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Questions tagged [chaos-theory]

For questions in chaos theory.

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Why Lorenz attractor can be embedded by a 3-step time delay map?

I'm investigating attractor reconstruction of Lorenz system. I saw a bunch of work claiming that the time delay map $[x(t), x(t -\tau), x(t - 2\tau)]$ is sufficient to reconstruct the attracotr, e.g. ...
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Stability Analysis after finding derivative [closed]

How do I analyse the stability of a function once I have found its derivative, which is a linear equation"
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34 views

Question about Chaos Theory and Relation to Game…

I am working on a project (for social sciences) and I found a game, Parable of Polygons (link) I observe chaotic behavior, but am unable to mathematically explain this. Are there any ideas on how the ...
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If the solutions to Lorentz equation diverge exponentially, how can they be confined in a strange attractor? [duplicate]

In the book of Chaosbook, at the beginning of chapter 6, it is given that [...]if the attractor is strange, any two trajectories $x(t) = f^t(x_0)$ and $x(t)+\delta x(t) = f^t(x_0 + \delta x_0)$ ...
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Does the existence of a superstable fixed point imply a $-\infty$ Lyapunov exponent?

This is the Lyapunov exponent as a function of $r$ for the logistic map ($x_{n+1}=f(x_n)=r(x_{n}-x_{n}^2)$) The big dips are centered around points where $f'(x)=0$ for some $x$ in the trajectory used ...
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Application of Correlation Dimension to Fractals other than Sets of Points

In Chaos Theory, the Correlation Dimension is defined to calculate the dimension of fractals. At least in the context where I've learnt it, it is applied to fractals made up of sets of points. Is it ...
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Id $f,g$ orientation preserving cricle-diffeomorphisms, then $\rho(g^{-1}\circ f \circ g) = \rho(f)$.

Let $F, G : \mathbb{R} \rightarrow \mathbb{R}$ be a lift of $f$ and $G$ of $g$. That is, $ \pi \circ F = f \circ \pi$ with $\pi(x) = e^{2\pi i}$. We define $$\rho_{0}(F) = \lim_{n\rightarrow \infty}\...
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1answer
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Chaotic system?

I'm currently working on an individual project. In which I have created a program to simulate a particle bouncing around a finite bordered region. Where I need help in understanding how I might show ...
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1answer
34 views

Can a map be both nonchaotic and chaotic?

In Wikipedia’s article ‘List of chaotic maps’ (https://en.wikipedia.org/wiki/List_of_chaotic_maps), one of the entries is: Feigenbaum strange nonchaotic map So, how is this possible?
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Sard's theorem for orientation preserving diffeomorphism of the circle

thanks in advance for helping me. First I'll introduce some definitions: (1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving ...
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Dynamics $\delta x(t)=\delta x(0) e^{\lambda t}$ of Henon Attractor

Recall the question I asked before: Linearized perturbation dynamics of Henon Attractor So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$\delta x(t)=\delta x(0)...
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Fixed Points and their Stability and Finally Bifurcation diagram

Find and Classify all fixed points and sketch vector field. What's the bifurcation? $$ 𝑥̇ = 𝑟𝑥 − \tan(𝑥)$$
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2answers
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Dynamic system $f(x) = 2x$ mod $1$

I am reading the following paper: ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded) My question is from an example on p....
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1answer
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Time dependence in chaos theory [closed]

One of the best, more original examples of chaos theory comes from Ray Bradbury’s A Sound of Thunder. People in the chaos-theory community have always said that the hunters that time safari allowed to ...
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31 views

Hypercyclic operators in $L_p (0,\infty)$

I'm pretty new to this area of study so if there are logical lacune in my proof (I'm sure there are many) please let me know. This is material I'm self studying. I'm trying to adapt the methods used ...
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Although Lorenz system is a deterministic system, can it shows locally stochastic behavior?

The Lorenz system is a system of ordinary differential equations $$\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x),\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y,\\{...
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Perron-Frobenius operator definition

According to http://mathworld.wolfram.com/Perron-FrobeniusOperator.html, the P-F operator describes the time evolution of densities in phase space: $$\rho_{n+1} = P (\rho_n).$$ But for the ...
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1answer
52 views

Hypercyclicity examples

Does anyone have simple practical examples of hypercyclicity they use in explaining the concept (graphically or numerically)? This appears often in texts about chaos in infinite dimensional linear ...
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1answer
39 views

Does a greater Lyapunov exponent result in a more chaotic system?

