(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the families of iterating functions F_{λ}(x) = λ(x^{3}- x). Fλ(x) undergoes a bifurcation at λ=1, about the fixed point x=0. Figure out what ilk of bifurcation is occurring for F_{λ}(x) and prove your assertion rigorously.

2. Relevant equations

My book says this about period-doubling bifurcations, and I need to prove all four of these to prove the problem.

Definition: A one-parameter family of functions F_{λ}undergoes a period-doubling bifurcation at the parameter value λ=λ_{0}if there is an open interval and an ε such that:

1. For each λ in the interval [λ_{0}- ε, λ_{0}+ ε], there is a unique fixed point p_{λ}for F_{λ}in I.

2.For λ_{0}- ε < λ < λ_{0}, F_{λ}has no cycles of period 2 in I and p_{λ}is attracting (resp. repelling).

3.For λ_{0}< λ < λ_{0}+ ε, there is a unique 2-cycle q^{1}_{λ},q^{2}_{λ}in I with F_{λ}(q^{1}_{λ})=q^{2}_{λ}. This 2-cycle is attracting (resp. repelling). Meanwhile, the fixed point p_{λ}is repelling (resp. attracting).

4.As λ -> λ_{0}, we have q^{i}_{λ}-> p_{λ0}

Also, these theorems are necessary (I think).

Chain Rule Along A Cycle: Suppose x_{0}, x_{2}, ..., x_{n-1}lie on a cycle of period n for F with x_{i}= F^{i}(x_{0}). Then

(F^{n})'(x_{0}) = F'(x_{n-1}) * ... * F'(x_{1}) * F'(x_{0}).

The corollary for this is:

Suppose x_{0}, x_{1}, ..., x_{n-1}lie on an n-cycle for F. Then

(F^{n})'(x_{0}) = (F^{n})'(x_{1}) = ... = (F^{n})'(x_{n-1})

3. The attempt at a solution\

I have almost no idea where to start the proof. So far I have:

The interval I can = (-1, 1).

To answer 1., I know that there are fixed points at 0, +sqrt(2), -sqrt(2). So to have a unique fixed point p_{λ}in the interval [λ_{0}- ε, λ_{0}+ ε], 0<ε<sqrt(2). Can I just choose an arbitrary ε, like ε=1?

To answer 2., if ε=1, λ_{0}- ε < λ < λ_{0}, so 0 - 1 < λ < 0, so -1<λ<0. I used algebra and got 2-cycles for x = -1, -.7548777, 0, 1 1.4655712, but all but x=1.4655712 lie in the interval I. Could someone check this? I may have made a mistake in my algebra. I don't see why else I would get this.

I really appreciate the help!!

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# Homework Help: Help with a proof with discrete dynamical sysmtes / chaos theory.

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