For questions in chaos theory.

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2answers
25 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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0answers
33 views

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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0answers
30 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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0answers
36 views

I'm having trouble with solving this discrete logistic dynamics problem [closed]

Show that a solution to the discrete logistic dynamics $$x_{n+1} = 4x_n(1-x_n) $$ can have the form of $$x_n= A\sin^\nu b^n. $$ Determine the $A,\nu, b$.
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0answers
27 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
31 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
48 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
36 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
33 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
69 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
37 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 ...
4
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1answer
52 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 ...
1
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1answer
29 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 ...
2
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0answers
29 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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0answers
25 views

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 ...
2
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0answers
169 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
42 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
74 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
86 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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0answers
16 views

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
46 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
39 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
23 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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0answers
60 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 ...
2
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2answers
355 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 $f(\mathbf{v}\in \Bbb{R}^3-\{\mathbf{0}\})$ { ...
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0answers
15 views

How can I find 2-cycles for sine function?

I am studying bifurcation of function. I am given $f(x)=a\sin x$ and this is period-doubling at $a=-1$. I would like to find $2$ cycles of this function. So, $f(f(x))=a\sin(a\sin x)$ How can I ...
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0answers
33 views

Logistic map chaos theory experiment, need advice on interpretation of results

Recently while studying this lecture notes to learn about numerical techniques, I performed the following experiment in Mathematica as the lecture note encourage the reader to explore the logistic map ...
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1answer
43 views

Tripling function and its periodic points [closed]

Given tripling function $$F(x)= \begin{cases}3x & 0<x<1/3 \\ 3x-1 & 1/3<x<2/3 \\ 3x-2 & 2/3<x<1 \end{cases}$$ Suppose $x=\frac{p}{q}$ is rational number. ...
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1answer
79 views

Question About Filled Julia and Julia Sets

Question: Let $Q_{c}(z) = z^{2} +c $ which $ c \in \mathbb{C}$ and suppose that $z_{0} \in K _{c}$ for the filled Julia Set, $K_{c}$ of $Q_{c}$. Suppose further that $z_{1} = Q_{c}(z_{0})$ and it ...
4
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1answer
370 views

Prove that the Mandelbrot Set Is A Closed Set

The Problem: Suppose we define the Mandelbrot Set as the following For $c \in \mathbb{C}$ , $\mathbb{M}$ = $({c:|c| \leq 2}) \cap ({c: |c^2 + c| \leq 2}) \cap ({c: |(c^2+c)^2 +2| \leq 2}) \cap ...
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1answer
66 views

Is this a valid example of a non-euclidean Sierpinski attractor?

I am learning the basic concepts about the Chaos Game (I did a previous question about the same topic here), the method to create fractals elaborated by professor Michael Barnsley. The basic example ...
1
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1answer
79 views

If the Chaos Game result is a Sierpinski attractor when the random seed is a sequence (Möbius function), does it imply that the sequence is random?

The Chaos Game is the famous method to create fractals elaborated by professor Michael Barnsley. As Wikipedia explains: "The fractal is created by iteratively creating a sequence of points, starting ...
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0answers
22 views

Inverse evolution of a dynamical system

Background Considering a one-dimensional chaotic map $T$. The domain of the map (the closed interval $[0,\,1]=I_1 \cup I_1$) is partitioned into two regions (i.e. $I_0\cap I_1=\emptyset$): $[0,\,A] = ...
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0answers
52 views

A continuous function with a dense set of periodic points but without sensitive dependence on initial conditions

The Question: Give an example of a continuous function, $f$, on the interval, $I$, such that the set of periodic points of $f$ is dense in $I$, but f does not have sensitive dependence on initial ...
2
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0answers
136 views

Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? [closed]

Let $H$ be a separable, infinite-dimensional Hilbert space, and $B(H) = \{T : H \to H, T \space \text {is non-bounded and linear operator} \}$. We say An operator $T \in B(H)$ is chaotic if $T$ is ...
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0answers
10 views

Highly optimized systems

I'm interested in the idea that highly selected causal systems exhibit general behaviors and properties. By causal systems I mean formal systems that progress from a defined set of starting conditions ...
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1answer
29 views

Find a vector field on the plane such that $\omega$-limit set of a single point is two parallel lines.

