For questions in chaos theory.

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2answers
44 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 ...
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1answer
52 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
95 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
57 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
21 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 ...
36
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5answers
479 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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0answers
34 views

Recommend Courses before taking Chaotic Dynamics

I am a junior CS major at my school and am interested in taking a chaotic dynamics course. The requirements are an intro CS course and Calculus 3 (both of which I have taken), but they also recommend ...
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1answer
41 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
35 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
47 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}} \\ ...
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1answer
48 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
52 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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0answers
12 views

Time series of Complex numbers and Lyapunav exponent calculation

I have a time series of length $n$ of complex numbers that is $\{(x_i,y_i)\}$. $i=1,2,\dots,n$ where $(x_i, y_i)$ is a complex number. I would like to calculate the Lyapunav exponent of the series. ...
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1answer
37 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
35 views

Are chaotic function one way?

Are chaotic functions also one way functions? Can they be used in cryptography?
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1answer
54 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
142 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
26 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
53 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
30 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
490 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
46 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
74 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
20 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
102 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
92 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
57 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 ...
3
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0answers
56 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
60 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
80 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
25 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
22 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
35 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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0answers
43 views

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
32 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 ...
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1answer
78 views

Differential Equation: Periodicity of a circle with zero radius in polar coordinates

I am given the following diff. equation in polar coordinates: $$\dfrac{dr}{dt} = r(1 + a~\cos \theta - r^2) \\ \dfrac{d \theta}{dt} = 1$$ where $a$ is a positive number and is less than $1$. I am ...
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1answer
55 views

Onset of n-cycles in the logistic function

The logistic function f(x) = r x (1 - x) is well known in dynamical systems theory. So is the famous bifurcation diagram for it. I recently came upon an article in Mathematics Magazine from 1996 ...
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1answer
149 views

Chaos Theory and Sum Subset Problem

In respect to the "P versus NP" controversy, can't chaos theory be used to solve problems like the Sum Subset Problem with non exponential performance? Like, chaotic equations, are like paths to very ...
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1answer
47 views

Periodic points of topologically conjugated functions in dynamical systems?

I'm working on a homework problem which seems obvious, but I am having a hard time proving/completing. The problem can be stated as follows: Let $f,g:$ $\mathbb R$ $\rightarrow$ $\mathbb R$ be ...
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0answers
55 views

Why do we want the Periodic Points to be dense for a Chaotic Map?

Devaney defines a dynamical system on $S$ with an iterator function $f:S\rightarrow S$ as being chaotic if we have sensitivity to initial conditions; topological mixing; and the set of period points ...
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1answer
77 views

Are there chaotic systems with explicit solutions?

Are there such continuous chaotic systems, for which an explicit solution exists, which would allow to practically compute state at any given position in time just knowing the initial conditions? Or ...
4
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1answer
297 views

Are continuous chaotic systems necessarily uncomputable?

I have seen the claim in a recent unpublished paper that chaotic dynamics are necessarily uncomputable. This follows, they argue, from the sensitivity to initial conditions shown in chaotic systems. ...
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3answers
109 views

Prove sensitivity to initial conditions numerically?

How can I prove sensitivity to initial conditions numerically? I mean directly from the computed data and neglecting the dynamical system that originated the data. The data comes from hybrid ...
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0answers
51 views

Decision Making Algorithms - Chaos Theory

I'm doing research on decision making algorithms on robotics. And recently I've read a lot of about Chaos Theory. I've searched all over the web, in IEEEXplore, ACM Digital Library, but can't find any ...
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0answers
15 views

Can correlation dimension of an attractor exceed the dimension of the space?

Here is the definition of the correlation dimension: http://en.wikipedia.org/wiki/Correlation_dimension Is there a proof that the correlation dimension cannon exceed the dimension of the space?
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1answer
64 views

Are deterministic RNGs chaotic systems?

Deterministic random number generators (RNG) are designed to provide faithful approximations of a uniform distribution. Given that a deterministic RNG always gives the same sequence for a given ...
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0answers
62 views

How influential was Lorenz' work?

I've recently read an article in Pour la Science (a French equivalent of the Scientific American, with an overall very good quality) on the history of Chaos theory. Essentially, the article goes ...
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0answers
49 views

Explicit form of strange attractors

Are there any examples of continuous-time dynamical systems possessing strange attractors for which there exist explicit formulas describing these attractors? Many thanks in advance and apologies if ...
2
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1answer
43 views

What does the phrase “invariant under f” mean?

Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(x,y)=(\frac{x}{3}+11y^2,-2y)$. Let $A=\{(3y^2,y)|y\in \mathbb{R}\}$. Show that $A$ is invariant under $f$. What is it asking?