For questions in chaos theory.

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Can this kernel function exist?

I have a sample set ${(\mathbf{x_i})}_{I=1}^N$, each $\mathbf{x_i}$ $\in R^d$ and $\mathbf{x_i}$ is a column vector with $d$ dimensions. Webpage gives an idea that a kernel function can be build ...
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What methods can I use to model a seemingly chaotic set of values? [on hold]

If I have a graph/set of data, what methods can I use to model the data to predict a likely next set of values? For example, given the following numbers [ 2, 4, 5, 2, 1, -1, -2, 0 ], are there some ...
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54 views

Finding a parametrization of the solutions of $\frac{dx}{dt}=\frac{\sinh y}{\cosh y+A\cos x}$, $\frac{dy}{dt}=\frac{A\sin x}{\cosh y+A\cos x}$

I am trying desperately to find a parametrization for the following: $\frac{dx}{dt}=\frac{\sinh y}{\cosh y+A\cos x}$ $\frac{dy}{dt}=\frac{A\sin x}{\cosh y+A\cos x}$ I tried to devide the equation ...
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How to prove that probability for different initial conditions to yield similar trajectory is very small?

For $\epsilon > 0$, suppose $f$ is a function describing chaotic dynamics. Then, for any two different initial conditions, $x,y$, the trajectory obtained is by repeated application of the function $...
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2answers
30 views

Permutations in an Infinite List of Random Numbers

In an infinite list of random numbers from a to b, prove that in this list, there are all possible permutations of n numbers from the list, where n can be any number. Here are some versions of the ...
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generating functions for catastrophe theory

I am studying Thom's theorem in catastrophe theory and am having a hard time understanding what the "generating functions" actually do. How exactly are they used to classify generic caustics? The ...
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1answer
9 views

How to know whether a solution for a set of coupled non-linear differential equations are stable or not

I have 3-coupled ordinary non-linear differential equations, one second order in time and other two first order in time, with 3-dependant variables say x(t), y(t) and z(t). I have a particular ...
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1answer
42 views

Is Fractal Dimension Really Enough to Diagnose Chaos?

I've been reading a fair amount about the relationship between fractality and chaos. While approximating the dominant Lyapunov exponent seems to be the preferred method for diagnosing chaos, it seems ...
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1answer
54 views

What is the difference between high dimensional and low dimensional chaos?

Often I read of high and low dimensional chaos. But, I don't know what is their difference. I have thought the following answer. Let us consider a time series $\{x_i\}_{i\in\mathbb N}$. According to ...
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1answer
1k views

Does an iterated exponential $z^{z^{z^{…}}}$ always have a finite period

Let $z \in \mathbb{C}.$ Let $t = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $...
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1answer
76 views

Hyperbolicity without ergodicity?

I have a question concerning the ergodic properties of hyperbolic Hamiltonian flows. Let $\Phi_{H}^{t}$ be a Hamiltonian flow on a symplectic manifold $\mathcal{M}$. If $\Phi_{H}^{t}$ is Anosov on a ...
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26 views

Conjugacy tent map and Bernoulli shift map

I'm studying chaos and I got stuck on following: 'the Bernoulli shift map is topologically conjugate to the tent map' e.g. Wikipedia. I tried finding the conjugacy function C(x) such that: C(f(x))=...
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1answer
35 views

Conceptual Question : Relationship between entropy and a technique for source coding

I want to encode the messages to a sequence of 1s and 0s (subsequently called "bits"). This is called "source coding". Shannon's source coding theory states that the entropy of a source that emits a ...
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1answer
79 views

Strange behavior of $\sin(x^3)$ and $\tan(x^3)$

I noticed this behavior a long time ago and never really figured this out but if you take the $\sin$ or $\cos$ or $\tan$ etc. of a cubic polynomial you get a very strange and erratic behavior. It ...
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19 views

Transcritical bifurcation for map function

Question: Determine the transcritical bifurcation for $x_{n+1}=\alpha x_{n}\left ( 1-x_{n} \right )^{2}$ I have determined the fixed point to be $x^{\ast}=0$ and $x^{\ast}=1$ Also, for values $...
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0answers
30 views

What is computational complexity of a coding technique

In my previous Question Help in understanding a coding technique based on inverse mapping of a dynamical system I learnt how to apply chaotic map in coding theory in communications. Steps: (1) The ...
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1answer
36 views

Very different starting conditions into similar outcomes within Chaos Theory.

