I have to prove that the function $F(x) = 8x^{4} - 8x^{2} + 1$ is chaotic. I would like to use the definition of a chaotic function which is:
Let $F$ be $F: V$ -> $V$.
1) Sensitive dependance on initial conditions: it exists $\delta > 0$ such that it exists $y \in N$ and $n \ge 0$ such that $|f_{n}(x) - f_{n}(y)| > \delta$ for any $x \in V$ and any neighborhood $N$ of $x$.
2) Topologically transitive: it exists $k > 0$ such that $f_{k}(J) \cap I \ne 0$ for any pair of open sets $J$ and $J \subset V$.
3) Periodic points are dense in $V$.
I wanted to try the following method:
Find $G$ and $\phi$ such that $F o \phi = \phi o G$, prove that G is chaotic and then as a matter of fact that F is chaotic.
Here, Use the definition of "topologically conjugate" to prove that $F(x)=4x^3-3x$ is chaotic., you can see the method for the function $F(x) = 4x^{3} - 3x$.
The problem I have is to find a function $G$.
Thanks for reading,
Sign009
Let's take a look at the two graphs you mention.
It looks to me like the first is conjugate to $\theta \mapsto 4\theta \mod 2\pi$ while the second is conjugate to $\theta \mapsto 3\theta \mod 2\pi$. In fact, the conjugacy functions are both given by $\varphi(\theta) = \cos(n\theta)$ for $n=3,4$. This is because these are both Chebyshev polynomials and, in that context, their defining characteristic is that $T_n(\cos(\theta))=\cos(n\theta))$. Of course, $\theta \mapsto n\theta \mod 2\pi$ is well known to be chaotic.