Questions tagged [chaos-theory]
For questions in chaos theory.
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Why Lorenz attractor can be embedded by a 3-step time delay map?
I'm investigating attractor reconstruction of Lorenz system. I saw a bunch of work claiming that the time delay map $[x(t), x(t -\tau), x(t - 2\tau)]$ is sufficient to reconstruct the attracotr, e.g. ...
-1
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0answers
18 views
Stability Analysis after finding derivative [closed]
How do I analyse the stability of a function once I have found its derivative, which is a linear equation"
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1answer
34 views
Question about Chaos Theory and Relation to Game…
I am working on a project (for social sciences) and I found a game, Parable of Polygons (link)
I observe chaotic behavior, but am unable to mathematically explain this. Are there any ideas on how the ...
0
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0answers
39 views
If the solutions to Lorentz equation diverge exponentially, how can they be confined in a strange attractor? [duplicate]
In the book of Chaosbook, at the beginning of chapter 6, it is given that
[...]if the attractor is strange, any two trajectories $x(t) =
f^t(x_0)$ and $x(t)+\delta x(t) = f^t(x_0 + \delta x_0)$ ...
7
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1answer
105 views
Does the existence of a superstable fixed point imply a $-\infty$ Lyapunov exponent?
This is the Lyapunov exponent as a function of $r$ for the logistic map ($x_{n+1}=f(x_n)=r(x_{n}-x_{n}^2)$)
The big dips are centered around points where $f'(x)=0$ for some $x$ in the trajectory used ...
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0answers
11 views
Application of Correlation Dimension to Fractals other than Sets of Points
In Chaos Theory, the Correlation Dimension is defined to calculate the dimension of fractals.
At least in the context where I've learnt it, it is applied to fractals made up of sets of points.
Is it ...
0
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0answers
9 views
Id $f,g$ orientation preserving cricle-diffeomorphisms, then $\rho(g^{-1}\circ f \circ g) = \rho(f)$.
Let $F, G : \mathbb{R} \rightarrow \mathbb{R}$ be a lift of $f$ and $G$ of $g$. That is, $ \pi \circ F = f \circ \pi$ with $\pi(x) = e^{2\pi i}$. We define $$\rho_{0}(F) = \lim_{n\rightarrow \infty}\...
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1answer
47 views
Chaotic system?
I'm currently working on an individual project. In which I have created a program to simulate a particle bouncing around a finite bordered region.
Where I need help in understanding how I might show ...
0
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1answer
34 views
Can a map be both nonchaotic and chaotic?
In Wikipedia’s article ‘List of chaotic maps’ (https://en.wikipedia.org/wiki/List_of_chaotic_maps), one of the entries is:
Feigenbaum strange nonchaotic map
So, how is this possible?
2
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0answers
42 views
Sard's theorem for orientation preserving diffeomorphism of the circle
thanks in advance for helping me. First I'll introduce some definitions:
(1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving ...
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0answers
33 views
Dynamics $\delta x(t)=\delta x(0) e^{\lambda t}$ of Henon Attractor
Recall the question I asked before:
Linearized perturbation dynamics of Henon Attractor
So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$\delta x(t)=\delta x(0)...
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0answers
37 views
Fixed Points and their Stability and Finally Bifurcation diagram
Find and Classify all fixed points and sketch vector field. What's the bifurcation?
$$ 𝑥̇ = 𝑟𝑥 − \tan(𝑥)$$
4
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2answers
54 views
Dynamic system $f(x) = 2x$ mod $1$
I am reading the following paper:
ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded)
My question is from an example on p....
0
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1answer
32 views
Time dependence in chaos theory [closed]
One of the best, more original examples of chaos theory comes from Ray Bradbury’s A Sound of Thunder. People in the chaos-theory community have always said that the hunters that time safari allowed to ...
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0answers
31 views
Hypercyclic operators in $L_p (0,\infty)$
I'm pretty new to this area of study so if there are logical lacune in my proof (I'm sure there are many) please let me know.
This is material I'm self studying. I'm trying to adapt the methods used ...
2
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0answers
46 views
Although Lorenz system is a deterministic system, can it shows locally stochastic behavior?
The Lorenz system is a system of ordinary differential equations
$$\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x),\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y,\\{...
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0answers
36 views
Perron-Frobenius operator definition
According to
http://mathworld.wolfram.com/Perron-FrobeniusOperator.html, the P-F operator describes the time evolution of densities in phase space:
$$\rho_{n+1} = P (\rho_n).$$
But for the ...
0
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1answer
52 views
Hypercyclicity examples
Does anyone have simple practical examples of hypercyclicity they use in explaining the concept (graphically or numerically)? This appears often in texts about chaos in infinite dimensional linear ...
1
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1answer
39 views
Does a greater Lyapunov exponent result in a more chaotic system?
