For questions in chaos theory.

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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
30 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
19 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
9 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 ...
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2answers
43 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
47 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
15 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
52 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 ...
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0answers
37 views

what does it mean when it says find an exact, closed form solution?

What does it mean when it says find an exact, closed form solution? I'm currently trying to solve a problem nonlinear difference equation which states that at the end "This is one of the rare cases ...
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2answers
102 views

Does randomness exist? [closed]

I've been plagued with this question for a few years now and wanted to know what others think. Does true randomness really exist? In mathematics, a random process is based on the concept of random ...
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1answer
58 views

Prove that if $f$ and $g$ are topologically conjugate functions and $f$ has chaos, then $g$ has chaos.

Question: Prove that if $f$ and $g$ are topologically conjugate functions and $f$ has chaos, then $g$ has chaos. I can see a bit of the reason behind of the claim but I can't prove it. To prove ...
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1answer
36 views

What are properties of dynamical systems in non-integer dimension spaces?

A 1D dynamical system (R1) exhibits convergence to a fixed point, or escapes to infinity. A 2D dynamical system (R3) can produce oscillations, spiral-shaped trajectories, etc. A 3D dynamical system ...
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0answers
11 views

Show that the following functions are topologically conjugate?

$T(x)=\begin{array}{lr}\ 2x, x \in [0,\frac{1}{2}] \\ 2-2x, x\in [\frac{1}{2},1] \end{array}$ and $F_4(x)=4x(1-x)$ with conjugacy $h(x)=\sin^2(\frac{\pi x}{2})$ I am completely lost, these two things ...
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1answer
34 views

Detection of Cycles without a Center in an ODE

In my classes in dynamical systems theory, we were taught how to detect cycles or cyclic behavior in an ODE (be it dampened, sustained or growing) around a fixed point by looking at the eigenvalues ...
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1answer
16 views

$f_\alpha:S^1\rightarrow S^1, \alpha\geq 2$ be given by$ f_\alpha (\theta)=2\theta, \forall \theta \in [0,2\pi]$ prove f is chaotic?

$f_\alpha:S^1\rightarrow S^1, \alpha\geq 2$ be given by$ f_\alpha (\theta)=2\theta, \forall \theta \in [0,2\pi]$ prove f is chaotic? Now I know I need to prove that 1) F is topologically transitive ...
2
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2answers
80 views

Logistic family and chaos

It is a well known fact that the map $f(x)=4x(1-x)$ is chaotic on $[0,1]$. By chaotic I mean the usual definition, i.e.: a) the periodic points of $f$ are dense in $[0,1]$, b) $f$ is topologically ...
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2answers
50 views

For $c<1$, let $f_c(x)=x^2+c$, determine the period-2 points?

Ok so I know I need to set $$f_c^2(x)=x$$ so: $$(x_0^2+c)^2+c = x \iff x_0^4+2cx_0^2-x_0+c^2+c=0$$ But how do I then solve this? Ok I have solved for the four set of roots, now you can see there are ...
3
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1answer
74 views

Prove that $F(x) = 8x^{4} - 8x^{2} + 1$ is chaotic

I have to prove that the function $F(x) = 8x^{4} - 8x^{2} + 1$ is chaotic. I would like to use the definition of a chaotic function which is: Let $F$ be $F: V$ -> $V$. 1) Sensitive dependance on ...
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0answers
109 views

Classical perturbation theory + KAM theory

In classical canonical perturbation theory of many degrees of freedom we encounter the problem of small divisors when attempting to find a solution for the generating function of the canonical ...
2
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1answer
60 views

Construct a continuous function $F:[0,1]\rightarrow[0,1]$ that has a point with period 2015

Construct a continuous function $F:[0,1] \to [0,1]$ that has a point with period 2015. I think I should do it with Sharkovskii's theorem, but then? Where can I start with? Or try to find a point ...
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1answer
36 views

Solving inverse problem related to Iterated function systems?

I generated a Barnsley's fern fractal using details in this link with the aid of MATLAB. My doubts are as follows : How do we justify the shape generated from those equations? Is it possible to ...
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5answers
504 views

If $f(x)=x^2-x-1$ and $f^n(x)=f(f(\cdots f(x)\cdots))$, find all $x$ for which $f^{3n}(x)$ converges.

Let $f:\mathbb{R}\to\mathbb{R}$ be the polynomial defined by $$f(x)=x^2-x-1$$ and let $$g_0(x)=f(x),\quad g_1(x)=f(f(x)),\quad\ldots\quad g_n(x)=f(f(f(\cdots f(x)\cdots)))$$ The positive root of ...
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1answer
63 views

Definition of Markov partition?

My teacher handed out an excerpt from a book by Robinson on chaotic dynamical systems. The excerpt is from a chapter on Markov partitions, and the following part has me confused: Let $$f(x)= ...
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1answer
58 views

Definition of Butterfly Effect

The Wikipedia definition of the Butterfly Effect seems to imply that linear functions can exhibit the Butterfly Effect. In particular if the state space is $\mathbb{R}$ with the usual metric then if ...
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0answers
49 views

Rational dynamical system with nonnegative paramaters and with nonnegative initial conditions

let $A$ be a rational system of the form :\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
4
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1answer
110 views

Chaos in Newtons Method

Im trying to prove that Newtons method applied to ${\rm f}\left(\, x\,\right) =x^{2} + c$, is chaotic for $c > 0$. I know I need to prove: (a) The periodic points of ${\rm f}$ are dense in $X$, ...
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0answers
75 views

Find all sinks/sources/saddles for a certain diffeomorphism

I'm trying to do the following exercise from Devaney's Introduction to Chaotic Dynamical Systems, exercise 2.6.1. The problem is this: Consider the diffeomorphism $Q_\lambda$ of the plane given by ...
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1answer
41 views

Showing that the linear twist map is sensitive dependent

Choose $\Delta=\frac{1}{2}$ (I believe this value should work). let $\delta > 0$ and let $\textbf{x}_1=(x_1,y_1) \in X$. I assuming that $d$ is the Euclidean distance. Somehow I think we ...
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1answer
40 views

Are chaotic function one way?

