# Questions tagged [chaos]

The chaos tag has no usage guidance.

42
questions

**6**

votes

**0**answers

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+\...

**2**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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 ...

**0**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**3**answers

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**

vote

**1**answer

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**

votes

**0**answers

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 ...

**0**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**2**answers

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

**1**answer

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**

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^...

**1**

vote

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**2**answers

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 ...

**1**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**4**answers

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**

votes

**0**answers

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**

votes

**2**answers

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**

vote

**2**answers

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**

votes

**5**answers

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.