Questions tagged [chaos-theory]
For questions in chaos theory.
679
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If a diffeomorphism has a dense orbit, are almost all of its orbits dense?
Let $M$ be a closed manifold. Suppose that a diffeomorphism $f:M\to M$ has a dense orbit. Is it true that almost every ofbit of $f$ is dense in $M$? Or, maybe, if the orbit of $x_0$ is dense, then all ...
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A chaotic function related to the $3x+1$ problem? (Li-Yorke and the Collatz problem)
Let $ x $ be an infinite binary string. Define the function $ f(x) $ mapping $ x $ to the Cantor set of $ I = [0,1] $ as:
$$
f(x) = \sum_{n=0}^{\infty} \frac{2 x_n}{3^{n+1}}
$$
where $ x_n $ are the ...
- 689
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Chaotic one-dimensional system
Why can the solution of a one-dimensional equation of the form $$m\ddot{x}=F(x)$$ not be chaotic if $F$ is not explicitly time-dependent?
Multiplying by $\dot{x}$ and integrating with respect to time, ...
- 6,277
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Nonlinear Dynamics and Chaos - Convergence of a Map to the Logistic Map
The cosine–map is defined as:xn+1 = r/4((a+1)cos[k(xn - 1/2)]-a), with k = 2arccos(a/a+1), a > 0.
Show that in the limit a → ∞, the cosine–map is the logistic map.
I am really struggling in where ...
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Can probe trajectories to compute Lyapunov exponents get "stuck in more regular orbits" after rescaling?
I am computing Lyapunov exponents, and there is something that I do not understand about the data.
The model has a regime for $\delta \approx 1$ (in some units) where it is fully chaotic, and the ...
- 31
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Does the inverted single pendulum have a positive Lyapunov exponent?
I'm doing some numerical experiments to test an integrator, and I got this plot, for the motion of 5 pendula, whose initial displacement differ by $10^{-6}$ radians away from straight up ($\theta_0 = \...
- 121
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Reflecting a fractal
Assume you have a certain IFS (iterated function system - a finite set of contractions) given by affine transformations.
As reflections are affine transformations, any reflection of an IFS fractal ...
- 139
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How to identify heteroclinic (or homeoclinic) points on a mapping?
Identify the heteroclinic points of the following map:
\begin{equation}
F\begin{pmatrix}
\theta_1\\ \theta_2
\end{pmatrix}=\begin{pmatrix}
\theta_1+\epsilon\sin\theta_1\\
...
- 2,973
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Show that the circle $r=\sqrt{(1-\lambda/\beta)}$ is invariant under a given map $F$
With the given map
\begin{equation}
F\begin{pmatrix}
r\\
\theta
\end{pmatrix}=\begin{cases}
&\lambda r+\beta r^3\\
&\theta+\frac{2\pi}{n}+\epsilon\sin{n\...
- 2,973
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Analog to Feigenbaum constant for even and odd cycles
A picture is worth a thousand words, so starting off there's this peculiar aspect to the logistic map where it seems to have consistently-placed even and odd cycles after the onset of chaos:
Where ...
- 137
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How to find fixed points of this radial map $f:\mathbb{R}^2\to\mathbb{R}^2$
I have learned that finding fixed points of a map is usually done by setting the map $f(x)=x$. However, for this radial map $f:\mathbb{R}^2\to\mathbb{R}^2$
$$F\begin{pmatrix}\theta_1\\ \theta_2\end{...
- 2,973
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Logistic map: bifurcation and domain of attraction
Let $f(x) = \mu x(1-x)$ be the logistic map, the question is divided into 3 parts:
Part (1): what can you say about the domain of attraction of the 2-cycle in $3<\mu<1+\sqrt 6$?
My attempt: let $...
- 305
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Sarkovskiis theorem and the Cantor set
Can we prove the following as such (with relevance with the Sarkovskiis theorem)?
Suppose that $f$ is continuous and that $A_0 , A_1 ,\dots, A_n $ are
closed intervals and $f(A_i) \supset A_{i+1}$ ...
- 2,973
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Looking for an efficient (computerized) way to convert large ternary sequences into their decimal equivalents.
I am doing a project on Cantor Sets for my undergraduate and I need an efficient (computerized) way to convert large ternary expansions such as the following to their decimal equivalent.
$$0....
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set of orthogonal quasiperiodic functions
I hope you are well.
Could you help me with the following?
I want to build a set of quasi-periodic functions that are also orthonormal. This set will help me with a chaos problem. I tried with $\sin (\...
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