Questions tagged [chaos-theory]
For questions in chaos theory.
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How can I calculate Feigenbaum's constant?
So I am trying to calculate Feigenbaum's constant for the logistic map: $$ x_{n+1} = 4 \lambda x_n (1-x_n) $$
I am writing this through python and the main pieces I have for my code that are relevant ...
0
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0answers
26 views
Why does Chaos Improve Evolutionary Algorithms? [on hold]
I have presented an evolutionary algorithm using chaos theory (chaotic numbers) to solve an optimization problem. The results of the experiments show that this algorithm is much better than the same ...
0
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0answers
32 views
How to generate chaos?
I would like to generate chaotic behavior that would have reastic 1/f noise properties.
For example, a time-series of price fluctuations or pendulum oscillations that would have chaotic properties.
...
0
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2answers
31 views
Bernoulli Map: $f'(x) = 2$ Almost Everywhere and “Local Separation” Increases as $2^n$
The Bernoulli map is
$$x_{n + 1} = f(x_n)=
\begin{cases}
2x_n, & 0 \leq x_n < 0.5\\
2x_n - 1, & 0.5 \le x_n \le 1
\end{cases}$$
I am told that (1) $f'(x) = 2$ almost everywhere and so (...
2
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1answer
57 views
The Unpredictability of Chaos: An Intrinsic Property, or a Matter of Computing Power and/or Mathematical Knowledge?
I was reading that Chaos (in the mathematical sense) is deterministic, but not predictable. Is this unpredictability an intrinsic property of Chaos, or is it a practical matter of computing power and/...
1
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1answer
37 views
Applying a linear transformation to a system of differential equations
I am reading the book: Nonlinear Oscillations, Dynamical System and Bifurcations of Vector Fields (Guckenheimer and Holmes), chapter 2: An introduction to chaos.
About Van der Pol's equation, it can ...
2
votes
1answer
34 views
Equivalent condition of a diffeomorphism having a dense orbit
Say $M$ is a manifold and $f: M \to M$ is a diffeomorphism. Assume also that, if we are given any nonempty open subsets $U$ and $V$, then there is $n \in \mathbb{Z}$ such that $f^n(U)$ intersects $V$....
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Does the property of uniquely ergodic imply that the map has a unique chaotic attractor for all $c$ in $D$
Definition: Let $(X,B)$ be a measurable space and let $T:X→X$ be a measurable transformation. If there is a unique $T$-invariant probability measure then we say that $T$ is uniquely ergodic.
Consider ...
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0answers
37 views
How to find bifurcation value from a potential
Just needed some guidance to see if my approach is correct.
The following is how I proceeded to do this problem. What seems to be confusing me is that it is in the potential form and not $\dot{x} = ......
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0answers
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Is the infinite product map $(∏_{i=1}^{∞}S_{i})×f$ topologically transitive
Let $f:X→X$ be a map. We say that $f$ is topologically mixing if for every open subsets $U,V$ of $X$, there exists $N$ such that for every $n≥N$ the set $f^{n}(U)∩V$ is non-empty.
Let $S : X → X$ and ...
2
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1answer
22 views
Is the Lorenz system well-posed in the Hadamard sense?
Apologies if this has already been discussed, but I searched the site and I couldn't find an answer.
For the sake of simplicity, consider only ODEs, possibly depending on some vector of parameters $\...
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0answers
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Can a family of functions $F_{\lambda}$ experience a saddle-node bifurcation and also experience a period -doubling bifurcation?
Suppose $F_{C}=(x-C)^{2}$ is a family of functions. I found that at $C=-\frac{1}{4}$ there is a saddle-node bifurcation. So for $C<-\frac{1}{4}$ there are $0$ fixed points, $C=-\frac{1}{4}$ there ...
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0answers
44 views
About topological conjugacy
The Smale horseshoe map $f$ is desribed in this page: What's the point of a Horseshoe map?
A striking feature of this system is the stability of its dynamics: given any diffeomorphism $g$ ...
