Help with a proof with discrete dynamical sysmtes / chaos theory.

[goosefrabbas]

Homework Statement

Consider the families of iterating functions F_λ(x) = λ(x³ - 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.

Homework 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=λ₀ if there is an open interval and an ε such that:
1. For each λ in the interval [λ₀ - ε, λ₀ + ε], there is a unique fixed point p_λ for F_λ in I.
2.For λ₀ - ε < λ < λ₀, F_λ has no cycles of period 2 in I and p_λ is attracting (resp. repelling).
3.For λ₀ < λ < λ₀ + ε, there is a unique 2-cycle q¹_λ,q²_λ in I with F_λ(q¹_λ)=q²_λ. This 2-cycle is attracting (resp. repelling). Meanwhile, the fixed point p_λ is repelling (resp. attracting).
4.As λ -> λ₀, we have qⁱ_λ -> p_λ0
Also, these theorems are necessary (I think).
Chain Rule Along A Cycle: Suppose x₀, x₂, ..., x_n-1 lie on a cycle of period n for F with x_i = Fⁱ(x₀). Then
(Fⁿ)'(x₀) = F'(x_n-1) * ... * F'(x₁) * F'(x₀).
The corollary for this is:
Suppose x₀, x₁, ..., x_n-1 lie on an n-cycle for F. Then
(Fⁿ)'(x₀) = (Fⁿ)'(x₁) = ... = (Fⁿ)'(x_n-1)

The Attempt at a Solution

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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 [λ₀ - ε, λ₀ + ε], 0<ε<sqrt(2). Can I just choose an arbitrary ε, like ε=1?

To answer 2., if ε=1, λ₀ - ε < λ < λ₀, 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!

 

[goosefrabbas]

I changed my interval I to (-.5, .5) so that part 2 would work.

And for part 3, I got this far.
F_λ(x)=λ(x³-x) is an odd function, so the second iteration of F_λ(x₀) = x₀. Using algebra I got x = (1 +- sqrt(1 - 4λ))/(2λ), so this is the 2-cycle for F_λ, but only for 0<λ<¹/₄.

For part 4, As λ -> λ₀, we have qⁱ_λ -> p_λ0. So substituting, I get as λ -> 0, qⁱ_λ -> 0.

Is everything right so far?
I'm not sure how to prove part 4.