# Questions tagged [chaos-theory]

For questions in chaos theory.

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### Example of Topologically Transitive Maps on $d$-dimensional Euclidean Space

What are some standard examples of topologically transitive maps on $\mathbb{R}^d$? I know of many on compacta but not on the entire space itself...
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### Are chaotic systems generally non-differentiable w.r.t. initial conditions?

There is some important background to go over before this question will make any kind of sense, so before calling me out on my poor understanding of dynamical systems, chaos, and differentiabilty, ...
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### Finding stable and unstable manifolds

Consider the system $x ^ { \prime } = 4 x + 2y ^ { 3 } \\ y ^ { \prime } = - 3 x$ Question: Find the stable and unstable manifolds around the fixed point $(0,0)$ and and sketch the phase portrait ...
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### Prove that non-attracting periodic orbits of $x \mapsto x^2 + c$ are in the Julia set [closed]

Let $c$ be complex number and let \begin{align} f_c : \mathbb{C} &\to \mathbb{C} \\ x &\mapsto x^2 + c \end{align} be the quadratic map. It is to be shown that if $O$ is a periodic orbit that ...
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### How to find the Interval $[a,b]$ for a contraction mapping for $f(x)=x-\cos x$ where $a,b$ exists in $[0,1]$?

So, using Newton's method, I know the fixed point for the contraction mapping is approximately $0.739$. However, I'm not sure how to go about finding the interval for the contraction mapping. Every ...
Show that for $1 <\mu< 3$ and $\mu\neq2$, the Lyapunov exponent of the logistic map $F_\mu(x)=\mu x(1-x)$ is given by $\lambda(x)=\ln|2-\mu|$. My work: We have $F_\mu(x(k))=\mu x(k)(1-x(k))$. ...