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0answers
17 views

Subshifts of finite type; No fixed or period 2 points

I'm working out of Devaney's Introduction to Chaotic Systems, and one of the problems I'm working on is to construct a subshift of finite type in $\Sigma_3$ with no fixed or period two points, but ...
4
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1answer
38 views

How to show no periodic orbits exist

I am trying to show that no periodic orbits exist for the system: $$ x_1'=y+x^2+xy^3$$ $$y'=-2x-y^3$$ I have tried using Dulac's criterion to find a function $g(x,y)$ such that $\Phi(x,y)$ given by ...
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3answers
97 views

Showing $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \log(\cos(\phi))\cos(\phi) \ d\phi = \log(4) - 2 $

This is a minor detail of a proof in 'Chaotic Billiards' by Chernov and Markarian which I foolishly decided to verify. It's page 44 of the book, during the proof that lyapunov exponents exist almost ...
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2answers
106 views

Logistic map bifurcation

Ok I am trying to do this on matlab, but I need to understand how to find the bifurcation values for logistic map by hand first. So here is the logistic map: $$ x_{i+1} = f(x_i) \qquad \text{where} ...
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1answer
119 views

What's the name of this chaotic system? (Cool pics included.)

I found this playing with a 2D-ODE-system plotter I'm writing. Surely, since it's so simple, it's been found and extensively studied by someone. What's it called? I'd like to look it up and learn a ...
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0answers
48 views

How to numerically find Floquet multipliers (e.g., characteristic multipliers or Lyapunov exponents for periodic orbits from chaotic systems)?

I understand the theory (c.f., Perko or Nayfeh and Balachandran, Ch.3), but I do not understand how this is accomplished numerically: Given a (chaotic) dynamical system (for example, I am using the ...
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0answers
11 views

Generalized Bernoulli map (dyadic transformation)

The dyadic tranformaiton is defined as $f(x)=2x \mod 1$. Does a generalization $f(x)=nx \mod 1$ exist? How is it called in the literature? Many thanks!
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1answer
50 views

Period 2 orbit of logistic map?

Suppose that G: $\mathbb R$ $\rightarrow$ $\mathbb R$ such that G(x)=4x(1-x). I need to find the period 2 orbit(s?) of this map and decide if it's a sink or a source. If I could find the fixed points ...
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0answers
52 views

Example of a continuous function with only one fixed point and no periodic points

Does anyone know how to go about finding an example of a function with no periodic point and only one fixed point. This f is continuous in an interval I, and I c f(I). As an addendum, can this fixed ...
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1answer
434 views

Prove that the tent map has exactly nine 6-cycles.

Prove that the tent map T(x)= {2x if 0<=x<=1/2 and 2-2x if 1/2
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0answers
126 views

Finding periodic points of every period, Sharkovskii's Theorem

Consider the map: $C_c(x)=c\cos(x)$ (a) Find a value of the parameter c for which this map has prime periodic points of every period, and provide an explanation with graphs supporting your argument. ...
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1answer
457 views

Isolated Versus Non-Isolated Fixed Point, 2D Dynamics

I am trying to understand the classification of fixed points in a dynamical systems context (fixed points of a system of two linear differential equations are places where both $x_1' = x_2' = 0$). ...
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2answers
122 views

Cat map like maps without period

Is there any area-preserving chaotic map other than Arnold cat map which can be applied on a rectangle as well as being reversible but not periodic?
3
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1answer
100 views

The extent of chaos

In chaotic systems the typical situation is that at a low level trajectories of points are wild, but overall there is a nice statistical description of the system. For example, consider the ...
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1answer
2k views

Why isn't the 3 body problem solvable?

I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of ...
8
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1answer
141 views

Feedback loop in real-time voting of TV show?

Today, a German TV casting show ("Unser Star Fur Baku") introduced a new "real-time" voting system that works as follows: 10 contestants take part in a song competition. Viewers can call in and vote ...
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1answer
103 views

Examples when Resonance Overlap fails to predict the onset of Chaos

In a Hamiltonian system Chirikov's resonance overlap criterion approximately predicts the onset of chaotic behavior. Furthermore in a system where resonances overlap, the strengths of the resonances ...
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0answers
151 views

What are the intial conditions for Chaotic Pendulum behavior?

I've been assigned to do a computer model of a simple pendulum. The model uses Runge-Kutta to do the integration to find the position and velocity of the pendulum. I am currently having problems with ...
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2answers
96 views

Ring-shaped mirror chaos points

I came across this problem in the context of spectroscopy today. Because of it's simplicity, I'm sure it's a question that's been posed by some mathematician ages ago, I just can't figure out how to ...