Questions tagged [chaos]
The chaos tag has no usage guidance.
67 questions
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Analysis of sensitivity to initial conditions in dynamic systems
Consider the iterative function defined by:
$$
x_{n+1} = f(x_n)
$$
where $x_0\in [0, 1]$ and
$$
f(x) = \sin\left(\pi \left(b^{rx(x-1)}\mod 1 \right)\right)
$$
with $b, r > 0$. We aim to demonstrate ...
- 11
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170
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Bounding proportion of phase space which is chaotic
There are dynamical systems which have regions of phase space that are both chaotic and integrable, e.g. small perturbations of integrable systems as in KAM theory. Are there any tools for bounding ...
- 509
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207
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Inverse problems and chaos theory
In the classical theory of inverse problems we want to recover an unknown $u \in U$ from its noisy measurements $y \in L^2$, where $U$ is a Banach space. In particular, we study the following problem:
...
- 43
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202
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Finding a two point scrambled set for the function $g:[0,1] \rightarrow [0,1], x \mapsto \min_{n\in \mathbb{Z}} |3x-2n|$?
Let $I=[0,1]$ be the unit interval and $g$ as defined below.
Then $x \neq y$ with $x,y \in I$ are called "two point scrambled set"=$\{x,y\}$, if
$\lim\inf_{n \rightarrow \infty} | g^{(n)}(x)...
- 3,032
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120
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coupled discrete dynamical system -- bifurcation analysis
Suppose you have the following coupled discrete dynamical system:
\begin{align*}
e_{k+1}&=e_k - 2~\alpha~e_k~\lambda^2~\alpha_k^2 + \alpha^2~e_k^2~\lambda^3 \alpha_k^3\\
\alpha_{k+1}&= \...
- 519
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18
views
Nature of unbounded initials in polynomial symplectic maps
Is the following statement true? How it can be proved/rejected?
Initial conditions that correspond to unbounded orbits in polynomial symplectic mappings, which exhibit chaotic behavior (exponential ...
- 101
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100
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Chaotic behaviour of the secant method for $\sin(x)$
For not very serious reasons I was trying to understand the behaviour of the secant method for solving $\sin(x)=0$ starting with $x_0=2$ and $x_1=18$, so
$$ x_{n+2}=x_{n+1}-\sin(x_{n+1})\frac{x_{n+1}-...
- 56.9k
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0
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71
views
How to analyze a nonlinear time series dataset?
I have a time series that appears chaotic that I would like to analyze with Python. To draw its logistic map, I must use the logistic equation: $$x_{t+1}=rx_{t}(1-x_{t})$$
I have the data in a text ...
- 1
3
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0
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49
views
Kaplan-Yorke dimension when all Lyapunov exponents are positive or the system is 1-dimensional
The Lyapunov/Kaplan-Yorke dimension $D_{KY}$ can be calculated using the Lyapunov Exponents of an n-dimensional dynamical system, as shown in the relevant Scholarpedia entry.
If $\lambda_1>\...
2
votes
2
answers
252
views
Devaney chaos and topological entropy
I am searching for dynamical systems on compact spaces which are Devaney chaotic but have topological entropy zero. On the interval such systems do not exist. I think on the Cantor space and on the ...
- 1,648
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votes
6
answers
1k
views
Radial behavior of dynamical map $x_{n+1}=2x_ny_n$, $y_{n+1}=1-2x_n^2$
Consider the sequence in the unit disk $D=\{(x,y)\,|\,x^2+y^2\leq 1\}$ iteratively defined by the quadratic map $$\begin{aligned} x_{n+1}&=2x_ny_n\\y_{n+1}&=1-2x_n^2\end{aligned},$$
starting ...
2
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0
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66
views
Iterated chaos expansion
Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2
random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$,
$$E[X(h)X(g)] = \...
- 623
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Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero
Using the following definition of Dirichlet L-function
$$
L(1,\chi)=\begin{cases}
\dfrac{2\pi h}{w\sqrt{m}} & \textit{if}\ \chi(-1)=-1 \\\\
\dfrac{2 h \log{|\epsilon|}}{w\sqrt{m}} & \textit{...
6
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1
answer
328
views
Solution of an ODE upon singular perturbation
The following question originates from a Physics problem, so I apologize if I am not using a suitable mathematical jargon.
The original system involves $N$ massless electric charges at position $\...
- 255
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128
views
When is a composition of homeomorphisms topologically transitive provided one of the two is?
Suppose that $S$ is a (connected) regular surface, possibly with boundary, for instance an annulus. Suppose that both $f$ and $g$ are homeomorphisms whose restriction to $\partial S$ is the identity. ...
- 4,459
8
votes
1
answer
355
views
State of the art on: "If a dynamical system is Li-Yorke chaotic, does there exist a Cantor scrambled set?"
The following problem is presented in the paper Recent development of chaos theory in topological dynamics - by Jian Li and Xiangdong Ye :
"If a dynamical system [$(X,f)$, $X$ metric space, $f$ ...
- 339
2
votes
0
answers
55
views
Search for period N logistic map
The logistic map is a period doubling bifurcation system.
