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Questions tagged [chaos]

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About topologically mixing functions

Let us consider a sequence of continuous functions $g_{q}:R^2\to R^2$. Let $(A_{q})_{q\geq 1}$ be a sequence of compact sets in $R^2$. Assuming that each function $g_{q}$ is topologically mixing in $...
2
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0answers
63 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
25 views

(Upper) bounding the MGF of a semi-decoupled non-homogeneous Rademacher chaos of order 4

Let $(\xi_i)_{1\leq i\leq n}$, $(\xi'_i)_{1\leq i\leq n}$ be two independent vectors of independent Rademacher random variables, and $(a_{ij})_{1\leq i,j\leq n}$, $(b_{ijk})_{1\leq i,j,k\leq n}$, $(c_{...
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0answers
73 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
797 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 ...
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0answers
67 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 ...
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0answers
118 views

Why according to Takens' theorem (1981) we need 2m+1 observation functions (or time series) to reconstruct the attractor?

In the paper [1], page 369 we have Theorem 1 which says: Let $M$ be a compact manifold of dimension $m$. For pairs $(\phi,y),\;\phi: M\to M $ a smooth diffeomorphism and $y: M \to \mathbb{R}$ is a ...
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0answers
130 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 ...
38
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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
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1answer
111 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
361 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
190 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
130 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
315 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
39 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
48 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 $...
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0answers
141 views

Is the Rossler attractor globally stable?

The standard form of the Rossler system is $\frac{dx}{dt}=-y-z$, $\frac{dy}{dt}=-x+ay$, $\frac{dz}{dt}=b+z(x-c)$. For simplicity, consider the well known chaotic attractor that exists at a=b=0.2, c=5....
7
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2answers
222 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
137 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 ...
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1answer
286 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
215 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
184 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 ...
8
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2answers
1k 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
307 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 ...
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4answers
1k 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
144 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 ...
11
votes
2answers
685 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.