not allow
a_list := seq(1..10);
b_list := seq(1..10);a_list
#Now the looping through discrete values of a and b
#for i in a_list do print(SFP(3-x,i*x));
#od;
#We are getting some sort of error. we cant have a product of an integer and a variable for a 2nd argument
#So i need to do more than not leaving something just as a symbol
#What is wrong with the diff() command
#This is the error: diff(x,i*x) is not radable for the computer because of i*x being a symbol.
#What happens if we put "solve" on i*x to get an actual value of something
#diff(x,solve(2*x));
#Nope, nothing good happened :(
#What about
#diff(x,solve(3*5));
#diff((x,2x),x);
#I figured out exactly whats wrong. the 2nd paramater of diff is the differential, and regular maple needs a single variable like x as the differential. No other fancy symbols as the differential?
#Even though I do not know what i am talking about, if we keep an x*i in the first paramater, and let our second paramater be just x this code should be runnable
for i in a_list do print(i); print(SFP((0.3)*i*x*(1-x),x));
#I was just putting the three on there to work with entries closer to each other in 1
od;OrbOrb(2*x*(1-x),x,1,1000,1020);
#Having an x as paramater 2 will give us discrete values in our outputdiff(x,i*x)#To find the answer to -1<C(a,b)<1
#It might be better to just experiment and play around with the values of a and b to make 3 cases.
#C(a,b) < -1
#-1 < C(a,b) <1
#C(a,b) > 1
#Just to be clear, C(a,b) is what the function?for a in a_list do;
for b in b_list do;
print("a=" + a + "b=" + b);
print(SFP((x+a)/(x+b),x));
end do:
end do:
#3. for an arbitrary value of k between 1 and 4, find equilibrium points of x->kx(1-x),
eqns := {1 <= k <= 4};
#I think the error occured because this calling of a set was in text format.
#Arent the equilibrium points the same thing as orbits?
#SFP(solve(kx
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#I think eqns is only symbolic, which me