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 http://ifts.zju.edu.cn/profiles/xingangwang/Course2010/download/Yorke-chaos.pdf page 208, […]

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 […]

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

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 […]

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 […]

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) […]

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

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 […]

$\DeclareMathOperator{\Arg}{Arg}$ 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 […]

Intereting Posts

A help to understand the generalized version of the associative law of union
$a^m+k=b^n$ Finite or infinite solutions?
A problem with my reasoning in a problem about combinations
If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$
Interesting circles hidden in Poncelet's porism configuration
A space curve is planar if and only if its torsion is everywhere 0
If $f$ and $g$ are continuous and for every $q\in \mathbb{Q}$ we have $f(q)=g(q)$, then $f(x)=g(x)$ for every $x\in \mathbb{R}$
How many weakly-connected digraphs of n vertices are there without loops and whose vertices all have indegree 1?
Simplify series involving derivatives
Fundamental group of projective plane is $C_{2}$???
Are these 2 graphs isomorphic?
Calculation of a strange series
How find this limit $\lim_{x\to 0^{+}}\dfrac{\sin{(\tan{x})}-\tan{(\sin{x})}}{x^7}$
Convergence in measure implies convergence in $L^p$ under the hypothesis of domination
Vitali-type set with given outer measure