Let $\sigma_1$ be the Lyapunov exponent of a one-dimensional system $I_1$ and $\sigma_2$ be Lyapunov exponent of one-dimensional system $I_2$. Can I say that if $\sigma_1 > \sigma_2$, then $I_1$ is ...
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1answer
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Periodic solutions of the double pendulum

I'm stuck: Are there periodic solutions of the double pendulum, or not? The question is four-fold: Does every double pendulum - defined by two masses $m_1$, $m_2$ and lengths $l_1$, $l_2$ - have ...
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1answer
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How to calculate the envelope of the trajectory of a double pendulum?

Consider a double pendulum: Background For the angles $\varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$): $\dot{\varphi_1} = 6\frac{...
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Project ideas on Chaos theory, Cellular Automata, Fractals, Games, IA [closed]

I'm a computer science student and I need to find a final year project. What interests me the most is Chaos, IA, Games, Fractals, CA.. Something I liked was the chaos theory within sudoku. The ...
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How to calculte the Fourier Transform of a sovable chaos waveform?

Recently I am stucking in frequency estimation of a solvable chaos waveform. Its analytic expression in time domain is $$ z(t)=s_m(u_m-s_m)e^{\beta(t-mT)}\cos(\omega_0 t+\varphi) $$ where $u_m \sim U(-...
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1answer
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Let $M$ be a shift space over a finite alphabet $\mathcal{A}$. Prove that $M$ is compact in the metric topology $\tau_{\rho}$.

GIVEN Define a map \begin{equation*} \label{eq1} \begin{split} \rho(x,y) = \begin{cases} 2^{-k} \ \ \ &\text{if } x \neq y, \text{ and } k \text{ is maximal so that } x_{[-k.k]} = y_{[-k,k]...
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Is $\tan(x)$ chaotic on the entire real line?

In Robert L. Devaney's "An Introduction to Chaotic Dynamical Systems" it defines a function $f:J\longrightarrow J$ to be expansive if there exists $\nu>0$ such that, for any $x,y\in J$, there ...
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Existence of infinite iteration of functions $f_\infty$?

Given a sequence of functions $\{f_n\}$ satisifying an iterated relation such as $f_n(x)=g(x+f_{n-1}(x))$ $f_n(x)=g(xf_{n-1}(x))$ $f_n(x)=g(x/f_{n-1}(x))$ Where $g:=f_1$ is ...
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1answer
105 views

Diffeomorphism with only hyperbolic periodic points has finitely many periodic points (Morse-Smale)

Came across the following question during a course Chaotic Dynamical Systems: If a diffeomorphism $f:I\to I$ is Morse-Smale (i.e. has only hyperbolic periodic points), then it has finitely many ...
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1answer
116 views

How reliable a measure of chaos is the largest Lyapunov exponent?

I’m dealing with a system of ODEs where I think I might have found chaotic solutions. I’ve calculated the largest Lyapunov exponent, and found it to be approximately 0.02. It’s positive, which ...
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1answer
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Are there non-chaotic systems which exhibits topological mixing?

I was recently reading about chaos theory. Chaos is commonly defined to exhibit 3 properties: Sensitivity to initial conditions/Lyapunov coefficient is positive Exhibits topological mixing Dense ...
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1answer
303 views

Fractal Dream Attractor behavior

This question is similar to this one: Properties of King's Dream fractal The fractal is described here: Softology - Visions Of Chaos 2D Strange Attractor Tutorial I accidentally modefied the formula ...
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2answers
127 views

Can Brownian motion be regarded as chaos?

I want to know if it is included in chaos. Does it have boundedness, deterministic, initial value sensitivity that is characteristic of chaos?
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What is the name of this chaotic sequence?

A long time ago I came across a chaotic sequence of the form x[n+1] = f(x[n], y[n]) y[n+1] = g(x[n], y[n]) If I remember correctly, ...
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Examples of chaotic motion in non-deterministic dynamical systems?

