For questions in chaos theory.

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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 ...
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110 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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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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44 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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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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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
39 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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45 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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17 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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49 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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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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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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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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51 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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44 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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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
18 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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27 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
36 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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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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68 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
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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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137 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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40 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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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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74 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 ...
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271 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
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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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43 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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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
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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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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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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 ...
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1answer
40 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?
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where did the term $\omega$-limit set originate from?

What it says on the tin. I've always used the phrase 'in the limit of all things' but hearing '$\omega$-limit' in a chaos theory class has me wanting to use the term. That said, I'd feel really ...
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Practical applications of chaos theory in engineering or physics

Can anyone give me some examples of practical applications of chaos theory in engineering or physics? Do you know any good books about chaos theory or its applications?
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33 views

Chaoticity and randomness in a time series

Suppose we have a time series : $X=\{X_t,t\in T\}$. How can we check if the data $X_t$ are random or they are the result of some chaotic behaviour of a nonlinear dynamical system? Is there some test ...
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“is topologically mixing” vs. “is topologically transitive” in the defition of chaos

Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits" as the definition of chaos, and this paper shows that its (the paper's) definition of chaos is equivalent ...
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Discrete vs Continuous Dynamics

Where does the problem arise concretely in considering a discrete process as a sampling of a continuous process? Can we use continuous methods of solving say differential equations and find the ...
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132 views

Coupling in the circle map

I'm currently investigating Arnold tongues (areas in parameter space with rational rotation numbers $\rho$, ie. $\rho(\Omega, K) \in \mathbb{Q}$) arising when iterating the circle map $$ ...
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3answers
246 views

Discuss the convergence of $ \left \{ a_n \right\} $ where $ a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1 $

Let $$ a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2} $$ where $ a_1 = \dfrac{a_0}{2} $ and $ n\geq 1 $ Discuss the convergence of $ \left\{a_n\right\} $
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How are definitions of chaos related?

Chaotic systems can be defined in many ways. One definition is that the system has a positive Lyapunov exponent, that is, two trajectories starting near each other will diverge exponentially quickly. ...
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170 views

How does chaos arise in Hamiltonian systems?

I have a question about how chaos arises in Hamiltonian systems. I've been taking a upper year undergrad class on dynamics and it has focused on chaos a lot. So far however we have only seen the more ...
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1answer
84 views

Possible to make a flow that forms horseshoes on a 2-dimensional manifold?

It it possible to have a flow $\phi(t,x)$ on a 2-dimensional manifold where for some $t > 0$, the map $g(x) := \phi(t,x)$ creates a horseshoe? By $\phi(t,x)$ I mean the solution to the ODE ...
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109 views

How can the lyapunov exponents for the Mandelbrot Set be computed?

I am trying to find a way to calculate the Lyapunov exponents of the Mandelbrot set. There are some very nice diagrams that you can find on Flickr of a plot of the Lyapunov exponents of the Mandelbrot ...
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Generating Bifurcation Animations

https://en.wikipedia.org/wiki/File:Hopf-bif.gif Does anyone know how this animation was produced? I could make it by stitching together snapshots (what I'm doing) but this seems primitive, especially ...
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Testing for chaos in data

I have data for 3 variables ,each with respect to the discrete time values. How do I check for the existence of chaos for this 3D discrete system?(I don't have the analytic eqs.,just the data). MY ...
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Chaos (and logistic functions); what is it and is it truly chaotic?

I'm currently studying discrete dynamic models and I am now reading about the logistic function $x_{n+1} = ax_n(1-x_n)$. Below there is a picture what happens with different values of a: These are ...
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Chaotic iterative example needed

I'm using a very simple numerical method to find solutions to an equation. Start with an equation $\operatorname{f}(x)=0$ that you need to solve. Rearrange to give $x=\operatorname{g}(x)$ and then use ...