I came across this puzzling question in "Chaos" by K. T. Alligood. Sketch a vector field in the plane for which the $\omega$-limit set of a single trajectory is two (unbounded) parallel lines. ...
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0answers
18 views

Are binary sequences generated from ergodic maps chaotic?

Chaotic Sequences of IID Binary Random variables with their applications to Communications and related papers by the same Author, Tohru Kohda talks about the statistical properties of symbolic ...
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0answers
43 views

Is this a bijective mapping?

Considering a one-dimensional chaotic map $T$. The domain of the map (the closed interval $[0,\,1]=I_1 \cup I_1$) is partitioned into two regions (i.e. $I_0\cap I_1=\emptyset$): $[0,\,A] = I_0$ and ...
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1answer
75 views

Help in understanding a conjugacy problem

I am studying the book Applied Symbolic Dynamics and Chaos By Bai-lin Hao, Wei-Mou Zheng The basic premise of the concept of Symbolic Dynamics is : "Symbolic ...
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0answers
35 views

Is there a connection between topological mixing and squashing functions used in neural networks?

Sigmoid, ReLU, tanh, logistic -type "squashing" functions are popular in neural networks to introduce nonlinearity into the transformations of the input vector, allowing the network to fit complex ...
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0answers
46 views

Suspension flow and topological equivalence(s)

Let $M$ be a compact smooth manifold, $\tau:M\to\mathbb{R}_{\geq 0}$. Let $f:M\to M$ is a surjective piecewise-smooth map. There is a standard construction of suspension allowing to extend $f$ to a ...
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2answers
127 views

Chaos theory and fractal geometry: Constructing from data

I understand that fractal geometry represents behaviour of 'chaotic' system, if I am not wrong. And also, fractals are generated by a recursive function. But, lets say I have random data lying with ...
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1answer
41 views

Newton Iteration Function

So I'm having trouble figuring this problem out so if someone can help me out that'd be great. Find all the fixed points for the associated Newton iteration function for $$ f(x) = \frac{x}{(x-1)^n} ...
2
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1answer
56 views

Expanding maps of the circle and their coding

A continuous smooth ($C^{10}$-smooth, for example) map $f:S^1\to S^1$ of the circle is called expanding if $\inf_{x\in S^1} f'(x) > 1 $. Here $S^1 = [0,1]/\sim$, the segment with identified ...
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0answers
18 views

transformation of variables in Melnikov method

Supposing there is a system of non-autonomous non-linear differential equations with small damping and small forcing. The unperturbed system (zero damping and forcing) is Hamiltonian but neither has a ...
2
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2answers
55 views

Can there be an interval where $F(x)=4 x^2-\frac{1}{2}$ is chaotic?

The function $$F(x)=4 x^2-\frac{1}{2}$$ has two repelling fixed points. Now, I wonder, can there be an interval $I$ where it is chaotic? I think not, because of the repulsiveness of the fixed points. ...
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1answer
50 views

Prove that if $x<0$ then $F_\mu^n(x)\rightarrow -\infty$ as $n\rightarrow \infty$, where $F_\mu=\mu x(1-x)$.

Logistic map is given as $$x_{n+1}=\mu x_n(1-x_n)$$ Let $F_\mu=\mu x(1-x)$. Therefore, for $\mu>1$, prove that if $x<0$ then $F_\mu^n(x)\rightarrow -\infty$ as $n\rightarrow \infty$.
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0answers
21 views

What is the name of this chaos theoretic transformation?

I once read an article about chaos theory, which I can no longer find, that went through the following example: Suppose you take a rectangular picture of Henri Poincare, and stretch it out vertically ...
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0answers
98 views

How can we construct Markov partitions for Smale Horseshoe and Solenoid?

I was reading the book Ergodic Theory, Hyperbolic Dynamics and Dimension Theory by Luis Barreira. In Chapter 7, it is asked to construct Markov partitions for the Smale Horseshoe and the solenoid. In ...