I was reading this article on Hyperbolic dynamics, in which I came across the line "deterministic chaos - the appearance of chaotic motions in purely deterministic dynamical systems". Hence I wonder ...
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Proof check: Show that $x_{n + 1} = f(x_n) = 15 - x_n^2$ is a (Smale) horseshoe

A solution is given here: However, I believe it to be incorrect because $f(K_1) = f(K_2) \neq (-\sqrt{15}, \sqrt{15})$. Attempt at a solution: Referring to the diagram from the question... By the ...
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To classify bifurcations by calculating partial derivatives evaluated at the bifurcation point

Question: Given a one dimensional system $\dot{x} = f(x;\; \mu)$ for which a bifurcation of fixed points occurs at $(x^\star, \mu_c)$, where $x^\star$ is a fixed point and $\mu_c$ is the ...
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1answer
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Determining Rate of Convergence for Chaotic Behavior

I've got a set of data that's been manipulated solely numerically. But anyway, for a certain part of that data set, I see that it's convergent (image). But I'm sort of at a loss for how depict that ...
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If a system is defined in $n$-dimension then it has $n$ Lyapunov exponents

I wanted to know if there is a formal proof to show that if a system is defined in $n$-dimension then it has $n$ Lyapunov exponents. Any links regarding information about this and Kaplan-Yorke ...
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1answer
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To find the approximate period and approximate equation of the limit cycle for a system with a Hopf bifurcation

Question: The system $$ \dot{x} = 3y + 3x^3 + xy $$ $$\dot{y} = -3x + \mu y + 2xy^2 - y^3$$ undergoes a Hopf bifurcation at $(x, y) = (0, 0)$ as $\mu$ passes through 0. Calculate the approximated ...
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Do all systems of differential equations have a solution?

I'm not really sure that I have the required knowledge to pose this question properly so I'll just pose it poorly. The rise of numerics has allowed us to study systems of differential equations for ...
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2answers
133 views

Show that the origin of the following system is stable (non-linear vs linear stability)

$$\dot{x} = -2y -x^3 + x^2 y^2$$ $$\dot{y} = x - x^2 y$$ Jacobian evaluation at $(0, 0)$: $$\begin{pmatrix}0 & -2\\ 1 & 0\end{pmatrix}$$ Clearly the determinant is greater than zero but the ...
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1answer
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What is the “bifurcation of a fixed point”?

Isn't the bifurcation of a system, say $\dot{x} = \mu \sin(x) - 2x$, related to the system rather than a fixed point, say $x^{\star} = 0$?. My understanding is that a bifurcation occurs when there's ...
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Understanding the definition of Sensitive dependence on initial conditions?

I was trying to understand the rigorous definition of sensitive dependence on initial conditions which is as follows - $f : X \mapsto X$ where $X$ is a metric space. If there exists $\epsilon > ...
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Lorenz Attractor, its Geometric Model, and 14th Smale's problem.

I've found a post with a beautifully animated video that states the following: In 2001 mathematician Warwick Tucker proved that the paper model accurately describes the motion on the Lorenz ...
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Understanding fixed point of Arnold's Cat Map

I'm currently performing a small investigation into the properties of Arnold's Cat Map, a function which is defined as: C:[0,1) -> [0,1) C(x,y) = (2x+y, x+y) (mod 1) This function has exactly one ...
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A question on Smale-Birkhoff homoclinic theorem

Does Smale-Birkhoff homoclinic theorem ("given a transverse homoclinic intersection for a diffeomorphism $f$, there exists an integer $n \geq 1$ such that $f^{n}$ has an invariant Cantor set on which ...
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50 views

Measure of the periodic orbits of this simple dynamical system

Consider the dynamical system $\phi$ on a phase space $M =\{(x,y): x \in [0, 2\pi), y \in [0,\pi)\}$ where we take $x(mod 2\pi)$, i.e. $M$ is a cylinder. Define $(x_{1}, y_{1}) = \phi(x_{0}, y_{0})$ ...
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1answer
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Does positive topological entropy imply horseshoes?

I am aware that positive topological entropy implies chaos in the sense of Li-Yorke. I want to get an idea of what consequences positive topological entropy has for the the presence (or lack of) ...
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Peaks in the circle chaos game

The chaos game is a random walk that steps halfway between your current position and a set of predefined points. If the points are on an equilateral triangle, the resulting set is Sierpinski's ...
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What can linear stability analysis say about trajectories starting in the neighbourhood of a fixed point? The role of Arnold diffusion

Consider a Hamiltonian dynamical system having more degrees of freedom than invariant quantities (Suppose that the number of degrees of freedom is $D>1$ and that the only invariant quantity is the ...
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Differences in the eigenvalue distribution between “regular” and chaotic domains

Weyl's conjecture gives us the asymptotic distribution of the eigenvalues $\lambda$ of the Laplace operator in a domain $\Omega$ with homogeneous Dirichlet boundary condition on $\partial \Omega$: $$ ...