Questions tagged [chaos]

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6
votes
0answers
266 views

How to analytically prove chaos

Consider the following map \begin{align*} T \colon \mathbb{R}\times\mathbb{S}^1 \to & \mathbb{R}\times\mathbb{S}^1 \\ (x,\theta) \mapsto & \left(\frac{x}{4}+ \sin^2\left(\pi\left(\theta+\...
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0answers
90 views

Birth of chaos due to nonautonomous perturbation

Let $\sigma, b>0$. I want to study the dynamics of the map $$ T \colon \mathbb{N} \times \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1 \times \mathbb{R}$$ such that $$T_{\sigma,b}(n,\theta,y) = (\...
0
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0answers
54 views

Li-Yorke sensitivity Vs Li-Yorke dense chaos

Let $X$ be a compact metric space, $X*X$ its cartesian product, and $A$ a subset of $X*X$. Are the following two properties the same, or e.g. one is stronger than the other? $A$ is dense and residual ...
0
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0answers
48 views

Implications for a simple deterministic chaos definition

Among many others, one definition of deterministic chaos terms "chaotic" a classical dynamical system that satisfies the following three properties: sensitive dependence to initial ...
1
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0answers
28 views

Nonintegrable classical dynamical systems and deterministic chaos

I'm trying to delineate a minimal (and informal) "taxonomy" for classical continuous dynamical systems that could be interested by the phenomenon of "chaos" - unfortunately the ...
0
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0answers
28 views

Applications of piecewise-smooth dynamical system with closed switching curves

There are hundreds of applications of the theory of piecewise dynamical systems where the switching curve are straight lines. See for example this picture from the paper "Global dynamics of a ...
2
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0answers
69 views

Oscillator with discrete number of amplitudes?

I know the question is a little vague, but I would like to know if someone can direct me to what kind of oscillator (if exist), that can follow the next behavior. I manually create the gif to try to ...
1
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1answer
68 views

Starting vector in Lyapunov exponents evaluation

Let us consider the equation: $$ \dot{x}_i = F_i(x) $$ with $x\in \mathbb{R}^n$ and $i=1\dots n$, and the equation for small displacements: $$ \dot{\delta x} = \sum_j \frac{\partial}{\partial x_j} F_i(...
7
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0answers
166 views

Rigorous results on chaos in a driven damped pendulum

The harmonically driven damped pendulum is often used as a simple example of a chaotic system, the equation is just $\ddot{\phi}+\frac1q\dot{\phi}+\sin\phi=A\cos(\omega t)$. As long as $A$ and $\omega$...
4
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1answer
372 views

Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?

It is known that for $r=-2,2,4$ the logistic map $x_{n+1}=r x_n (1-x_n)$ has exact solutions of the form $$ x_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x_0)\right)\right\} \qquad \qquad{(*)} $$ for ...
2
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2answers
295 views

Why are attempts to define chaos with discrete states so scarce?

Interestingly, the theory of nested recurrence relations has been correlated with “discrete chaos” by Golomb (1991) and Tanny (1992). And in literature, there are very few studies that have different ...
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78 views

Question regarding Ito representation theorem

Let $H$ be a Gaussian Hilbert space and $H^{:n:}$ be the homogeneous chaos of order $n$. and let $D_n:=\{(t_1,\cdots,t_n):t_1<t_2<\cdots <t_n\}$. For each $n\geq 0$ there exists an isometry \...
0
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1answer
106 views

Consequences of invariant-subspace problem to Li–Yorke chaos [closed]

The invariant-subspace problem is probably an open problem for reflexive spaces which asks: Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial ...
7
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2answers
712 views

Is this a new strange attractor?

I recently made some experiments in programming strange attractors, and I found this (very simple) equations, which create a nice strange attractor: ...
4
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1answer
292 views

Questions about a return map

Consider the following map in the interval $u\in[-1,1]$ ($U\in[-1,1]$ also) $$U=u-\frac{64}{3}u^{3}+64u^{5}-64u^{7}+\frac{64}{3}u^{9}$$ It has 3 fixed points at $u=0,\pm 1$. If we compute the ...
3
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3answers
182 views

Example of a Chaotic discrete dynamical system in dimension 2

I am looking for examples of discrete dynamical systems in dimension 2 that are : 1) Chaotic dynamical system in Devaney's sense in dimension 2 ? 2) Chaotic dynamical system in Li-Yorke sense but ...
1
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1answer
166 views

Formal justification of the Chaos game in the Sierpinski triangle

I want to justify why the Chaos game works to produce Sierpinski triangle. I use a theorem taken from Massopust Interpolation and Approximation with Splines and Fractals. Suppose that $(X,d)$ is a ...
3
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0answers
84 views

Curious if this chaotic recurrence relation has a name and/or interesting properties

I was playing around with 2-dimensional recurrence relations and I stumbled upon this: $$ x_{n+1} = \sin(k(y_n + x_n))\\ y_{n+1} = \sin(k(y_n - x_n)) $$ I was wondering if this was a known ...
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0answers
82 views

On the measure of regular and chaotic regions in a phase space

Consider a Hamiltonian, non-linear, dynamical system associated to $H(\vec{q},\vec{p})$. Assume that the number of effective degrees of freedom is relatively small, say $D=3,4,5$. Now choose a certain ...
3
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2answers
1k views

