Questions tagged [chaos]
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65 questions
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Chaos in the Kepler map
Consider the Kepler map $f:\mathbb{R}\times \mathbb{S}^{1}\to\mathbb{R}\times \mathbb{S}^{1}$ given by
$$f_{a}\left(\begin{array}{c} x\\ y \end{array}\right)=\left(\begin{array}{c} x+a\sin(4\pi y)\\ y+...
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2
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0
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173
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Spectral distributions in Gram matrices built from unit random vectors in high dimensions (and relation to LLM embeddings)
I am wondering if any rigorous mathematical results are known for the following:
${V}_i \in \mathbb{R}^d$ ($i=1,2,\ldots N$) are randomly-distributed unit vectors, $\| V_i\| =1$.
Consider the ...
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6
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1
answer
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Bounding proportion of phase space which is chaotic
There are dynamical systems which have regions of phase space that are both chaotic and integrable, e.g. small perturbations of integrable systems as in KAM theory. Are there any tools for bounding ...
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3
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1
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Finding a two point scrambled set for the function $g:[0,1] \rightarrow [0,1], x \mapsto \min_{n\in \mathbb{Z}} |3x-2n|$?
Let $I=[0,1]$ be the unit interval and $g$ as defined below.
Then $x \neq y$ with $x,y \in I$ are called "two point scrambled set"=$\{x,y\}$, if
$\lim\inf_{n \rightarrow \infty} | g^{(n)}(x)...
1
vote
0
answers
138
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Chaotic behaviour of the secant method for $\sin(x)$
For not very serious reasons I was trying to understand the behaviour of the secant method for solving $\sin(x)=0$ starting with $x_0=2$ and $x_1=18$, so
$$ x_{n+2}=x_{n+1}-\sin(x_{n+1})\frac{x_{n+1}-...
- 58.8k
0
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0
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136
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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
3
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0
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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
2
answers
433
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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 ...
- 1,738
12
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6
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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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105
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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 $X(h)$,
$$E[X(h)X(g)] = \...
- 623
0
votes
0
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2k
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Discrete dynamical system described by Dirichlet L-function using Yitang latest results on Landau–Siegel zero
Using the following definition of Dirichlet L-function
$$
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
votes
1
answer
466
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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 $\...
- 285
2
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0
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191
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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. ...
- 4,801
8
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1
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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
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0
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74
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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 ...
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