For questions in chaos theory.

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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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47 views

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 ...
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31 views

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
20 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 ...
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1answer
18 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 ...
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1answer
62 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} = ...
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3answers
235 views

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

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 $z_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the ...
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1answer
34 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 ...
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1answer
37 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
29 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
23 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
30 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 ...
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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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25 views

'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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0answers
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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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20 views

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
29 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
47 views

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
23 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 ...
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1answer
91 views

Is the distribution of prime numbers chaotic? [duplicate]

The distribution of prime numbers, at first glance, appears to be somewhat random. It is however, deeply structured and deterministic. Does this qualify it as chaotic. As chaotic is somewhat of a ...
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2answers
45 views

Lyapunov Exponent sensitivity to initial conditions

I am plotting the Lyapunov exponent as a function of a parameter $r$ with an initial condition $x_0$. The equation looks like this: $$x_{n+1} =4rx_n (1-x_n)$$ When I try different initial conditions ...
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0answers
64 views

$f(x) = 2x \mod 1$ not equal to zero for all $x$?

If any number $\mod 1$ is zero, then how can $f(x) = 2x \mod 1$ be a Baker's map? For any $x\in \mathbb{R}$, shouldn't $f(x)=0$?
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1answer
31 views

Ergodicity of the natural measure implies uniqueness of the invariant density?

Consider a dynamical system $x_{t+1} = F(x_t)$ defined in $\Omega$ and its natural measure to be $\mu$. The Perron-Frobenius operator $F$ maps the density $f$ in time according to $f_{t+1} = F(f_t)$. ...
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2answers
68 views

$2$-dim dynamical system IVP

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
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Limits of two fixed points of $E_\mu(x) = \mu e^x$

Please let me know if this proof is OK. Problem statement: Given that $E_\mu(x) = \mu e^x$, where $0 < \mu < 1/e$, show that if $q_\mu < p_\mu$ are fixed points, where $q_\mu$ is attractive ...
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32 views

Show that $E_\mu$ has no periodic points that are not fixed points

Problem statement: Consider $E_\mu(x)=\mu e^x$, where $0<\mu<1/e$. Show that $E_\mu$ has no periodic points that are not fixed points. It is in my understanding that what we need to show is ...
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28 views

$\lim\limits_{n\to\infty}$ of iterated function $e^{x-1}$

Do you think there's a better way to do this? Given that $E(x)=e^{x-1}$, $n$ iterates of $E(x)$ are defined as $E(\underbrace{E(E(...(x))...)}_{n-1\text{ times}}=E^{\circ n}(x)$. Here's my attempt ...
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3answers
33 views

$e^{x-1}$ has only one fixed point

How does one show that the function $E(x)=e^{x-1}$ has only one fixed point? We know that there is only one integer solution for $e^{1-x} = x$, which is $x=1$, but Wolfram Alpha also gives a second ...
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0answers
54 views

converting to math from economics major

Recently, i'm majoring in honour track of economics taking econometrics statistics courses and minoring in mathematics taking advanced calculus, real analysis ,linear algebra courses. Upon research on ...
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1answer
44 views

Proving that a sequence is unbounded without knowing the sequence explicitly

Given that $f(x)=x^2+\frac{1}{4}$, there exists the iterated sequence ${f^{\circ n}(x)}_{n=1}^\infty$ (where $f^{\circ n}(x)$ is defined as $\underbrace{f(f(f...(x)...))}_{n\text{ times}}$), which is ...
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2answers
41 views

Proving that a sequence is bounded without knowing the sequence explicitly

Given that $f(x)=x^2+\frac{1}{4}$, there exists the iterated sequence ${f^{\circ n}(x)}_{n=1}^\infty$ (where $f^{\circ n}(x)$ is defined as $\underbrace{f(f(f...(x)...))}_{n\text{ times}}$), which is ...
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4answers
92 views

Book on Chaos Theory

Please suggest some good chaos theory as general read, which can be enjoyed while on beach has patterns. I am a electrical Eng Post Graduate in communication theory and signal processing so can ...
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0answers
47 views

Are the vertices of a Voronoi diagram obtained from a Sierpinski attractor also a kind of attractor?

