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Questions tagged [chaos-theory]

For questions in chaos theory.

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29 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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39 views

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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30 views

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
50 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
33 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
32 views

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
60 views

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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6 views

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
35 views

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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35 views

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
164 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
91 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
101 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
48 views

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
286 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
122 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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52 views

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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41 views

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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101 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 ...
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1answer
51 views

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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1answer
126 views

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
126 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
49 views

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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0answers
97 views

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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1answer
138 views

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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91 views

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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53 views

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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43 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
53 views

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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40 views

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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157 views

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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31 views

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 $\partial \Omega$: $$ ...
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1answer
48 views

A question on Lyapunov exponents associated with a fixed point of a vector field.

The Lyapunov exponent $\chi(x_{0}, e)$ in a direction $e \in \mathbb{R}^{n}$ along the trajectory $x(t,x_{0})$ of a vector field $\dot{x} = f(x)$ through a point $x_{0}$ is defined to be $$\chi(x_{0},...
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0answers
182 views

The surface of Aizawa attractor

Aizawa attractor creates a closed surface. Let's ignore the transient state in the beginning. I am wondering how to calculate its surface? I even don't know where to start. The dynamic model of ...
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78 views

Inverse of Baker's map

Consider a matrix $A$ whose size is $n\times n$, and the following Baker's map, that maps $A$ to $B$, whose size is $\frac{n}{2}\times2n$: $$ B(i,j) = \begin{cases} \frac{1}{2}\left(A(2i,\lceil\frac{...
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0answers
28 views

measure of the chaotic region of the logistic map

The logistic map has infinitely many regions with periodic behavior after the onset of chaos, called "islands of stability". I'm interested how large the chaotic region actually is, after you subtract ...
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1answer
34 views

logistic map with lamba greater than 4

I was doing some recreational math about the logistic map. (If you're not familiar with what the logistic map is, here are some links you can check out) https://www.youtube.com/watch?v=ETrYE4MdoLQ ...
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1answer
105 views

Chaos of Irrational Numbers

I just finished a course on chaos and fractals, and it got me thinking about irrational numbers, and I was thinking about whether or not the distribution of the digits are chaotic. Consider a map $\...
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1answer
66 views

How are Chaotic systems different from Ill-posed systems?

I got puzzled when I thought of two concepts that is 1) Chaotic Systems 2)Well posed and ill-posed systems. Chaotic systems are those dynamical systems which have the sensitive dependence to the ...
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1answer
61 views

How do I find the fixed points of this 2-d map?

Consider the two-dimensional map $$ \begin{aligned} X_{n+1} &= X_n + Y_n + r\left(X_n - X^3_n\right)\\ Y_{n+1} &= Y_n + r\left(X_n - X^3_n\right) \end{aligned} $$ where $r$ is a control ...
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0answers
70 views

Poincaré-Bendixson criterion applied to non-autonomous dynamical systems

I'm fairly new to dynamical systems and chaos. I'm dealing with a second order non-autonomous system of differential equations: $$\dot{x} = y $$ $$\dot{y} = f(x,y,t)$$ I can further split $f$ as ...
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0answers
28 views

Newhouse Ruelle Takens theorem and three frequency quasiperiodicty

NRT theorem suggests that there exist arbitrarily small perturbations of a vector field on an $m$-torus for $m\geq3$ leading to strange axiom A attractors. Although, three frequency quasi periodic ...
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1answer
47 views

Periodic points of $F(x)=\lambda x + \beta x^3$ where $\lambda >1$ and $\beta <0$

So, when the coefficients are somewhat arbitary, I'm quite confused. Obviously there are 3 fixed points. There are other quite obvious ones, by letting $\lambda x+ \beta x^3 = -x$. However when we ...
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2answers
135 views

Solving $F(x) = F(e^x) - F(-e^x)$

What are non-trivial functions such that for all $x \in \mathbb{R}$, $$F(x) = F(e^x) - F(-e^x)?$$ I obtained this equations after seeking for random variable $X$ with CDF $F$ such that $\log |X|$ is ...
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2answers
85 views

Are multiple kinds of attractors (chaotic and otherwise) possible within a single system of differential equations?

I’m looking for any $n$-dimensional system of first order differential equations where depending on the initial conditions you can end up in a number of attractors, for example multiple chaotic orbits,...
2
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1answer
141 views

How to prove a fixed point is stable?

\begin{align*} \dot{x} &= 2 x - \frac{8}{5} x^2 - xy\\ \dot{y} &= \frac{5}{2} y - y^2 - 2 xy \end{align*} So I have this dynamical system. I linearized it and found that the fixed point (at $...