For questions in chaos theory.

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Question About Filled Julia and Julia Sets

Question: Let $Q_{c}(z) = z^{2} +c $ which $ c \in \mathbb{C}$ and suppose that $z_{0} \in K _{c}$ for the filled Julia Set, $K_{c}$ of $Q_{c}$. Suppose further that $z_{1} = Q_{c}(z_{0})$ and it ...
4
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1answer
252 views

Prove that the Mandelbrot Set Is A Closed Set

The Problem: Suppose we define the Mandelbrot Set as the following For $c \in \mathbb{C}$ , $\mathbb{M}$ = $({c:|c| \leq 2}) \cap ({c: |c^2 + c| \leq 2}) \cap ({c: |(c^2+c)^2 +2| \leq 2}) \cap ...
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1answer
53 views

Is this a valid example of a non-euclidean Sierpinski attractor?

I am learning the basic concepts about the Chaos Game (I did a previous question about the same topic here), the method to create fractals elaborated by professor Michael Barnsley. The basic example ...
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52 views

Chaos theory in stock market

I am doing an IB Extended Essay on chaos theory and fractals in the consumer stock market. It is a high school level essay (4000 words) and should be understandable for a calculus student. I'm having ...
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1answer
59 views

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 ...
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19 views

Inverse evolution of a dynamical system

Background Considering a one-dimensional chaotic map $T$. The domain of the map (the closed interval $[0,\,1]=I_1 \cup I_1$) is partitioned into two regions (i.e. $I_0\cap I_1=\emptyset$): $[0,\,A] = ...
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37 views

A continuous function with a dense set of periodic points but without sensitive dependence on initial conditions

The Question: Give an example of a continuous function, $f$, on the interval, $I$, such that the set of periodic points of $f$ is dense in $I$, but f does not have sensitive dependence on initial ...
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128 views

Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? [closed]

Let $H$ be a separable, infinite-dimensional Hilbert space, and $B(H) = \{T : H \to H, T \space \text {is non-bounded and linear operator} \}$. We say An operator $T \in B(H)$ is chaotic if $T$ is ...
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9 views

Highly optimized systems

I'm interested in the idea that highly selected causal systems exhibit general behaviors and properties. By causal systems I mean formal systems that progress from a defined set of starting conditions ...
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1answer
21 views

Find a vector field on the plane such that $\omega$-limit set of a single point is two parallel lines.

I came across this puzzling question in "Chaos" by K. T. Alligood. Sketch a vector field in the plane for which the $\omega$-limit set of a single trajectory is two (unbounded) parallel lines. ...
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18 views

Are binary sequences generated from ergodic maps chaotic?

Chaotic Sequences of IID Binary Random variables with their applications to Communications and related papers by the same Author, Tohru Kohda talks about the statistical properties of symbolic ...
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41 views

Is this a bijective mapping?

Considering a one-dimensional chaotic map $T$. The domain of the map (the closed interval $[0,\,1]=I_1 \cup I_1$) is partitioned into two regions (i.e. $I_0\cap I_1=\emptyset$): $[0,\,A] = I_0$ and ...
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1answer
62 views

Help in understanding a conjugacy problem

I am studying the book Applied Symbolic Dynamics and Chaos By Bai-lin Hao, Wei-Mou Zheng The basic premise of the concept of Symbolic Dynamics is : "Symbolic ...
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0answers
23 views

Is there a connection between topological mixing and squashing functions used in neural networks?

Sigmoid, ReLU, tanh, logistic -type "squashing" functions are popular in neural networks to introduce nonlinearity into the transformations of the input vector, allowing the network to fit complex ...
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27 views

Suspension flow and topological equivalence(s)

Let $M$ be a compact smooth manifold, $\tau:M\to\mathbb{R}_{\geq 0}$. Let $f:M\to M$ is a surjective piecewise-smooth map. There is a standard construction of suspension allowing to extend $f$ to a ...
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2answers
84 views

Chaos theory and fractal geometry: Constructing from data

I understand that fractal geometry represents behaviour of 'chaotic' system, if I am not wrong. And also, fractals are generated by a recursive function. But, lets say I have random data lying with ...
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1answer
38 views

Newton Iteration Function

So I'm having trouble figuring this problem out so if someone can help me out that'd be great. Find all the fixed points for the associated Newton iteration function for $$ f(x) = \frac{x}{(x-1)^n} ...
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1answer
37 views