Let $\sigma_1$ be the Lyapunov exponent of a one-dimensional system $I_1$ and $\sigma_2$ be Lyapunov exponent of one-dimensional system $I_2$. Can I say that if $\sigma_1 > \sigma_2$, then $I_1$ is ...
4
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1answer
41 views
Periodic solutions of the double pendulum
I'm stuck: Are there periodic solutions of the double pendulum, or not?
The question is four-fold:
Does every double pendulum - defined by two masses $m_1$, $m_2$ and lengths $l_1$, $l_2$ - have ...
5
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1answer
65 views
How to calculate the envelope of the trajectory of a double pendulum?
Consider a double pendulum:
Background
For the angles $\varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$):
$\dot{\varphi_1} = 6\frac{...
2
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0answers
66 views
Project ideas on Chaos theory, Cellular Automata, Fractals, Games, IA [closed]
I'm a computer science student and I need to find a final year project. What interests me the most is Chaos, IA, Games, Fractals, CA..
Something I liked was the chaos theory within sudoku.
The ...
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0answers
9 views
How to calculte the Fourier Transform of a sovable chaos waveform?
Recently I am stucking in frequency estimation of a solvable chaos waveform.
Its analytic expression in time domain is
$$
z(t)=s_m(u_m-s_m)e^{\beta(t-mT)}\cos(\omega_0 t+\varphi)
$$
where $u_m \sim U(-...
0
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1answer
41 views
Let $M$ be a shift space over a finite alphabet $\mathcal{A}$. Prove that $M$ is compact in the metric topology $\tau_{\rho}$.
GIVEN
Define a map
\begin{equation*} \label{eq1}
\begin{split}
\rho(x,y) =
\begin{cases}
2^{-k} \ \ \ &\text{if } x \neq y, \text{ and } k \text{ is maximal so that } x_{[-k.k]} = y_{[-k,k]...
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0answers
39 views
Is $\tan(x)$ chaotic on the entire real line?
In Robert L. Devaney's "An Introduction to Chaotic Dynamical Systems" it defines a function $f:J\longrightarrow J$ to be expansive if there exists $\nu>0$ such that, for any $x,y\in J$, there ...
5
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2answers
175 views
Existence of infinite iteration of functions $f_\infty$?
Given a sequence of functions $\{f_n\}$ satisifying an iterated relation such as
$f_n(x)=g(x+f_{n-1}(x))$
$f_n(x)=g(xf_{n-1}(x))$
$f_n(x)=g(x/f_{n-1}(x))$
Where $g:=f_1$ is ...
1
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1answer
105 views
Diffeomorphism with only hyperbolic periodic points has finitely many periodic points (Morse-Smale)
Came across the following question during a course Chaotic Dynamical Systems:
If a diffeomorphism $f:I\to I$ is Morse-Smale (i.e. has only hyperbolic periodic points), then it has finitely many ...
1
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1answer
116 views
How reliable a measure of chaos is the largest Lyapunov exponent?
I’m dealing with a system of ODEs where I think I might have found chaotic solutions. I’ve calculated the largest Lyapunov exponent, and found it to be approximately 0.02. It’s positive, which ...
3
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1answer
57 views
Are there non-chaotic systems which exhibits topological mixing?
I was recently reading about chaos theory. Chaos is commonly defined to exhibit 3 properties:
Sensitivity to initial conditions/Lyapunov coefficient is positive
Exhibits topological mixing
Dense ...
4
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1answer
303 views
Fractal Dream Attractor behavior
This question is similar to this one: Properties of King's Dream fractal
The fractal is described here: Softology - Visions Of Chaos 2D Strange Attractor Tutorial
I accidentally modefied the formula ...
2
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2answers
127 views
Can Brownian motion be regarded as chaos?
I want to know if it is included in chaos. Does it have boundedness, deterministic, initial value sensitivity that is characteristic of chaos?
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0answers
53 views
What is the name of this chaotic sequence?
A long time ago I came across a chaotic sequence of the form
x[n+1] = f(x[n], y[n])
y[n+1] = g(x[n], y[n])
If I remember correctly, ...
0
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0answers
43 views
Examples of chaotic motion in non-deterministic dynamical systems?
I was reading this article on Hyperbolic dynamics, in which I came across the line "deterministic chaos - the appearance of chaotic motions in purely deterministic dynamical systems".
Hence I wonder ...
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0answers
117 views
Proof check: Show that $x_{n + 1} = f(x_n) = 15 - x_n^2$ is a (Smale) horseshoe
A solution is given here:
However, I believe it to be incorrect because $f(K_1) = f(K_2) \neq (-\sqrt{15}, \sqrt{15})$.
Attempt at a solution:
Referring to the diagram from the question...
