Articles of chaos theory

Winding Numbers and Fixed Point Theorems

It is known that winding numbers can be used to prove the existence of fixed point theorems in two dimensions. We look at the vector $f(x)-x$ and if the corresponding winding number does not equal $0$ then we can establish the existence of a fixed point. For example, in the following book page 208, […]

If the Chaos Game result is a Sierpinski attractor when the random seed is a sequence (Möbius function), does it imply that the sequence is random?

The Chaos Game is the famous method to create fractals elaborated by professor Michael Barnsley. As Wikipedia explains: “The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between the previous point and one […]

Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the following picture: Picture File:Mandel zoom 00 mandelbrot set.jpg by Wolfgang Beyer licensed under CC-BY-SA 3.0

Where can I learn “everything” about strange attractors?

I have a (possibly very) limited math background, but I would like to find a book, website, or other type of work that will teach almost everything about strange attractors. As a motivating example, I found a recurrence of the following form: $$x_{n+1} = a x_n + y_n$$ $$y_{n+1} = b + (x_n)^2$$ …at this […]

Are the vertices of a Voronoi diagram obtained from a Sierpinski attractor also a kind of attractor?

Trying to understand how the Voronoi Diagrams work I did a test generating the Voronoi diagram of the points obtained from The Chaos Game algorithm when it is applied to a $3$-gon. The result is a set of points part of the Sierpinski attractor, the Voronoi cells and the vertices of each cell. This is […]

Prove that Anosov Automorphisms are chaotic

Let me add some detail first. An Anosov automorphism on $R^2$ is a mapping from the unit square $S$ onto $S$ of the form $\begin{bmatrix}x \\y\end{bmatrix}\rightarrow\begin{bmatrix}a && b \\c && d \end{bmatrix}\begin{bmatrix}x \\y\end{bmatrix}mod \hspace{1mm}1$ in which (i) $a, b, c, and \hspace{1mm} d$ are integers, (ii) the determinant of the matrix is $\pm$1, and (iii) […]

Zero Lyapunov exponent for chaotic systems

In addition to a positive Lyapunov exponent (for sensitivity to ICs), why do continuous chaotic dynamical systems also require a zero Lyapunov exponent?

If $f(x)=x^2-x-1$ and $f^n(x)=f(f(\cdots f(x)\cdots))$, find all $x$ for which $f^{3n}(x)$ converges.

Let $f:\mathbb{R}\to\mathbb{R}$ be the polynomial defined by $$f(x)=x^2-x-1$$ and let $$g_0(x)=f(x),\quad g_1(x)=f(f(x)),\quad\ldots\quad g_n(x)=f(f(f(\cdots f(x)\cdots)))$$ The positive root of $f(x)$ is the famous golden ratio $$\varphi=\frac{1+\sqrt{5}}{2}.$$ Problem: Find all $x\in\mathbb{R}$ for which the limit $$\lim_{n\to\infty}g_{3n}(x)$$ exists. The answer appear to be not at all trivial. Based on numerical computations, I conjecture that the limit exists if […]

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

$\DeclareMathOperator{\Arg}{Arg}$ Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $z, z^z, z^{z^z} …$ I am trying to classify the sequence $a_n$ as convergent […]