CHAPTER 6, Chaos.

6.1 Some Chaotic Maps.

Consider the piecewise linear map defined below. It has one fixed point and we notice that {0,1/2,1} is a 3-cycle.

f[x_]:=Piecewise[{{x+1/2,0<x<1/2},{2-2x,1/2<x<1}}]

Plot[{x,f[x]},{x,0,1},AspectRatio->Automatic]

Plot[{x,f[f[x]]},{x,0,1},AspectRatio->Automatic]

Plot[{x,f[f[f[x]]]},{x,0,1},AspectRatio->Automatic]

6.2 Sensitive Dependence on Initial Conditions.

Consider what happens when we iterate the full tent map nine times for two points within .001 of each other. Due to the sensitive dependence on initial conditions, we see that (T^9)(x) can be widely different for each of the randomly chosen points.

T[x_]:=Piecewise[{{2x,0<x<1/2},{2x-1,1/2<x<1}}];

n=9;
a=Random[Real,{0,1}];
A=NestList[T,a,n];
B=NestList[T,a+.001,n];
{A[[n]],B[[n]]}

6.3 Probability Distributions.

If we iterate a randomly chosen point and count how often the orbit enters each small interval, we obtain what we call a probability distribution:

Clear[f];
f[x_]:=4x(1-x);

n=50;
R=100000;
s=Table[0,{i,1,R}];
For[t=1,t≤R,t++,
s[[t]]=NestList[f,Random[Real,{0,1}],n][[n]];]

prob=(1/R)*BinCounts[s,{0,1,0.01}];
probUnitinterval=Table[{i/100,prob[[i]]*100},{i,1,100}];
pdf=ListPlot[probUnitinterval,Axes->False,
Frame->True,PlotJoined->True,PlotRange->All]

6.4 Exercises.

1.    Consider sensitivity of initial conditions for the logistic map (Subscript[L, 4])(x)=4x(1-x), and also for the map f(x) defined in 6.1 above.

2.    Check to see whether the irrational rotation (Subscript[T, α(x)])=x+α mod 1, has sensitive dependence, for α=1-π/5.

3.    Consider the probability distribution for the map f(x) defined in 6.1 above, and for the function f(x)=sin(π x), 0≤x≤1.

4.    Consider the probability distributions of f(x)=r x(1-x), 0≤x≤1, for various values of 3<r<4. Can you explain what you are seeing?