Questions tagged [chaos]

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How do I know if my time series are chaotic?

For my PhD in physics I'm currently analyzing simulation results to identify chaotic time series, but learning chaos theory is a bit challenging for me.. What would be the most effective method to ...
Elijah14's user avatar
4 votes
0 answers
53 views

Can the Wiener Chaos expansion converge in $L^p$?

For a function $F$ satisfying $|F(x)|\le C(1+|x|)^M$ for some $C,M>0$, and a mollifier $\rho$, we define $F_{\epsilon}=F\ast\rho_{\epsilon}$, where $\rho_{\epsilon}(x)=\epsilon^{-1}\rho(\frac{x}{\...
why's user avatar
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12 votes
6 answers
1k views

Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$

Consider the sequence in the unit disk $D=\{(x,y)\,|\,x^2+y^2\leq 1\}$ iteratively defined by the quadratic map $$\begin{aligned} x_{n+1}&=2x_ny_n\\y_{n+1}&=1-2x_n^2\end{aligned},$$ starting ...
Saúl Pilatowsky-Cameo's user avatar
2 votes
0 answers
60 views

Iterated chaos expansion

Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2 random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$, $$E[X(h)X(g)] = \...
Julian's user avatar
  • 613
1 vote
0 answers
1k views

Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero

Using the following definition of Dirichlet L-function $$ L(1,\chi)=\begin{cases} \dfrac{2\pi h}{w\sqrt{m}} & \textit{if}\ \chi(-1)=-1 \\\\ \dfrac{2 h \log{|\epsilon|}}{w\sqrt{m}} & \textit{...
zeraoulia rafik's user avatar
6 votes
1 answer
285 views

Solution of an ODE upon singular perturbation

The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon. The original system involves $N$ massless electric charges at position $\...
AndreaPaco's user avatar
2 votes
0 answers
117 views

When is a composition of homeomorphisms topologically transitive provided one of the two is?

Suppose that $S$ is a (connected) regular surface, possibly with boundary, for instance an annulus. Suppose that both $f$ and $g$ are homeomorphisms whose restriction to $\partial S$ is the identity. ...
Alessandro Della Corte's user avatar
8 votes
1 answer
280 views

State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"

The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye : "If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...
Marco Farotti's user avatar
2 votes
0 answers
52 views

Search for period N logistic map

The logistic map is a period doubling bifurcation system. Are there known dynamical maps, which oscillate between $N$ points where $N$ is a prime number, like 2, 3, or 5, or 7... , where each ...
Sean's user avatar
  • 155
2 votes
1 answer
130 views

Critical Reynolds numbers for turbulence in 3D and 2D planar Couette flows

In 3 spatial dimensions, the incompressible Navier-Stokes equations are: $$ \begin{split} \frac{\partial u_i}{\partial t} + \sum_{j=1}^3 u_j \frac{\partial u_i}{\partial x_j} &= - \frac{\partial p}...
bobuhito's user avatar
  • 1,537
2 votes
1 answer
189 views

Chaotic complex dynamics and Newton's method

I'm trying to understand Harold E. Benzinger, Scott A. Burns, and Julian I. Palmore's work on one-parameter families of Julia sets arising from Newton's method in the complex domain. They show the ...
Il Padrino's user avatar
3 votes
1 answer
140 views

Stable periodic orbits for three equal masses

For three equal masses in any number of dimensions (this might not be important, but 2D or 3D or 4D is fine) under just classical gravity (i.e., inverse-square force law), what stable periodic orbits ...
bobuhito's user avatar
  • 1,537
0 votes
0 answers
78 views

Cyclicity of composition operators

Let $E=C([0,1]^m,\mathbb{R}^n)$ where $K=[0,1]^m$ where $E$ has the compact convergence topology. Recall that for a function $f:[0,1]^m\rightarrow [0,1]^m$ the associated composition operator $C_f$ ...
ABIM's user avatar
  • 5,047
5 votes
0 answers
285 views