2
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0answers
173 views
Is there a meaningful link between the golden ratio and chaos theory?
I heard it casually mentioned by strangers but am unable to find any information about this.
Is there a meaningful link between the golden ratio and chaos theory?
The closest thing I could find is ...
1
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2answers
49 views
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. ...
0
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1answer
37 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 ...
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0answers
40 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
votes
1answer
127 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 ...
0
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0answers
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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 ...
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0answers
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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 $$\rho_{0}(F) = \lim_{n\rightarrow \infty}\...
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1answer
48 views
Chaotic system?
I'm currently working on an individual project. In which I have created a program to simulate a particle bouncing around a finite bordered region.
Where I need help in understanding how I might show ...
0
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1answer
35 views
Can a map be both nonchaotic and chaotic?
In Wikipedia’s article ‘List of chaotic maps’ (https://en.wikipedia.org/wiki/List_of_chaotic_maps), one of the entries is:
Feigenbaum strange nonchaotic map
So, how is this possible?
2
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0answers
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Sard's theorem for orientation preserving diffeomorphism of the circle
thanks in advance for helping me. First I'll introduce some definitions:
(1) Suppose that $f : \mathbb{S}^{1} \rightarrow \mathbb{S}^{1} = \mathbb{R} / \mathbb{Z}$ is an orientation preserving ...
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0answers
35 views
Dynamics $\delta x(t)=\delta x(0) e^{\lambda t}$ of Henon Attractor
Recall the question I asked before:
Linearized perturbation dynamics of Henon Attractor
So, I have the following separation dynamics for Henon attractor ($a = 1.4. b = 0.3$) $$\delta x(t)=\delta x(0)...
0
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0answers
40 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
votes
2answers
65 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
votes
1answer
35 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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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 ...
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0answers
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Although Lorenz system is a deterministic system, can it shows locally stochastic behavior?
The Lorenz system is a system of ordinary differential equations
$$\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x),\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y,\\{...
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0answers
42 views
Perron-Frobenius operator definition
According to
http://mathworld.wolfram.com/Perron-FrobeniusOperator.html, the P-F operator describes the time evolution of densities in phase space:
$$\rho_{n+1} = P (\rho_n).$$
But for the ...
0
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1answer
53 views
Hypercyclicity examples
Does anyone have simple practical examples of hypercyclicity they use in explaining the concept (graphically or numerically)? This appears often in texts about chaos in infinite dimensional linear ...
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1answer
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Does a greater Lyapunov exponent result in a more chaotic system?
Let $\sigma_1$ be the Lyapunov exponent of a one-dimensional system $I_1$ and $\sigma_2$ be Lyapunov exponent of one-dimensional system $I_2$. Can I say that if $\sigma_1 > \sigma_2$, then $I_1$ is ...
4
votes
1answer
51 views
Periodic solutions of the double pendulum
I'm stuck: Are there periodic solutions of the double pendulum, or not?
The question is four-fold:
Does every double pendulum - defined by two masses $m_1$, $m_2$ and lengths $l_1$, $l_2$ - have ...
5
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1answer
78 views
How to calculate the envelope of the trajectory of a double pendulum?
Consider a double pendulum:
Background
For the angles $\varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$):
$\dot{\varphi_1} = 6\frac{...
2
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0answers
76 views
Project ideas on Chaos theory, Cellular Automata, Fractals, Games, IA [closed]
I'm a computer science student and I need to find a final year project. What interests me the most is Chaos, IA, Games, Fractals, CA..
Something I liked was the chaos theory within sudoku.
The ...
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0answers
12 views
How to calculte the Fourier Transform of a sovable chaos waveform?
Recently I am stucking in frequency estimation of a solvable chaos waveform.
Its analytic expression in time domain is
$$
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
49 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]...
1
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0answers
43 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
votes
2answers
182 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
vote
1answer
119 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
134 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
votes
1answer
70 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
votes
1answer
326 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
votes
2answers
143 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, ...
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0answers
47 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
130 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
35 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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votes
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 ...
0
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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 ...
