For questions in chaos theory.

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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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33 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
28 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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37 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
43 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
29 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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1answer
22 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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32 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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1answer
65 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
24 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
133 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
35 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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69 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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252 views

Are 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
78 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
39 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
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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
37 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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872 views

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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0answers
132 views

“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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1answer
118 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
244 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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1answer
160 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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1answer
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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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3answers
87 views

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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3answers
142 views

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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2answers
127 views

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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4answers
94 views

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 ...
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3answers
459 views

What could be some applications of Chaos Theory in Computer Science?

I am a Computer Science student. While going through some random maths topics I came across Chaos Theory. I wanted to know if there are any applications of it in CS. I tried searching on the internet ...
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2answers
74 views

Chaos in finite field

Let's think about some finite field $\mathbb{F}$. Is it possible to construct a map $x[n+1] = \mathcal{P}(x[n], x[n-1],...,x[n-k]), \ \ \ \forall x\in\mathbb{F} $ where $\mathcal{P}$ - ...
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1answer
60 views

Examples of systems conforming the Lorentz Attractor

Might sound like a trivial question but would you please show me some examples of real systems conforming the Lorentz Attractor? It can be any kind of system, just a little list. It can be a system ...
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3answers
261 views

Does Differential Topology or Differential Geometry play a larger role in Chaos Theory?

I'm an undergraduate on somewhat of a time constraint in school. I have room in my remaining schedule for a semester of either Differential Geometry or Differential Topology. I understand the ...
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1answer
37 views

Lyapunov Exponent of a Trigonometric Solution

I am attempting to work through the following problem. I have no problem with part (a) and have included it only for context. (a) Verify that $u_n = \sin^2(2^n)$ is a solution of the map ...
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179 views

Chaos without period doubling

I have been studying the Duffing oscillator rather intensively lately, mainly based on the theory in of the book by Guckenheimer and Holmes. From all that I have gathered, it seems that most dynamical ...
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1answer
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Chaos/Fixed points. I was reading a book by Strogatz and I encountered this.

Now, I always thought a fixed point implied $f(x)=x$, so somebody tell me, what is he talking about here? Thank you.
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1answer
132 views

How important are the following undergrad courses when trying to pursue studies in chaos theory/dynamical systems?

I'm currently a physics major with a year left, and deciding whether to switch into mathematical physics, mathematics or applied mathematics. I'm definitely switching into one of them, as I can meet ...
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2answers
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Attractors and stability.

Does anyone who studied chaos get this? I get everything preceding the red box, but I don't understand how it makes x* an attractor. The notes we have don't explain the theory very well, nor ...