Expanding maps of the circle and their coding

A continuous smooth ($C^{10}$-smooth, for example) map $f:S^1\to S^1$ of the circle is called expanding if $\inf_{x\in S^1} f'(x) > 1 $. Here $S^1 = [0,1]/\sim$, the segment with identified ...
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13 views

transformation of variables in Melnikov method

Supposing there is a system of non-autonomous non-linear differential equations with small damping and small forcing. The unperturbed system (zero damping and forcing) is Hamiltonian but neither has a ...
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2answers
49 views

Can there be an interval where $F(x)=4 x^2-\frac{1}{2}$ is chaotic?

The function $$F(x)=4 x^2-\frac{1}{2}$$ has two repelling fixed points. Now, I wonder, can there be an interval $I$ where it is chaotic? I think not, because of the repulsiveness of the fixed points. ...
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1answer
48 views

Prove that if $x<0$ then $F_\mu^n(x)\rightarrow -\infty$ as $n\rightarrow \infty$, where $F_\mu=\mu x(1-x)$.

Logistic map is given as $$x_{n+1}=\mu x_n(1-x_n)$$ Let $F_\mu=\mu x(1-x)$. Therefore, for $\mu>1$, prove that if $x<0$ then $F_\mu^n(x)\rightarrow -\infty$ as $n\rightarrow \infty$.
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What is the name of this chaos theoretic transformation?

I once read an article about chaos theory, which I can no longer find, that went through the following example: Suppose you take a rectangular picture of Henri Poincare, and stretch it out vertically ...
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69 views

How can we construct Markov partitions for Smale Horseshoe and Solenoid?

I was reading the book Ergodic Theory, Hyperbolic Dynamics and Dimension Theory by Luis Barreira. In Chapter 7, it is asked to construct Markov partitions for the Smale Horseshoe and the solenoid. In ...
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43 views

what does it mean when it says find an exact, closed form solution?

What does it mean when it says find an exact, closed form solution? I'm currently trying to solve a problem nonlinear difference equation which states that at the end "This is one of the rare cases ...
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2answers
123 views

Does randomness exist? [closed]

I've been plagued with this question for a few years now and wanted to know what others think. Does true randomness really exist? In mathematics, a random process is based on the concept of random ...
2
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1answer
85 views

Prove that if $f$ and $g$ are topologically conjugate functions and $f$ has chaos, then $g$ has chaos.

Question: Prove that if $f$ and $g$ are topologically conjugate functions and $f$ has chaos, then $g$ has chaos. I can see a bit of the reason behind of the claim but I can't prove it. To prove ...
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1answer
41 views

What are properties of dynamical systems in non-integer dimension spaces?

A 1D dynamical system (R1) exhibits convergence to a fixed point, or escapes to infinity. A 2D dynamical system (R3) can produce oscillations, spiral-shaped trajectories, etc. A 3D dynamical system ...
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Show that the following functions are topologically conjugate?

$T(x)=\begin{array}{lr}\ 2x, x \in [0,\frac{1}{2}] \\ 2-2x, x\in [\frac{1}{2},1] \end{array}$ and $F_4(x)=4x(1-x)$ with conjugacy $h(x)=\sin^2(\frac{\pi x}{2})$ I am completely lost, these two things ...
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1answer
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Detection of Cycles without a Center in an ODE

In my classes in dynamical systems theory, we were taught how to detect cycles or cyclic behavior in an ODE (be it dampened, sustained or growing) around a fixed point by looking at the eigenvalues ...
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1answer
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$f_\alpha:S^1\rightarrow S^1, \alpha\geq 2$ be given by$ f_\alpha (\theta)=2\theta, \forall \theta \in [0,2\pi]$ prove f is chaotic?

$f_\alpha:S^1\rightarrow S^1, \alpha\geq 2$ be given by$ f_\alpha (\theta)=2\theta, \forall \theta \in [0,2\pi]$ prove f is chaotic? Now I know I need to prove that 1) F is topologically transitive ...
2
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2answers
84 views

Logistic family and chaos

It is a well known fact that the map $f(x)=4x(1-x)$ is chaotic on $[0,1]$. By chaotic I mean the usual definition, i.e.: a) the periodic points of $f$ are dense in $[0,1]$, b) $f$ is topologically ...
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2answers
50 views

For $c<1$, let $f_c(x)=x^2+c$, determine the period-2 points?

