Fixed Point Iteration Ideas....

Andy Long, Spring, 2024

Here’s our author’s picture of two scenarios:  on the left, convergence; on the right, failure to converge. (He has a link to a cobwebbing example, too.)

Danny suggested a variation on my fixed point function -- scaling it by a parameter. That can be beneficial (but not at the tricky endpoint...).

f[x_,a_,b_]:=a(-Sin[x]+b)+x
fp[x_,a_]:=-a Cos[x]+1
Manipulate[
Module[{},
Plot[{x,f[x,a,b]},{x,0,Pi/2},PlotLabel->fp[Pi/2,a]]
],
{a,0.01,2},
{b,0.01,1}
]

Let’s figure out where the first bifurcation point occurs in the chaotic logistic system:

Clear[a,r]
fchaos[x_,a_]:=a* x (1-x)
D[fchaos[x,a],x]
solns=Solve[D[fchaos[x,a],x]==-1 && fchaos[x,a]==x, {x,a}]
bifurcation=solns[[1]];
rb=x/.bifurcation[[1]];
ab=a/.bifurcation[[2]];
Plot[{fchaos[x,ab],x},{x,0,1},AspectRatio->1.0]

Clear[x,a,r]
f2chaos[x_,a_]:=fchaos[fchaos[x,a],a]
D[f2chaos[x,a],x]
solns=Solve[D[f2chaos[x,a],x]==1 && f2chaos[x,a]==x, {x,a}]
bifurcation=solns[[3]];
rb=x/.bifurcation[[1]];
ab=a/.bifurcation[[2]];
Plot[{f2chaos[x,ab],x},{x,0,1},AspectRatio->1.0]

Manipulate[
Plot[{x,f2chaos[x,a]},{x,0,1},AspectRatio->1.0,PlotLabel->N[a]],
{{a,3},0.01,3.99}
]

Clear[x,a,r]
f2chaos[x_,a_]:=fchaos[fchaos[x,a],a]
D[f2chaos[x,a],x]
solns=Solve[D[f2chaos[x,a],x]==-1 && f2chaos[x,a]==x, {x,a}]
bifurcation=solns[[8]]
rb=x/.bifurcation[[1]]
ab=a/.bifurcation[[2]]
Plot[{f2chaos[x,ab],x},{x,0,1},AspectRatio->1.0,PlotLabel->N[ab]]