Are chaotic functions also one way functions? Can they be used in cryptography?
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1answer
73 views

Types of attractors

In studying dynamical systems and chaos theory, one usually gets across a classification that says that attractors can be of four basic types: -fixed point (equilibrium) -cyclic (periodic) -torus ...
4
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1answer
152 views

Is chaos theory really a theory? Why not just call it non-linear dynamics?

This may just be semantics, but it's always confused me. What is the thesis of Chaos Theory? I have read an entire book about it, and as far as I can tell, its just a bunch of analytical techniques, ...
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0answers
36 views

Question regarding graphing Standard Map and Circle Map (Arnold Tongue)

I'm attempting to program a python script that graphs two different mappings. My problem is I'm not even sure how to graph the mappings by hand since I'm unsure of the axis labels as well as how to ...
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2answers
59 views

Techniques for finding period points

Consider the tent function $f_2$ given by: $$f_2 = \begin{cases} 2x, & 0\leq x\leq \frac{1}{2} \\ 2-2x, & \frac{1}{2} < x \leq 1 \end{cases}$$ How do I find the periodic points of this ...
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1answer
48 views

Computing Feigenbaum Constant in Java

Is there a way to code a Java program that computes the Feigenbaum constant (which is around 4.66) for the function F(x) = r*sin(Pi*x)?
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3answers
560 views

Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the ...
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1answer
52 views

Fixed points and periodic orbits of $F(x)=x^2-1.1$

A question asked me to find the fixed points of $F(x)=x^2-1.1$, then use the fact that these points were also solutions of $F^2(x)=x$ to find the cycle of the prime period 2 for F. How do I go about ...
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2answers
96 views

Mathematical background for one wishing to study Chaos/Complexity Theory

I don't have a very strong mathematics background. In fact I quite abhorred mathematics during my Middle/High School years. I'm currently applying for PhD programs in the field of literature as that ...
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0answers
24 views

How to check if a point is on the attractor?

Consider a dissipative hyperbolic dynamical system defined on a set with a (strange) attractor. Given a point X on the phase-space, how do I (algorithmically) check if it is on the attractor? For ...
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1answer
149 views

Mandelbrot sets and radius of convergence

While watching this Numberphile video on Mandelbrot sets, it's more or less stated that the fractal will "blow up" if it's radius of convergence is greater than 2. What is the mathematical basis for ...
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0answers
118 views

How do you use R to find the box counting dimension of a two dimensional set of data, or scatter plot?

I'm using the software R to do some analysis on some data sets for a graduate project. R has a package called "fractaldim", in this package is a function for finding the box counting dimension. The ...
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1answer
75 views

Henon Map Parameter

In case of Hennon map two parameters $a$ and $b$ to be set.The Hénon map takes a point $(x_n, y_n)$ in the plane and maps it to a new point $x_{n+1} = 1-a x_n^2 + y_n$, $y_{n+1} = b x_n$. The map ...
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0answers
67 views

Period 3 window in chaos

Li and Yorke established the fact that period 3 implies chaos, which implies that if we have a period 3 orbit in a system then we have a chaotic system. I have seen that in bifurcation diagrams there ...
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1answer
81 views

An eventually periodic point must be an asymptotically periodic point?

Most definition can be found from Eventually periodic point and homeomorphism. And you can also find the definitions from the classic paper "Period Three Implies Chaos", my question is in the caption, ...
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1answer
138 views

How plot a bifurcation diagram ? or show find bifurcation points

I have a function $rx(3-x^2)$ How do I find the points it bifurcates and what does it mean ? I know how to find fixed points and check them for stability, how can I use that to answer this question ...
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0answers
26 views

what is the particle size effect in chaos?

I am studying dynamical system. As far as I know, chaotic behavior can be developed for one particle moving in a certain potential in 3D, and for this case, the position of the particle will be the ...
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1answer
26 views

Sensitivity Constants for Linear Expanding Maps

Let $E_m:S^1 \rightarrow S^1$ be the linear expanding map $E_m(x) = mx$ mod 1 (under the identification $[0,1] \sim S^1$). A sensitivity constant $\Delta$ is a positive real if for all $x, \in S^1$ ...
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1answer
39 views

Generlized Entropy compared to Generalized Dimension

I am currently reading the following paper by F.Takens: Multifractal analysis of dimensions and entropies. This paper discusses two different measures. One is generalized entropies and the other is ...
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Transient Behaviour Transient Property Lorenz Equation

Was reading Lorenz paper "Deterministic Nonperiodic Flow" and it says that if a trajectory is not a fixed point, periodic orbit or quasi-periodic orbit and no transient behaviour then it is a ...
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1answer
37 views

Is a cascaded chaotic system is still chaotic?

I am curious whether a new system which cascades two individual chaotic systems is always chaotic. My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that $${\rm ...
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0answers
35 views

Introductory book about economic models with deterministic chaos

I'm looking for introductory textbook about economic models (micro/macro/finance) which incorporate deterministic chaos. Models could be with or without random noise. By introductory I meant master ...