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4
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0answers
43 views

Quantifying Poincare map

I have a dynamical system which goes from chaos to ordered state (quasiperiodic state to be precise). I have represented this transition via a Poincare map. See the attached figures. Now, my question:...
4
votes
1answer
79 views

Strange behavior of $\sin(x^3)$ and $\tan(x^3)$

I noticed this behavior a long time ago and never really figured this out but if you take the $\sin$ or $\cos$ or $\tan$ etc. of a cubic polynomial you get a very strange and erratic behavior. It ...
2
votes
0answers
17 views

Chaotic neural network - help in understanding

I am having difficulty in understanding a technique for clustering and segmentation of biomedical images using the concept of time series. The paper on which the Question is based is : M. Lacomi et. ...
10
votes
2answers
173 views

How to know whether an Ordinary Differential Equation is Chaotic?

Assuming we have an ordinary differential equation (ODE) such as Lorenz system: $$ \dot x=\sigma(y-x)\\ \dot y=\gamma x-y-xz\\ \dot z=xy-bz $$ where $$ \sigma = 10\\ \gamma = 28\\ b = \frac{8}{3}\\...
18
votes
0answers
162 views

Has this chaotic map been studied?

I have recently been playing around with the discrete map $$z_{n+1} = z_n - \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some ...
0
votes
2answers
55 views

Lyapunov Exponent sensitivity to initial conditions

I am plotting the Lyapunov exponent as a function of a parameter $r$ with an initial condition $x_0$. The equation looks like this: $$x_{n+1} =4rx_n (1-x_n)$$ When I try different initial conditions ...
1
vote
1answer
56 views

Construct differential equation given the phase portrait (non-linear pendulum)

I am curious to know how to recover the differential equation that goes with a phase portrait. I have seen the following posts but the first one was a $y'$ (and easy enough to "do in my head") and ...
1
vote
1answer
49 views

Numerical integration of a system of stiff ODEs starting at a singular point

Good afternoon, I have a system of $3$ highly non linear differential equations, which I have to integrate form a starting singular point $x^1=[1,1,1]$, and theoretically I have to arrive to an ...
1
vote
1answer
34 views

Feigenbaum attractor is not an attractor?

I am reading about Feigenbaum attractor (FA) and am getting very confused with something that is described in some books. It is written that FA is not an attractor because in its neighbourhood however ...
-2
votes
1answer
94 views

Dynamics of Cubic Chaos [closed]

Consider the family of functions $f_{\lambda}(x) = \lambda x − x^3$. Describe the dynamics of this family of functions for all $\lambda < −1$.
1
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0answers
17 views

How to bound the input parameters to a chaotic function to obtain exact result in a finite precision setting?

While I was reading the paper entitled (http://dx.doi.org/10.1109/ISCAS.2003.1204947) Kocarev, Ljupco, and Zarko Tasev. "Public-key encryption based on Chebyshev maps." Circuits and Systems, ...
1
vote
0answers
13 views

Do theorems involving chaotic mappings hold in a finite precision context?

Chaotic mappings are known as highly sensitive to their initial state. It is well known that the first type of Chebyshev polynomials is chaotic, e.g. this. This mapping is defined recursively as $T_n(...
1
vote
0answers
45 views

How to draw bifurcation diagram:$\dot{x}=x^3-C*sin(\frac{\pi x}{2})$

I want to draw the bifurcation diagram but since I can't solve this equation by hand it is difficult. I can graph it by having $f1=x^3$ and $f2 = C*sin(\frac{\pi x}{2})$ and the intersection points ...
-1
votes
1answer
30 views

Devaney's definition of sensitive dependence on initial conditions

Devaney defines that a funtion $f:J \to J$ has sensitive dependence on initial conditions if there exists $\delta > 0$ such that, for every $x \in J$ and any neighborhood $N$ of $x$, there ...
0
votes
0answers
28 views

How to iterate a function 8 times about a given interval of x in a Discrete Dynamical System

This is Dynamical Systems, specifically a discrete system. We are using L and R as in Left and Right such as: L=[0,0.5] R=(0.5,1] and LL=[0,0.25] LR=(0.25,0.5] and so on like that. We keep ...
0
votes
1answer
60 views

Help in understanding the dynamics of the doubling transform

From the book: M.Amigó, Permutation Complexity in Dynamical Systems, Springer Verlag, 2010 and Based on my readings, a concept of symbolic dynamics exists www2.acqs.org/mathstat/personal_pages/...
0
votes
0answers
17 views

Determine points with bounded orbits of discrete dynamical system defined by quadratic polynomial and chaos

Consider the discrete dynamical system definded by the function $f(x) = ax^2+bx+c$ for real parameters $a,b,c$ with $a \neq 0$, $(b-1)^2 \geq 4ac$. How does the set $\Lambda$ of all $x \in \mathbb R$ ...
2
votes
1answer
50 views

Can You Help With This Tent Map Proof?

The question: Show that if $ x= \frac{k}{2^{n}}$ where k and n are positive integers with $ 0 < \frac{k}{2^{n}} <1 $, then x is eventually a fixed point of the tent map. My Attempt: If you ...
0
votes
1answer
91 views

Can You Help Me With This Logistic Difference Equation?

In population biology, the following equation is the Pielou Logistic Equation, is used to model population with non-overlapping generations $$x_{n+1} = \frac{\alpha x_{n}}{1+\beta x_{n}}$$ Show ...
0
votes
1answer
20 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, $...
1
vote
1answer
43 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
votes
1answer
89 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
votes
2answers
57 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. ...
1
vote
1answer
50 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 $n\rightarrow \infty$.
0
votes
0answers
142 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
votes
1answer
116 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 ...
4
votes
2answers
172 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
votes
0answers
48 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
votes
1answer
121 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
votes
2answers
328 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 \...
1
vote
0answers
19 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 ...
50
votes
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 $\Bbb{R}$.
6
votes
0answers
218 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 ...
0
votes
0answers
50 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. ...
1
vote
2answers
313 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" ...
1
vote
1answer
111 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
votes
1answer
162 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
votes
3answers
117 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 ...
1
vote
2answers
566 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} \...
8
votes
1answer
211 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
950 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
votes
1answer
568 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 ...
1
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0answers
95 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
1k 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
vote
0answers
237 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
votes
1answer
2k 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
209 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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2answers
173 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
votes
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
votes
1answer
161 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 ...