Chaos Theory demonstrates how systems with very similar starting conditions can end up in very different states, but can it demonstrate whether systems with very different starting conditions can end ...
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1answer
54 views

Upper and lower bound for the separation of two trajectories of a dynamical system

Consider trajectories $x(n)$ and $y(n)$ of the tent map, starting from initial conditions $x(0)$ and $y(0)$. Then the distance $δ$ between the trajectiories is: $δ = |x(n) - y(n)| = \exp (λ n)|x(0) - ...
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0answers
30 views

What it this theorem saying? - Regions of state space for which the flow eventually exists…

We have been given the following theorem to define regions in the state space for which the flow eventually exists. In questions, we use it to show that all trajectories eventually enter a bounded ...
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47 views

Poincare first return map, stability and bifurcations

Let $X= \mathbb R^3$ and consider the autonomous dynamical system $$\dot{x_1} = -x_2 + x_1 (1 - (x_1^2 +x_2^2)^2), \qquad{} \dot{x_2} = x_1 + x_2(1-(x_1^2 +x_2^2)^2), \qquad{} \dot{x_3}= \epsilon x_3$...
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1answer
34 views

Lyapunov number and Lyapunov exponent

Given two different points $x_{0}$ and$ x_{0}'$ where $x_{0}=x_{0}$ and $x'_{0}$=$x_{0}$+$\epsilon$ where $\epsilon_{n}$=$e^{n \lambda (x_{0})}$ is the n-th iteration of the separation distance ...
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1answer
26 views

Definition of a Period 2-orbit

In my text and notes, the term "period 2-orbit" and "period 2 doubling to chaos" is used but without a definition given. Can someone provide a definition for what period 2-orbit is? Thanks in ...
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1answer
69 views

$λ=log(2)$ for the tent map – which basis for the logarithm?

If $\lambda$ is the largest positive Lyapunov exponent of a piecewise linear dynamical chaotic discrete in time map, then is there a relationship between the entropy $h$ and its $\lambda$. According ...
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27 views

Finding simple bifurcation point

$\dot{x}=x-rx\left ( 1-x \right )$ I know where the fixed point occurs but what about the bifurcation points. I thought I knew how to find the bifurcation points. At least, for simple ODE where I can ...
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23 views

Show for a sufficiently large C the ellipsoid $rx^2 + σy^2 + σ(z − 2r) ^2 = C$ is the boundary of a trapping region for the Lorenz equations?

We're told that geometrically, inflow means that the dot product of the normal vector and the flow is negative which means $∇f [\dot{x}, \dot{y}, \dot{z}]^T$ is negative for all points on the boundary....
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1answer
35 views

What is the meaning of “smeared limit cycle”?

I'm reading the paper Phase dynamics of coupled oscillators reconstructed from data by Kralemann et. al. (2008), which is about representing phenomena that exhibit a stable limit cycle (i.e. non-...
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113 views

Poincaré surface of section of the kicked rotator

Wikipedia's article about the kicked rotator says that it's Hamiltonian is \begin{equation} H(p,q,t)=\frac{1}{2}p^2+K\cos(q)\sum_{n=-\infty}^{\infty}\delta(t-n) \end{equation} and it's Poincaré ...
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0answers
29 views

Applied nonlinear dynamics: the onset of chaos in biological cycles (reference request)

I have seen some applied research in the onset of chaos in the study of current regulation in the human heart and the transition into cardiac arrest. I would like to review any literature that exists ...
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Index of a curve is independent of the curve c?

Index of a curve C: $I_{C}$ is defined as the net number of counterclock wise revolutions made by the vector field as the vector field x moves once counterclockwise around the curve C. If C is a ...
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Idea behind Poincaré Bendixson theorem

The Poincaré Bendixson theorem states: If R is a closed bounded subset of $\mathbb{R}^{2}$ containing no fixed points and $\Psi_{t}\left ( x_{0} \right ) \in R$ for all $t\geq 0$, then, the omega-...
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Classifying the Trajectories of Pendulum

The equation of the pendulum is: $$\ddot{\theta}+\frac{g}{l}\sin\theta$$ After some manipulation, we get $$H=\frac{\dot{\theta}^{2}}{2}-\frac{g}{l}\cos\theta=\mathrm{positive\ constant}$$ ...
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1answer
22 views

omega-alpha limit set and manifold

Definition: The $\omega$-limit set $L_{\omega}\left ( x \right )$ of $x \in \mathbb{M}$ >is the set of $y \in \mathbb{M}$ which for each y there exists a strictly increasing unbounded sequence ...
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1answer
19 views

Geometric intuition of an invariant set, positively invariant and negatively invariant

Definition: Invariant set A set $S \subseteq \mathbb{M}$ is invariant if $\Psi_{t}\left ( S \right )\subseteq S,\forall t$ -if $\forall x \in S$, $\Psi_{t}\left ( x \right )\subseteq S,\...
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1answer
76 views

What exactly is a “chaotic” sequence?

