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1answer
10 views

Calculating the inverse of a continuous map for a certain interval in order to calculate the Perron-Frobenius operator.

Suppose we are observing chaotic continuos maps, the Perron-Frobenius operator $P$ satisfies: $P\phi_{n}(t) = \frac{d}{dt} \int_{f^{-1}([a,t])} \phi(x)dx$ I don't understand how for the shift map, ...
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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
24 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 ...
2
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2answers
45 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
58 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 ...
2
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1answer
67 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 ...
3
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2answers
49 views

How to find a superstable period-$2$ orbit of the logistic map.

Suppose that $G\colon R \to R$ such that $G(x)=rx(1-x)$. I need to find the value for $r$ at which the super stable period-$1$ and period-$2$ points exist. I think I know how to get the super stable ...
2
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0answers
28 views

Natural measure of a chaotic system and its prime orbits

Evidently, "the natural measure associated with a chaotic attractor gives the fraction of the time that the long orbit on the attractor spends in any given region of state space." An illustrative way ...
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 ...
2
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2answers
110 views

Proving a function is chaotic on an interval

I'm self studying from the book 'First course in chaotic dynamical systems' and am having a hard time grasping how to prove that a function is chaotic. For the function we have $T(x) = 2x$ for $x ...
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0answers
17 views

Show that $T:X\rightarrow X$ is chaotic iff every finite family of nonempty open sets shares a periodic orbit.

Here is a problem from Grosse-Erdmann and Peris' Linear Chaos book that I am trying to solve. Exercise 1.3.4. Show that $T:X\rightarrow X$ is chaotic iff every finite family of nonempty open sets ...
49
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1answer
1k views

Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?

Let $a_0=1,a_n=\tan{a_{n-1}}$. Then is $\{a_n\}_{n=0}^\infty$ dense in $\Bbb{R}$? I've drawn a map of this dynamical system and it seems that the sequence is dense on $\Bbb{R}$.
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0answers
169 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ...
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0answers
44 views

Chaotic dynamical systems with hard proposition

How I can prove this proposition ? proposition : Consider a network with a fixed connection topology, having in each node identical members of the family of m-modal functions f in the real interval. ...
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2answers
142 views

What's the intuition behind definition of chaotic function?

I read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. I want to understand which concepts of "chaos" ...
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1answer
55 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
121 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 ...
4
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3answers
114 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
298 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} ...
7
votes
1answer
176 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
552 views

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

Anyone have any suggestions for the following situation/question? (help wanted, please!) I understand the theory (c.f., Perko or Nayfeh and Balachandran, Ch.3), but I do not understand how this is ...
0
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1answer
286 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
70 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 ...
1
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1answer
692 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
1
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0answers
167 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. ...
0
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1answer
1k 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
165 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?
4
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2answers
156 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 ...
28
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1answer
3k 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
150 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 ...
2
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1answer
116 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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2answers
100 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 ...