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Questions tagged [chaos-theory]

For questions in chaos theory.

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Prove that the orbit of a semi-conjugacy is dense in a metric space

A semi-conjugacy $h$ is a surjective, continuous map for which the conjugacy relation $h\circ f=g\circ h$ holds. Assume that $X$ and $Y$ are metric space, and let \begin{equation} \begin{split} ...
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If a diffeomorphism has a dense orbit, are almost all of its orbits dense?

Let $M$ be a closed manifold. Suppose that a diffeomorphism $f:M\to M$ has a dense orbit. Is it true that almost every ofbit of $f$ is dense in $M$? Or, maybe, if the orbit of $x_0$ is dense, then all ...
Andrey Ryabichev's user avatar
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A chaotic function related to the $3x+1$ problem? (Li-Yorke and the Collatz problem)

Let $ x $ be an infinite binary string. Define the function $ f(x) $ mapping $ x $ to the Cantor set of $ I = [0,1] $ as: $$ f(x) = \sum_{n=0}^{\infty} \frac{2 x_n}{3^{n+1}} $$ where $ x_n $ are the ...
mathoverflowUser's user avatar
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Chaotic one-dimensional system

Why can the solution of a one-dimensional equation of the form $$m\ddot{x}=F(x)$$ not be chaotic if $F$ is not explicitly time-dependent? Multiplying by $\dot{x}$ and integrating with respect to time, ...
Diger's user avatar
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Nonlinear Dynamics and Chaos - Convergence of a Map to the Logistic Map

The cosine–map is defined as:xn+1 = r/4((a+1)cos[k(xn - 1/2)]-a), with k = 2arccos(a/a+1), a > 0. Show that in the limit a → ∞, the cosine–map is the logistic map. I am really struggling in where ...
bthomas28's user avatar
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Can probe trajectories to compute Lyapunov exponents get "stuck in more regular orbits" after rescaling?

I am computing Lyapunov exponents, and there is something that I do not understand about the data. The model has a regime for $\delta \approx 1$ (in some units) where it is fully chaotic, and the ...
Robin's user avatar
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Does the inverted single pendulum have a positive Lyapunov exponent?

I'm doing some numerical experiments to test an integrator, and I got this plot, for the motion of 5 pendula, whose initial displacement differ by $10^{-6}$ radians away from straight up ($\theta_0 = \...
David's user avatar
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Reflecting a fractal

Assume you have a certain IFS (iterated function system - a finite set of contractions) given by affine transformations. As reflections are affine transformations, any reflection of an IFS fractal ...
Psaro's user avatar
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How to identify heteroclinic (or homeoclinic) points on a mapping?

Identify the heteroclinic points of the following map: \begin{equation} F\begin{pmatrix} \theta_1\\ \theta_2 \end{pmatrix}=\begin{pmatrix} \theta_1+\epsilon\sin\theta_1\\ ...
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Show that the circle $r=\sqrt{(1-\lambda/\beta)}$ is invariant under a given map $F$

With the given map \begin{equation} F\begin{pmatrix} r\\ \theta \end{pmatrix}=\begin{cases} &\lambda r+\beta r^3\\ &\theta+\frac{2\pi}{n}+\epsilon\sin{n\...
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Analog to Feigenbaum constant for even and odd cycles

A picture is worth a thousand words, so starting off there's this peculiar aspect to the logistic map where it seems to have consistently-placed even and odd cycles after the onset of chaos: Where ...
mattrdowney's user avatar
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How to find fixed points of this radial map $f:\mathbb{R}^2\to\mathbb{R}^2$

I have learned that finding fixed points of a map is usually done by setting the map $f(x)=x$. However, for this radial map $f:\mathbb{R}^2\to\mathbb{R}^2$ $$F\begin{pmatrix}\theta_1\\ \theta_2\end{...
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Logistic map: bifurcation and domain of attraction

Let $f(x) = \mu x(1-x)$ be the logistic map, the question is divided into 3 parts: Part (1): what can you say about the domain of attraction of the 2-cycle in $3<\mu<1+\sqrt 6$? My attempt: let $...
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Sarkovskiis theorem and the Cantor set

Can we prove the following as such (with relevance with the Sarkovskiis theorem)? Suppose that $f$ is continuous and that $A_0 , A_1 ,\dots, A_n $ are closed intervals and $f(A_i) \supset A_{i+1}$ ...
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Looking for an efficient (computerized) way to convert large ternary sequences into their decimal equivalents.

