Help with any/all of the problems is appreciated! thanks! Im using \'y\' to repr

Question

Help with any/all of the problems is appreciated! thanks!

Im using 'y' to represent lamda

1. Show that if f has a stable fixed point x*, then y(f,x0) < 0.

2. Show that if f has a stable period-k point x0 (or equivalently has a stable k-cycle which contains x0), then y(f,x0)<0.

3. Calculate analytically the Lyapunov exponent of the tent map

T(x)= ax if 0<=x<=1/2

a(1-x) if 1/2<x<=1

interpret your results in terms of the values of a (0<=a<=2) for which the tent map is chaotic.

4. Compute numerically the Lyapunov exponent y for the logistic map f(x)=rx(1-x), using the following approach. Fix some value of r. Then starting from a random initial condition, iterate the map for, say 1000 iterates to allow transients to decay.

Explanation / Answer

I think that the following links should be helpful http://www.cds.caltech.edu/~murray/wiki/images/4/43/Cds140a-wi11-Week5Notes.pdf http://www.cmp.caltech.edu/~mcc/Chaos_Course/Lesson4/Maps1D.pdf http://www.math.harvard.edu/archive/118r_spring_05/handouts/lyapunov.pdf http://math.cityu.edu.hk/~mayylu/ma4528/notes.pdf

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