#M9.txt: Maple Code for "Dynamical models in Biology" (Math 336) taught by Dr. Z., Lecture 9

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: