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Questions tagged [chaos-theory]

For questions in chaos theory.

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How can I calculate Feigenbaum's constant?

So I am trying to calculate Feigenbaum's constant for the logistic map: $$ x_{n+1} = 4 \lambda x_n (1-x_n) $$ I am writing this through python and the main pieces I have for my code that are relevant ...
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Why does Chaos Improve Evolutionary Algorithms? [on hold]

I have presented an evolutionary algorithm using chaos theory (chaotic numbers) to solve an optimization problem. The results of the experiments show that this algorithm is much better than the same ...
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How to generate chaos?

I would like to generate chaotic behavior that would have reastic 1/f noise properties. For example, a time-series of price fluctuations or pendulum oscillations that would have chaotic properties. ...
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Bernoulli Map: $f'(x) = 2$ Almost Everywhere and “Local Separation” Increases as $2^n$

The Bernoulli map is $$x_{n + 1} = f(x_n)= \begin{cases} 2x_n, & 0 \leq x_n < 0.5\\ 2x_n - 1, & 0.5 \le x_n \le 1 \end{cases}$$ I am told that (1) $f'(x) = 2$ almost everywhere and so (...
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The Unpredictability of Chaos: An Intrinsic Property, or a Matter of Computing Power and/or Mathematical Knowledge?

I was reading that Chaos (in the mathematical sense) is deterministic, but not predictable. Is this unpredictability an intrinsic property of Chaos, or is it a practical matter of computing power and/...
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1answer
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Applying a linear transformation to a system of differential equations

I am reading the book: Nonlinear Oscillations, Dynamical System and Bifurcations of Vector Fields (Guckenheimer and Holmes), chapter 2: An introduction to chaos. About Van der Pol's equation, it can ...
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1answer
34 views

Equivalent condition of a diffeomorphism having a dense orbit

Say $M$ is a manifold and $f: M \to M$ is a diffeomorphism. Assume also that, if we are given any nonempty open subsets $U$ and $V$, then there is $n \in \mathbb{Z}$ such that $f^n(U)$ intersects $V$....
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Does the property of uniquely ergodic imply that the map has a unique chaotic attractor for all $c$ in $D$

Definition: Let $(X,B)$ be a measurable space and let $T:X→X$ be a measurable transformation. If there is a unique $T$-invariant probability measure then we say that $T$ is uniquely ergodic. Consider ...
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How to find bifurcation value from a potential

Just needed some guidance to see if my approach is correct. The following is how I proceeded to do this problem. What seems to be confusing me is that it is in the potential form and not $\dot{x} = ......
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Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive

Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty. Let $S : X → X$ and ...
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Is the Lorenz system well-posed in the Hadamard sense?

Apologies if this has already been discussed, but I searched the site and I couldn't find an answer. For the sake of simplicity, consider only ODEs, possibly depending on some vector of parameters $\...
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Can a family of functions $F_{\lambda}$ experience a saddle-node bifurcation and also experience a period -doubling bifurcation?

Suppose $F_{C}=(x-C)^{2}$ is a family of functions. I found that at $C=-\frac{1}{4}$ there is a saddle-node bifurcation. So for $C<-\frac{1}{4}$ there are $0$ fixed points, $C=-\frac{1}{4}$ there ...
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About topological conjugacy

The Smale horseshoe map $f$ is desribed in this page: What's the point of a Horseshoe map? A striking feature of this system is the stability of its dynamics: given any diffeomorphism $g$ ...
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Is there a meaningful link between the golden ratio and chaos theory?

I heard it casually mentioned by strangers but am unable to find any information about this. Is there a meaningful link between the golden ratio and chaos theory? The closest thing I could find is ...
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2answers
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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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1answer
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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
48 views

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
35 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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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
53 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
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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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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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2answers
182 views

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
119 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
134 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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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
326 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
143 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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130 views

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 ...