Questions tagged [chaos]
The chaos tag has no usage guidance.
26
questions with no upvoted or accepted answers
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answers
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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 $\omega$...
- 1,599
6
votes
0
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335
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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 ...
- 165
5
votes
0
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287
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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 ...
- 1,537
4
votes
0
answers
429
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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 ...
- 41
3
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0
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143
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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 ...
- 1,537
3
votes
0
answers
103
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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
3
votes
0
answers
46
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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 ...
- 3,771
2
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21
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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
0
answers
61
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 $X(h)$,
$$E[X(h)X(g)] = \...
- 613
2
votes
0
answers
121
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When is a composition of homeomorphisms topologically transitive provided one of the two is?
Suppose that $S$ is a (connected) regular surface, possibly with boundary, for instance an annulus. Suppose that both $f$ and $g$ are homeomorphisms whose restriction to $\partial S$ is the identity. ...
- 4,263
2
votes
0
answers
52
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Search for period N logistic map
The logistic map is a period doubling bifurcation system.
Are there known dynamical maps, which oscillate between $N$ points where $N$ is a prime number, like 2, 3, or 5, or 7... , where each ...
- 155
2
votes
0
answers
112
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Birth of chaos due to nonautonomous perturbation
Let $\sigma, b>0$. I want to study the dynamics of the map
$$ 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) = (\...
2
votes
0
answers
59
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
0
answers
96
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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
465
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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
2
votes
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66
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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 $...
- 3,554
1
vote
0
answers
86
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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
$$ x_{n+2}=x_{n+1}-\sin(x_{n+1})\frac{x_{n+1}-...
- 55k
1
vote
0
answers
1k
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Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero
Using the following definition of Dirichlet L-function
$$
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{...
- 2,236
1
vote
0
answers
196
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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 ...
- 111
0
votes
0
answers
45
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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
0
votes
0
answers
78
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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 $C_f$ ...
- 4,969
0
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65
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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
0
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61
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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
0
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228
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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
\...
- 485
0
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0
answers
89
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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 ...
- 235
0
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146
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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 ...
- 1