I am doing a project on Cantor Sets for my undergraduate and I need an efficient (computerized) way to convert large ternary expansions such as the following to their decimal equivalent. $$0....
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set of orthogonal quasiperiodic functions

I hope you are well. Could you help me with the following? I want to build a set of quasi-periodic functions that are also orthonormal. This set will help me with a chaos problem. I tried with $\sin (\...
Mathinho's user avatar
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Finding critical value of bifurcation

I have equations such as $\frac{dx}{dt} = y \\ \frac{dy}{dt} = \mu y + x - x^{2} + xy$ This system is known to be homoclinic bifurcation at the origin. To find the critical value of $\mu_{c}$, one ...
이영규's user avatar
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Duffing oscillator - Understanding the phase diagrams

I am trying to simulate a duffing oscillator. I am not sure how to read the phase diagrams I am obtaining. In the image, we can see at the top the phase diagram before the bifurcations; it has an ...
kimmil07's user avatar
5 votes
1 answer
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Why is the number 3 come up so often in Chaos theory and Undecidability as a boundry? Is it just a coincedence?

What I mean is that the number 3 comes up a lot in these fields as sort of a boundary between decidability and undecidability, or chaos and order. Examples: Quadratic Diophantine equations are always ...
Colonizor48's user avatar
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Density of points under dyadic transformation

My question concerns the Bernoulli map (a.k.a. the shift map and as dyadic transformation). I understand that most numbers $x \in [0,1)$ do not have a periodic orbit under this transformation, and ...
subset's user avatar
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Showing that the horseshoe set is locally minimal

I'm trying to prove the Smale's horseshoe set is locally minimal. More specifically, let $H$ be the horseshoe set described in Section 1.8 in the book "Introduction to Dynamical Systems" by ...
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Calculate Lyapunov Exponent for Double Pendulum

I have the x and y coordinates for two trajectories separated by a small difference in initial conditions (for a double pendulum). Since the pendulum is physically constrained, there can only be local ...
MaximeJaccon's user avatar
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Is this a viable way to calculate the Lyapunov Exponent?

Is this a viable way to compute the Lyapunov Exponent for a double pendulum? Here is the code for the double pendulum (You don't have to look at this part, it just returns the angles and angular ...
MaximeJaccon's user avatar
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189 views

How is Chaos defined in Math?

When I look at the English definition of the word "Chaos", I get the following definition : "Behavior so unpredictable as to appear random, owing to great sensitivity to small changes ...
stats_noob's user avatar
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Why does this distance estimator method render The Mandelbrot set incorrectly (non-divergent regions as divergent)?

I am using the following algorithm to render the Mandelbrot set and the exterior: • for each test point, calculate $c$ • initialise $z_{0}=(0+0i)$ • also initialise the gradient $dz_{0}=(0+0i)$ • ...
Penelope's user avatar
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Existence of a period three cycle for a continuous unimodal interval map (A question related to Li-Yorke theorem)

Let $f$ be a continuous unimodal map from a closed interval $I$ to itself. Suppose that there exist $x_1$, $x_2$, $x_3$, $x_4\in I$ such that $f(x_1)=x_2$, $f(x_2)=x_3$, $f(x_3)=x_4$, and $x_4<x_1&...
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Why more iterations benefit deeper Mandelbrot zooms over shallow zooms?

When rendering the Mandelbrot set fractal, we set a maximum number of iterations to test each point. If the escape criteria are met within the maximum iterations, we can stop further iterations ...
Penelope's user avatar
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Art historian trying to identify math diagrams in a painting

In a painting called 'Catastrophe Theory' from 1983, the German painter Sigmar Polke included the math diagrams shown in the linked image below (for another image see here: https://www.sfmoma.org/...
milque_toast's user avatar
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1 answer
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Lyapunov methods to show the trajectories that do not converge to the origin [closed]

Consider the map $f(z) = z^2$, where $z$ represents complex numbers. what is the function that $f$ corresponds to the map $p(r, \theta) = (r^2, \theta)$ in polar coordinates. And is it true that all ...
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Formal definition of a chaotic deterministic system

I have recently been interested in chaos theory, after reading many books and textbooks speaking of the butterfly effect. But what is the rigorous definition of a chaotic deterministic system? It is ...
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Proving this theorem about the skinny baker map

I'm working on a project for a class on Dynamical systems, and I've come to a point where even the professor was unable to help me. He excused me from having to prove it directly, but out of my own ...
Matthew's user avatar
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Best references for a more intuitive introduction to Dynamical Systems, Ergodic Theory and Quantum Chaos

As a graduate math and physics student, I am introducing myself to the study of Smooth Dynamical Systems and Ergodic Theory, with the aim of applying it to Quantum Chaos and Quantum Ergodic Theory. I ...
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Are there continuous chaotic systems where nearby paths diverge at rates other then exponential?

I know that in general, nearby paths in a chaotic system tend to diverge exponentially, but are there continuous systems where paths diverge at other rates? For example, is there a system where nearby ...
Colonizor48's user avatar
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Finding and studying Poincaré map

I know what the definition of a Poincaré map is but I am struggling a little bit with the exercises and with computing it. I am trying to solve the following problem: Given the system (in polar ...
Donatello's user avatar
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Doubt in a paper on Devaney Chaos and dense periodic points

I am reading from the paper here. In Lemma 5, the authors make the following assertion: Let $f : [0, 1] \to [0, 1]$ be a continuous function with a dense set of periodic points. If $f$ has no proper ...
Ajin Shaji Jose's user avatar
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Does an infinite number of points in a closed interval imply density?

