For questions in chaos theory.

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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 ...
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24 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
33 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
51 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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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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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 ...
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1answer
30 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
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$λ=log(2)$ 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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26 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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22 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 ...
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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. ...
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111 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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Chaotic Sequence $X_{n+1}=4X_n(1-X_n)$

I want to examine the chaotic sequence $$\begin{cases}X_{n+1}=4X_n(1-X_n),\\ X_0=0.2\end{cases}$$ More specifically, I want to prove that: The sequence does not converge It is non periodic It does ...
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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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17 views

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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50 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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33 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
21 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
66 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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380 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
36 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
51 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
31 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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11 views

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
28 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 ...
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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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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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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
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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
100 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
51 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
72 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
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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
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$2$-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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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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$\lim\limits_{n\to\infty}$ 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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$e^{x-1}$ 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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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
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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 ...