In response to this question, I was told that a certain sequence $a_n$ is "chaotic" and "wandering." The particular sequence in question is defined by $a_0 = z, z \in \mathbb{C}$ and $a_{n+1} = {a_0}^{...
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3answers
508 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

$\DeclareMathOperator{\Arg}{Arg}$ Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, ...
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1answer
37 views

Eigenvector for a non-linear system

Using the reversibility arguments alone, show that the system $\dot{x}=y$ $\dot{y}=x-x^{2}$ has a homoclinic orbit in the half-plane $x\leq 0$ This is a non-linear system. A linear ...
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1answer
56 views

Eigenvector of unstable and stable manifold of a non-linear system with non-linear center

Show that the system $\dot{x}=y-y^{3}$ $\dot{y}=-x-y^{2}$ has a non-linear center and plot the phase potrait. My attempt: The system is non-linear so we linearise it: The ...
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29 views

How was one derivied from the other?

In the geological paper entitled The power–law relationship between landslide occurrence and rainfall level by C. Li et al, a power-law cumulative probability distribution is derived. However, I don't ...
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1answer
32 views

How to determine constant $C$ in $p(x) = Cx^{-D}$?

Given a distribution obeying the power-law (fractal) relation, such as the cumulative distribution function $L_{cf}(> X) = CR^{-D}$, if $X$ is given, how does one find the constant $C$ from a given ...
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Symbolic dynamics of Lorenz system

Based on paper : Symbolic dynamics and periodic orbits of the Lorenz attractor download link and a previous post Symbolic dynamics of a multidimensional system From FIgure 1 in the paper says a ...
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1answer
36 views

Help in understanding a coding technique based on inverse mapping of a dynamical system

Based on paper titled : Simultaneous Arithmetic Coding and Encryption Using Chaotic Maps by Kwok-Wo Wong et. al The Authors use a non-linear dynamical system for generating keys to be used in ...
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1answer
31 views

Prove that the iteration of $\sin(x)$ goes to zero as $n$ goes to $\infty$ [duplicate]

Basically let $S(x)=\sin(x)$ such that $S^2(x)=\sin(\sin(x))$ and $S^3(x)=\sin(\sin(\sin(x)))$ and so on until $S^n(x)=\sin(\sin(\ldots\sin(x)\ldots))$ Prove that $S^n(x)\rightarrow 0$ as $n\...
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1answer
24 views

Meaning of the term “topologically mixing”

I understand that for a system to behave chaotically, it needs to be "topologically mixing". However, I am not sure what that term really means. There are several explanations of this online. ...
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'Fast' and 'slow' Eigendirection?

Can someone give an intuition and a definition of what a "fast" and "slow" eigendirection means? A reasonable google search reveals nothing that would help. Thanks in advance.
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Conceptual Question on Takens embedding Theorum

I am from signal processing background and so unaware of many details of Takens phase space reconstruction theorum. Reading the paper : A First Analysis of the Stability of Takens’ Embedding download ...
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Proof strategy - How to prove this modeling of time series

The question is based on a paper titled : Forecasting high waters at Venice Lagoon using chaotic time series analysis and nonlinear neural networks On page 2 right above Eq(1), the authors say ...
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0answers
32 views

Transform $dx/dt = r + x/2 - x/(x+1)$ into normal form $dX/dt = R +- X^2$ (saddle node bifurcation)

I am trying to show that $dx/dt = r + x/2 - x/(x+1)$ has a saddle node bifurcation by putting it into normal form. I've found that the bifurcation point must be at $$x=-1+-\sqrt(2)$$ and that from ...
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2answers
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Transform $dx/dt = 1 + r + x^2$ into normal form $dX/dt = R +- X^2$ (saddle node bifurcation)

I am trying to find a substitution $X(x,r)=?$ and $R(x,r)=?$ to allow me to transform $dx/dt = 1 + r + x^2$ into $dX/dt = R +- X^2$ (normal form for saddle node bifurcation) I'm sure it's a ...
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0answers
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Deriving convergence region of iterative formula

A year ago I asked this question about fractal icons, however I didn't get any wiser yet. Now I am trying to understand the convergence of a simplified version of the fractal, to learn more about the ...
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1answer
25 views

Why does the logistic map $x_{n+1}=rx_n(1-x_n)$ become unstable when $\lvert\frac{df(x^*)}{dx}\rvert=1$?

I'm having trouble understanding why the logistic map becomes unstable when $$\lvert df(x^*)/dx\rvert=1,$$ where $x^*$ are the fixed points of $f(x)=rx(1-x)=x$. I have read that it can be seen from ...