Trying to understand how the Voronoi Diagrams work I did a test generating the Voronoi diagram of the points obtained from The Chaos Game algorithm when it is applied to a $3$-gon. The result is a set ...
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1answer
55 views

Is there a known mathematical foundation to the concept of emergence?

I'm researching many topics including emergence and chaos theory, and I cannot for the life of me find strictly mathematical treatments of the idea of emergence. Is there any form or field of ...
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1answer
33 views

Feigenbaum attractor is not an attractor?

I am reading about Feigenbaum attractor (FA) and am getting very confused with something that is described in some books. It is written that FA is not an attractor because in its neighbourhood however ...
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0answers
31 views

Feigenbaum attractor is uncountable set?

How to prove that Feigenbaum attractor (appearing at the accumulation point in logistic map) is an uncountable set? (I am not a mathematician but know how to prove that a Cantor set is uncountable.)
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Poincare-Bendixon Theorem on a unit disk

If applying the Poincare-Bendixson theorem to a region $D$ nearly all the text books I've read say that this region is generally an annulus. Would it be possible to apply the theorem to a unit disk ...
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174 views

Is $a_n = f(a + b c^n)$ when $a_0 = \ln(2)$ and $a_{n+1} = (e+1) a_n (a_n - 1)$?

Definition Let $a_n$ be a real sequence. Assume there exists a continuous real-periodic function $f(x)$ such that $f(n) = a_n$ And $f(x)$ has the period $t$ , where $t$ is An irrational real ...
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1answer
47 views

What is the basin of attraction for the attracting fixed point $x_-$ of $f(x) = x^2+c$

Attempt: If $x_-^2+c=x_-$ then $x_-=\dfrac{1-\sqrt{1-4c}}{2}$ which is attracting for $|f(x)|<1$ i.e $-2<c<\dfrac14$. How do I find the set of points $x$ such that the orbit $f^n(x) \to x_-$ ...
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1answer
92 views

Dynamics of Cubic Chaos [closed]

Consider the family of functions $f_{\lambda}(x) = \lambda x − x^3$. Describe the dynamics of this family of functions for all $\lambda < −1$.
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2answers
90 views

conservation law for the trajectories

Is it possible to find the conservation equation as the form of $Q=h(x,y)$, given that $$\dot{x}=x-xy$$ $$\dot{y}=5xy-5y$$ I am not sure how to start with.
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How to bound the input parameters to a chaotic function to obtain exact result in a finite precision setting?

While I was reading the paper entitled (http://dx.doi.org/10.1109/ISCAS.2003.1204947) Kocarev, Ljupco, and Zarko Tasev. "Public-key encryption based on Chebyshev maps." Circuits and Systems, ...
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1answer
59 views

Stability type of logistic population equation's equilibrium points

I was working on this question: Here is what I have so far: $ (a) \ \ \text{We have the algebraic expression to find equilibrium points:} \\ \frac{dy}{dt} = ry \big[1 - \left(\frac{y}{K}\right) ...
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1answer
45 views

Chaos Theory Go With The Flow

I am confused about this question. Trajectories do not intersect. A trajectory in the state space M is the set of points one gets by evolving x ∈ M forwards and backwards in time: $$C_x = ...
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0answers
29 views

Chaos Theory Mathematical Knot

I have been watching this following video on Youtube: https://www.youtube.com/watch?v=aAJkLh76QnM It talks about what Edward Lorenz's ideas about chaos theory were in nonlinear dynamical systems. At ...
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68 views

Basins of attraction for Newton-Raphson fractal colouring

What's the general strategy/approach for defining the basins of attraction within the Newton-Raphson(NR) function in the complex plane? I would like to understand where these are to colour-in a NR ...
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2answers
357 views

A question about a fractal like iteratively defined function

I am trying to figure out what the following function $f:\Bbb{R}^3-\{\mathbf{0}\}\to\Bbb{R}$ defined below (in pseudocode) does: function $f(\mathbf{v}\in \Bbb{R}^3-\{\mathbf{0}\})$ { ...