By the ...
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0answers
30 views
To classify bifurcations by calculating partial derivatives evaluated at the bifurcation point
Question: Given a one dimensional system $\dot{x} = f(x;\; \mu)$ for which a bifurcation of fixed points occurs at $(x^\star, \mu_c)$, where $x^\star$ is a fixed point and $\mu_c$ is the ...
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1answer
26 views
Determining Rate of Convergence for Chaotic Behavior
I've got a set of data that's been manipulated solely numerically. But anyway, for a certain part of that data set, I see that it's convergent (image). But I'm sort of at a loss for how depict that ...
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0answers
16 views
If a system is defined in $n$-dimension then it has $n$ Lyapunov exponents
I wanted to know if there is a formal proof to show that if a system is defined in $n$-dimension then it has $n$ Lyapunov exponents. Any links regarding information about this and Kaplan-Yorke ...
2
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1answer
55 views
To find the approximate period and approximate equation of the limit cycle for a system with a Hopf bifurcation
Question: The system
$$ \dot{x} = 3y + 3x^3 + xy $$
$$\dot{y} = -3x + \mu y + 2xy^2 - y^3$$
undergoes a Hopf bifurcation at $(x, y) = (0, 0)$ as $\mu$ passes through 0. Calculate the approximated ...
5
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1answer
132 views
Do all systems of differential equations have a solution?
I'm not really sure that I have the required knowledge to pose this question properly so I'll just pose it poorly.
The rise of numerics has allowed us to study systems of differential equations for ...
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2answers
133 views
Show that the origin of the following system is stable (non-linear vs linear stability)
$$\dot{x} = -2y -x^3 + x^2 y^2$$
$$\dot{y} = x - x^2 y$$
Jacobian evaluation at $(0, 0)$:
$$\begin{pmatrix}0 & -2\\ 1 & 0\end{pmatrix}$$
Clearly the determinant is greater than zero but the ...
0
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1answer
67 views
What is the “bifurcation of a fixed point”?
Isn't the bifurcation of a system, say $\dot{x} = \mu \sin(x) - 2x$, related to the system rather than a fixed point, say $x^{\star} = 0$?.
My understanding is that a bifurcation occurs when there's ...
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0answers
112 views
Understanding the definition of Sensitive dependence on initial conditions?
I was trying to understand the rigorous definition of sensitive dependence on initial conditions which is as follows -
$f : X \mapsto X$ where $X$ is a metric space.
If there exists $\epsilon > ...
10
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1answer
167 views
Lorenz Attractor, its Geometric Model, and 14th Smale's problem.
I've found a post with a beautifully animated video that states the following:
In 2001 mathematician Warwick Tucker proved that the paper model accurately describes the motion on the Lorenz ...
0
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0answers
110 views
Understanding fixed point of Arnold's Cat Map
I'm currently performing a small investigation into the properties of Arnold's Cat Map, a function which is defined as:
C:[0,1) -> [0,1)
C(x,y) = (2x+y, x+y) (mod 1)
This function has exactly one ...
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0answers
63 views
A question on Smale-Birkhoff homoclinic theorem
Does Smale-Birkhoff homoclinic theorem ("given a transverse homoclinic intersection for a diffeomorphism $f$, there exists an integer $n \geq 1$ such that $f^{n}$ has an invariant Cantor set on which ...
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0answers
50 views
Measure of the periodic orbits of this simple dynamical system
Consider the dynamical system $\phi$ on a phase space $M =\{(x,y): x \in [0, 2\pi), y \in [0,\pi)\}$ where we take $x(mod 2\pi)$, i.e. $M$ is a cylinder. Define $(x_{1}, y_{1}) = \phi(x_{0}, y_{0})$ ...
0
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1answer
70 views
Does positive topological entropy imply horseshoes?
I am aware that positive topological entropy implies chaos in the sense of Li-Yorke. I want to get an idea of what consequences positive topological entropy has for the the presence (or lack of) ...
4
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0answers
53 views
Peaks in the circle chaos game
The chaos game is a random walk that steps halfway between your current position and a set of predefined points. If the points are on an equilateral triangle, the resulting set is Sierpinski's ...
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0answers
164 views
What can linear stability analysis say about trajectories starting in the neighbourhood of a fixed point? The role of Arnold diffusion
Consider a Hamiltonian dynamical system having more degrees of freedom than invariant quantities (Suppose that the number of degrees of freedom is $D>1$ and that the only invariant quantity is the ...
0
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0answers
31 views
Differences in the eigenvalue distribution between “regular” and chaotic domains
Weyl's conjecture gives us the asymptotic distribution of the eigenvalues $\lambda$ of the Laplace operator in a domain $\Omega$ with homogeneous Dirichlet boundary condition on $\partial \Omega$:
$$
...