Ok so I know I need to set $$f_c^2(x)=x$$ so: $$(x_0^2+c)^2+c = x \iff x_0^4+2cx_0^2-x_0+c^2+c=0$$ But how do I then solve this? Ok I have solved for the four set of roots, now you can see there are ...
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1answer
82 views

Prove that $F(x) = 8x^{4} - 8x^{2} + 1$ is chaotic

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 ...
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1answer
64 views

Construct a continuous function $F:[0,1]\rightarrow[0,1]$ that has a point with period 2015

Construct a continuous function $F:[0,1] \to [0,1]$ that has a point with period 2015. I think I should do it with Sharkovskii's theorem, but then? Where can I start with? Or try to find a point ...
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1answer
50 views

Solving inverse problem related to Iterated function systems?

I generated a Barnsley's fern fractal using details in this link with the aid of MATLAB. My doubts are as follows : How do we justify the shape generated from those equations? Is it possible to ...
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5answers
519 views

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 ...
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1answer
96 views

Definition of Markov partition?

My teacher handed out an excerpt from a book by Robinson on chaotic dynamical systems. The excerpt is from a chapter on Markov partitions, and the following part has me confused: Let $$f(x)= ...
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1answer
80 views

Definition of Butterfly Effect

The Wikipedia definition of the Butterfly Effect seems to imply that linear functions can exhibit the Butterfly Effect. In particular if the state space is $\mathbb{R}$ with the usual metric then if ...
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0answers
51 views

Rational dynamical system with nonnegative paramaters and with nonnegative initial conditions

let $A$ be a rational system of the form :\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
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1answer
114 views

Chaos in Newtons Method

Im trying to prove that Newtons method applied to ${\rm f}\left(\, x\,\right) =x^{2} + c$, is chaotic for $c > 0$. I know I need to prove: (a) The periodic points of ${\rm f}$ are dense in $X$, ...
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0answers
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Find all sinks/sources/saddles for a certain diffeomorphism

I'm trying to do the following exercise from Devaney's Introduction to Chaotic Dynamical Systems, exercise 2.6.1. The problem is this: Consider the diffeomorphism $Q_\lambda$ of the plane given by ...
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1answer
41 views

Showing that the linear twist map is sensitive dependent

Choose $\Delta=\frac{1}{2}$ (I believe this value should work). let $\delta > 0$ and let $\textbf{x}_1=(x_1,y_1) \in X$. I assuming that $d$ is the Euclidean distance. Somehow I think we ...
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1answer
68 views

Are chaotic function one way?

Are chaotic functions also one way functions? Can they be used in cryptography?
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2answers
113 views

Types of attractors

In studying dynamical systems and chaos theory, one usually gets across a classification that says that attractors can be of four basic types: -fixed point (equilibrium) -cyclic (periodic) -torus ...
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1answer
175 views

Is chaos theory really a theory? Why not just call it non-linear dynamics?

This may just be semantics, but it's always confused me. What is the thesis of Chaos Theory? I have read an entire book about it, and as far as I can tell, its just a bunch of analytical techniques, ...
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0answers
47 views

Question regarding graphing Standard Map and Circle Map (Arnold Tongue)

I'm attempting to program a python script that graphs two different mappings. My problem is I'm not even sure how to graph the mappings by hand since I'm unsure of the axis labels as well as how to ...
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2answers
62 views

Techniques for finding period points

Consider the tent function $f_2$ given by: $$f_2 = \begin{cases} 2x, & 0\leq x\leq \frac{1}{2} \\ 2-2x, & \frac{1}{2} < x \leq 1 \end{cases}$$ How do I find the periodic points of this ...
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1answer
69 views

Computing Feigenbaum Constant in Java

Is there a way to code a Java program that computes the Feigenbaum constant (which is around 4.66) for the function F(x) = r*sin(Pi*x)?
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3answers
636 views

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 ...
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1answer
56 views

Fixed points and periodic orbits of $F(x)=x^2-1.1$

A question asked me to find the fixed points of $F(x)=x^2-1.1$, then use the fact that these points were also solutions of $F^2(x)=x$ to find the cycle of the prime period 2 for F. How do I go about ...