# Questions tagged [chaos]

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### 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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### 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 $\omega$...
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### 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 ...
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### 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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### 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 \...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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 ...
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### 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 "...
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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 ...
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### 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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### 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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### 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 $H$=$L^2(\mathbb R)$...
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### 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 ...
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### 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 ...
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### “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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...