Fastest sine of a large power of 2

What is the fastest known way to calculate $\sin(2^{n})$ for large integer $n$? I only need the highest few bits to be correct. I suspect that the compute time required scales with $n$ (and actually ...
bobuhito's user avatar
  • 1,537
7 votes
3 answers
1k views

Proven chaos in logistic maps

For a logistic map using $f_r(x)=rx(1-x)$, what values of $r$ and starting $x$ are guaranteed (i.e., with an accepted proof) to be chaotic? I mean "chaotic" in the loosest sense: The ...
bobuhito's user avatar
  • 1,537
3 votes
0 answers
143 views

2-ball billiards in a circle

Consider a 2D circular billiards table with diameter 1m containing two balls with diameter 0.25m. Let each ball start with a speed of 1m/s. In general, this speed could change after the balls hit ...
bobuhito's user avatar
  • 1,537
5 votes
2 answers
227 views

Examples of different levels of the ergodic hierachy (specifically: weakly mixing & merely ergodic)

I am interested in generalizing some aspects of the ergodic hierarchy (of classical dynamical systems) to quantum theory. However, while I understand the definitions of the different levels of the ...
Victor Galitski's user avatar
0 votes
1 answer
246 views

Metric entropy and topological entropy

It is well known that, for a dynamical system $T$ on a metric space $(X,d)$, the variational principle connects the definition of metric entropy and topological entropy. In other words, if $$M(X,T) := ...
Giuseppe Tenaglia's user avatar
8 votes
1 answer
560 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+\...
Giuseppe Tenaglia's user avatar
2 votes
0 answers
110 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) = (\...
Giuseppe Tenaglia's user avatar
0 votes
0 answers
64 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 ...
Lo Scrondo's user avatar
0 votes
0 answers
61 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 ...
Lo Scrondo's user avatar
2 votes
0 answers
55 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 ...
Lo Scrondo's user avatar
2 votes
2 answers
204 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 ...
Carlos Giraldo's user avatar
1 vote
1 answer
107 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(...
Doriano Brogioli's user avatar
9 votes
0 answers
207 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 $\omega$...
Conifold's user avatar
  • 1,599
4 votes
1 answer
520 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 ...
visitor's user avatar
  • 43
2 votes
2 answers
355 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 ...
Alkan's user avatar
  • 681
0 votes
0 answers
209 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 \...
Chaos's user avatar
  • 485
0 votes
1 answer
128 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 ...
zeraoulia rafik's user avatar
7 votes
2 answers
821 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: ...
klangforscher's user avatar
4 votes
1 answer
333 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 ...
user2175783's user avatar
3 votes
3 answers
242 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 ...
kiki 's user avatar
  • 51
1 vote
1 answer
221 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 ...
user1868607's user avatar
3 votes
0 answers
100 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 ...
Matt Diamond's user avatar
0 votes
0 answers
89 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 ...
AndreaPaco's user avatar
4 votes
2 answers
2k 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 ...
Morteza Azad's user avatar
2 votes
0 answers
96 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 ...
Chopin's user avatar
  • 61
2 votes
0 answers
455 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 ...
Chopin's user avatar
  • 61
39 votes
2 answers
3k 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 "...
ShnitzelKiller's user avatar
2 votes
1 answer
140 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 ...
asldjk's user avatar
  • 318
4 votes
1 answer
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^...
Qijun Tan's user avatar
  • 577
1 vote
1 answer
271 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 ...
SKM's user avatar
  • 135
2 votes
0 answers
164 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 ...
BAYMAX's user avatar
  • 51
6 votes
0 answers
335 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 ...
Evgeny's user avatar
  • 165
3 votes
0 answers
46 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 ...
Arnold Neumaier's user avatar
2 votes
0 answers
66 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 $...
Amir Sagiv's user avatar
  • 3,536
7 votes
2 answers
360 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 ...
demitau's user avatar
  • 225
1 vote
0 answers
195 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 ...
np20's user avatar
  • 111
-3 votes
1 answer
316 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 $H$=$L^2(\mathbb R)$...
zeraoulia rafik's user avatar