Are there known dynamical maps, which oscillate between $N$ points where $N$ is a prime number, like 2, 3, or 5, or 7... , where each ...
- 155
2
votes
1
answer
148
views
Critical Reynolds numbers for turbulence in 3D and 2D planar Couette flows
In 3 spatial dimensions, the incompressible Navier-Stokes equations are:
$$
\begin{split}
\frac{\partial u_i}{\partial t} + \sum_{j=1}^3 u_j \frac{\partial u_i}{\partial x_j} &= - \frac{\partial p}...
- 1,547
2
votes
1
answer
197
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 ...
- 21
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1
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196
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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 ...
- 1,547
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80
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 $C_f$ ...
- 5,407
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votes
0
answers
292
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 ...
- 1,547
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 ...
- 1,547
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votes
0
answers
144
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 ...
- 1,547
5
votes
2
answers
277
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 ...
- 649
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votes
1
answer
280
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
$$M(X,T) := ...
8
votes
1
answer
647
views
How to analytically prove chaos
Consider the following map
\begin{align*}
T \colon \mathbb{R}\times\mathbb{S}^1 \to & \mathbb{R}\times\mathbb{S}^1 \\
(x,\theta) \mapsto & \left(\frac{x}{4}+ \sin^2\left(\pi\left(\theta+\...
2
votes
0
answers
116
views
Birth of chaos due to nonautonomous perturbation
Let $\sigma, b>0$. I want to study the dynamics of the map
$$ 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
72
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 ...
- 141
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0
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64
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 ...
- 141
2
votes
0
answers
73
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 ...
- 141
2
votes
2
answers
210
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
130
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:
$$
\dot{\delta x} = \sum_j \frac{\partial}{\partial x_j} F_i(...
- 469
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votes
0
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236
views
Rigorous results on chaos in a driven damped pendulum
The harmonically driven damped pendulum is often used as a simple example of a chaotic system, the equation is just $\ddot{\phi}+\frac1q\dot{\phi}+\sin\phi=A\cos(\omega t)$. As long as $A$ and $\omega$...
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votes
1
answer
551
views
Is the logistic map $x_{n+1}=r x_n (1-x_n)$ exactly solvable for any $r$ other than $-2,2,4$?
It is known that for $r=-2,2,4$ the logistic map $x_{n+1}=r x_n (1-x_n)$ has exact solutions of the form
$$
x_n=\frac12 \left\{ 1- f\left(r^n f^{-1}(1-2x_0)\right)\right\} \qquad \qquad{(*)}
$$
for ...
- 43
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votes
2
answers
421
views
Why are attempts to define chaos with discrete states so scarce?
Interestingly, the theory of nested recurrence relations has been correlated with “discrete chaos” by Golomb (1991) and Tanny (1992).
And in literature, there are very few studies that have different ...
- 701
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255
views
Question regarding Ito representation theorem
Let $H$ be a Gaussian Hilbert space and $H^{:n:}$ be the homogeneous chaos of order $n$.
and let $D_n:=\{(t_1,\cdots,t_n):t_1<t_2<\cdots <t_n\}$.
For each $n\geq 0$ there exists an isometry
\...
- 515
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votes
1
answer
134
views
Consequences of invariant-subspace problem to Li–Yorke chaos [closed]
The invariant-subspace problem is probably an open problem for reflexive spaces which asks:
Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial ...
7
votes
2
answers
901
views
Is this a new strange attractor?
I recently made some experiments in programming strange attractors, and I found this (very simple) equations, which create a nice strange attractor:
...
- 101
4
votes
1
answer
342
views
Questions about a return map
Consider the following map in the interval $u\in[-1,1]$ ($U\in[-1,1]$ also)
$$U=u-\frac{64}{3}u^{3}+64u^{5}-64u^{7}+\frac{64}{3}u^{9}$$
It has 3 fixed points at $u=0,\pm 1$. If we compute the ...
- 213
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votes
3
answers
278
views
Example of a Chaotic discrete dynamical system in dimension 2
I am looking for examples of discrete dynamical systems in dimension 2 that are :
1) Chaotic dynamical system in Devaney's sense in dimension 2 ?
2) Chaotic dynamical system in Li-Yorke sense but ...
- 51
1
vote
1
answer
237
views
Formal justification of the Chaos game in the Sierpinski triangle
I want to justify why the Chaos game works to produce Sierpinski triangle. I use a theorem taken from Massopust Interpolation and Approximation with Splines and Fractals.
Suppose that $(X,d)$ is a ...
- 439
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votes
0
answers
108
views
Curious if this chaotic recurrence relation has a name and/or interesting properties
I was playing around with 2-dimensional recurrence relations and I stumbled upon this:
$$
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 ...
- 131
0
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0
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94
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 ...
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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 ...
2
votes
0
answers
99
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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 ...
- 61
2
votes
0
answers
490
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 ...
- 61
39
votes
2
answers
3k
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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 "...
- 589
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1
answer
145
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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 ...
- 318
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votes
1
answer
2k
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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^...
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