On Mathematical Foundations of Football

Football (soccer) is arguably one of the most unpredictable sports. Countless variables play a role in determining the outcome of a certain football match. Due to the high complexity of the entire set ...
2
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0answers
84 views

On the convergence problem of box counting for the Rössler attractor

So 5 month ago i posted this Question: Rössler attractor, Convergence of box counting to estimate the fractal dimension. Since then i have assumed, that the rate of convergence of $n(ɛ,n)$ (sum ...
2
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0answers
280 views

Rössler attractor, Convergence of box counting to estimate the fractal dimension

At the moment I want to estimate the fractal dimension of the Rössler attractor. I have written a program, which is counting the boxes hitted N(ɛ) (with ɛ := side length of boxes) by a trajectory on ...
39
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2answers
2k views

3D Billiards problem inside a torus

I have been trying to simulate the behavior of a light particle being reflected inside of a torus (essentially a 3D billiards problem). I have found that after a few thousand bounces, it converges on "...
2
votes
1answer
131 views

Are all Torus Links in fact Lorenz links or not?

I'm currently trying to work through the material on Lorenz knots in the literature and there seems to be conflicting information. On p. 66, in the Birman-Williams' paper Knotted Periodic Orbits in ...
3
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1answer
1k views

Gutzwiller trace formula

I am reading A Proof of the Gutzwiller Semiclassical Trace Formula Using Coherent States Decomposition, Commmun. Math. Phys, 202, 463-480 (1990). Gutzwiller trace formula says where and $g$ is a $C^...
1
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1answer
227 views

Beginners level question : symbolic dynamics and notations

Let $f(.)$ be a chaotic 1 D Map which produces a scalar valued time series where the first iterate is obtained from an initial condition $x[0]$ as $x[1] = f(x[0],\mu)$ where $\mu$ is the control ...
2
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0answers
143 views

Advice on research ideas on Non Linear Dynamics [closed]

I am interested in Non Linear Dynamics. I am just a beginner to this research area in the sense that I am starting to read the research papers. My aim is to make my project to a research paper. But ...
6
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0answers
323 views

Had this theorem in Tresser's article been proven somewhere?

The article in question is About Some Theorems by L.P. Sil'nikov by Charles Tresser. I am interested in the theorem C from page 453 and a particular application of such theorem which is illustrated ...
3
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0answers
44 views

stochastic dynamics as approximate deterministic dynamics

Is there a rigorous sense in terms of which a stochastic process may be considered as an approximation to a chaotic but deterministic ODE in a higher-dimensional state space, in a manner that ...
2
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0answers
61 views

A proper class for smooth chaotic function

This might be a little, soft, but I'll try Consider the interval $I=[-1,1]$. We will define a chaotic function $f:\mathbb{R}_+ \times I \to \mathbb{C}$ in the following traditional way: For every $...
7
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2answers
278 views

List of Bernoulli chaotic systems

Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)? I am aware that it is known for some uniformly ...
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0answers
169 views

Quantification of the extent of periodicity in a time series using fractal analyses

I need metrics to quantify and compare the extent of periodicity between any two given time series, considering the time series were "almost periodic". By "almost periodic" I mean: if I were to take ...
-3
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1answer
300 views

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

I proposed this question in SE but no answer ,may I have a problem in my question, I would like to know when $\frac{\Bbb d}{\Bbb d x}$ does chaotic operator in Hilbert space ? Let $H$=$L^2(\mathbb R)$...
3
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0answers
320 views

Lorenz attractor power spectrum

If considered Lorenz attractor (with classical parameters $\sigma = 10, b = \frac{8}{3},r>25$), it is often noted, that while the spectral density (Fourier transformation of corresponding ...
2
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0answers
207 views

Fractional Derivatives [closed]

How far these Theories of "Fractional Derivatives" be rigorized ? I have few books and references on Fractional Differential Equations etc (mainly they stress on Applied Mathematics parts and similar ...
9
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2answers
2k views

“is topologically mixing” vs. “is topologically transitive” in the defition of chaos

This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours. Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits" as the ...
5
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1answer
357 views

What is the probability of an arbitrary nonlinear dynamical system to be chaotic?

Particularly, how to characterize a set of chaotic nonlinear dynamical systems as a subset of nonlinear dynamical systems with respect to the set cardinality? To explain the question more, a simple ...
22
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4answers
2k views

Non-chaotic bouncing-ball curves

I was surprised to learn from two Mathematica Demos by Enrique Zeleny that an elastic ball bouncing in a V or in a sinusoidal channel exhibits chaotic behavior:     (The Poincaré map ...
0
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0answers
145 views

cat map re-transformation

Hi, Is there any way of moving from one cat map transformation to the other without resetting parameters? For example, suppose you have two matrices '$A$'and '$B$' each permuted with different cat ...
12
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2answers
1k views

Is there any expression for the Feigenbaum constants ?

It has puzzled me for a long time that the Feigenbaum constant $\delta$ and reduction parameter $\alpha$ do not seem to be related to other constants (that is, numerically), even not to each other. In ...
1
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2answers
1k views

Lyapunov Exponent and degree of chaos

I am aware that having positive Lyapunov exponents in a system signifies that a system is chaotic. However, I would like to know if there is a means to know the degree of chaos in the system from the ...
3
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5answers
2k views

Recommended book for introduction to Chaotic dynamics? (application in probability distributions)

I'm just starting some research and I need a good introductory book in the topic of chaotic dynamics. Does anyone have a suggestion? Thanks.