It is not very important, but I am asked to prove that the periodic points in the tent map are dense on the closed interval $[0,1]$. Anyway, I can prove that there are an infinite number of periodic ...
Matthew's user avatar
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1 answer
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Henon map's sink fixed point and saddle fixed point.

It is noticeable that the sinking and saddling points of the Henon map are appropriately satisfied in both $a = 0.27, -0.09$, and $b=0.4$. As written in this fig: My question is, why the Henson map ...
simple1's user avatar
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2 votes
1 answer
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Lyapunov Exponents for $n$-Dimensional Matrix $A$

I am wondering whether my solve is correct. I know how to solve the 2, or 3 dimension of the state matrix. But what if the state matrix goes to n-dim? Here is what I tried: To find the Lyapunov ...
lulu's user avatar
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1 answer
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Exploring Sensitivity Dependence in Chaotic Systems

After analyzing the sensitivity dependence of orbits under the map $f(x) = 2x \mod 1$, we have found that every initial point has nearby points that eventually diverge by at least $1/2$ unit after ...
simple1's user avatar
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On the mathematical expression for the logistic map

Mathematically, the logistic map is written as follows $$x_{n+1}=rx_{n}(1-x_{n})$$ Here, $x_n\in[0,1]$ represents the ratio of the existing population at each step ($n$) to the maximum possible ...
haiku's user avatar
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For any chaotic system such as a chaotic attractor does there exist a higher dimensional representation in which the system is no longer chaotic? [closed]

For a given n dimensional chaotic system such as a chaotic attractor or really a time dependent chaotic dynamic system does there exist a higher dimensional representation where the behavior is in ...
Matthew Wander's user avatar
1 vote
1 answer
32 views

Action of the adjoint operator in tangent space

I'm reading a paper on the computation of covariant Lyapunov vectors (https://arxiv.org/pdf/1212.3961.pdf) and, as I have a Machine Learning background, I have some gaps concerning dynamical systems. ...
Pepper08's user avatar
2 votes
1 answer
94 views

Is this chaotic behaviour? Weakly coupled Van der Pol oscillators

I was investigating the following system of two weakly coupled identical Van der Pol oscillators $$\left\{\begin{array}{@{}l@{}} \ddot{x}_1 + x_1 + \epsilon(x_1^2 - 1)\dot{x}_1 = \epsilon k(x_2 - x_1)...
Hervé Schmit-Veiler's user avatar
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Connection between ergodicity and Lyapunov exponents

This will be a soft reference question in a sense, as I will not state any rigorous theorems/results. Whenever I happen to read about ergodic systems, be it a specific book, article or a blog post, a ...
Epsilon Away's user avatar
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15 votes
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Why does the Mandelbrot fractal appear when plotting $\underbrace{x\cos(x\cos( \cdots x\cos}_9(x))))$?

while plotting the function $x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x)))))))))$ using matplotlib in python I found the mandelbrot fractal. What is the reason that the mandelbrot fractal ...
intro's user avatar
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How do I change this to integration over a trajectory?

I'm working through CHAOS: An Introduction to Dynamical Systems by Alligood et. al. and I'm on Challenge 7 step 5. At this point we have a system of ODEs. (Note: a dot over a variable is its ...
roundsquare's user avatar
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How do you show that a chaotic system such as the logistic map acts transitively upon an infinite set?

The logistic map with parameter $r=4$ and the tent map with parameter $\mu=2$ are topologically conjugate chaotic dynamical systems. Question Given an infinite set of points $S$ within the domain, how ...
it's a hire car baby's user avatar
2 votes
0 answers
46 views

Pre-images of the critical point of of $3.83 x (1-x)$

This question may be easy; however, I have been unable to locate any references regarding the specific scenario described below. Let $T:[0,1]\to [0,1]$ be the quadratic map $T(x) = 3.83 x (1-x)$. It ...
Matheus Manzatto's user avatar
0 votes
1 answer
66 views

How does it follow from $(|f(x)-f(p)|)/|x-p| <a$ that $|f^k(x)-p| \leq a^k|x-p|$

The exercise is asking me to show the latter theorem $|f^k(x)-p| \leq a^k|x-P|$ for $k=2$. But I am confused as to how they even derived that theorem. In the proof of a theorem, that had: $(1)$ There ...
Matthew's user avatar
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1 vote
0 answers
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Precisely computing nth iterate of tent map

Suppose we have the tent map, $T_\mu(x) = \mu \min(x, 1-x)$ with parameter $\mu = 2$. Then note that if we pick an $x_0 \in [0, 1]\cap \mathbb{Q}$, so that if $x_0 = p/q$ with $p, q$ co-prime and $q$ ...
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