Questions tagged [chaos-theory]

For questions in chaos theory.

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Duffing oscillator - Understanding the phase diagrams

I am trying to simulate a duffing oscillator. I am not sure how to read the phase diagrams I am obtaining. In the image, we can see at the top the phase diagram before the bifurcations; it has an ...
kimmil07's user avatar
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Unable to synchronize two chaotic systems [closed]

The graph plots the X coordinate of the synchronized Lorenz chaotic system. I am self learning by reading research articles on how to synchronize identical chaotic systems. But as seen from the figure,...
Sm1's user avatar
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Why is the number 3 come up so often in Chaos theory and Undecidability as a boundry? Is it just a coincedence?

What I mean is that the number 3 comes up a lot in these fields as sort of a boundary between decidability and undecidability, or chaos and order. Examples: Quadratic Diophantine equations are always ...
Colonizor48's user avatar
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Density of points under dyadic transformation

My question concerns the Bernoulli map (a.k.a. the shift map and as dyadic transformation). I understand that most numbers $x \in [0,1)$ do not have a periodic orbit under this transformation, and ...
subset's user avatar
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3 votes
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Showing that the horseshoe set is locally minimal

I'm trying to prove the Smale's horseshoe set is locally minimal. More specifically, let $H$ be the horseshoe set described in Section 1.8 in the book "Introduction to Dynamical Systems" by ...
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Calculate Lyapunov Exponent for Double Pendulum

I have the x and y coordinates for two trajectories separated by a small difference in initial conditions (for a double pendulum). Since the pendulum is physically constrained, there can only be local ...
MaximeJaccon's user avatar
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Is this a viable way to calculate the Lyapunov Exponent?

Is this a viable way to compute the Lyapunov Exponent for a double pendulum? Here is the code for the double pendulum (You don't have to look at this part, it just returns the angles and angular ...
MaximeJaccon's user avatar
1 vote
1 answer
147 views

How is Chaos defined in Math?

When I look at the English definition of the word "Chaos", I get the following definition : "Behavior so unpredictable as to appear random, owing to great sensitivity to small changes ...
Uk rain troll's user avatar
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2 answers
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Why does this distance estimator method render The Mandelbrot set incorrectly (non-divergent regions as divergent)?

I am using the following algorithm to render the Mandelbrot set and the exterior: • for each test point, calculate $c$ • initialise $z_{0}=(0+0i)$ • also initialise the gradient $dz_{0}=(0+0i)$ • ...
Penelope's user avatar
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Existence of a period three cycle for a continuous unimodal interval map (A question related to Li-Yorke theorem)

Let $f$ be a continuous unimodal map from a closed interval $I$ to itself. Suppose that there exist $x_1$, $x_2$, $x_3$, $x_4\in I$ such that $f(x_1)=x_2$, $f(x_2)=x_3$, $f(x_3)=x_4$, and $x_4<x_1&...
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Why more iterations benefit deeper Mandelbrot zooms over shallow zooms?

When rendering the Mandelbrot set fractal, we set a maximum number of iterations to test each point. If the escape criteria are met within the maximum iterations, we can stop further iterations ...
Penelope's user avatar
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Art historian trying to identify math diagrams in a painting

In a painting called 'Catastrophe Theory' from 1983, the German painter Sigmar Polke included the math diagrams shown in the linked image below (for another image see here: https://www.sfmoma.org/...
milque_toast's user avatar
2 votes
1 answer
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Lyapunov methods to show the trajectories that do not converge to the origin [closed]

Consider the map $f(z) = z^2$, where $z$ represents complex numbers. what is the function that $f$ corresponds to the map $p(r, \theta) = (r^2, \theta)$ in polar coordinates. And is it true that all ...
lulu's user avatar
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Formal definition of a chaotic deterministic system

I have recently been interested in chaos theory, after reading many books and textbooks speaking of the butterfly effect. But what is the rigorous definition of a chaotic deterministic system? It is ...
user107952's user avatar
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Proving this theorem about the skinny baker map

I'm working on a project for a class on Dynamical systems, and I've come to a point where even the professor was unable to help me. He excused me from having to prove it directly, but out of my own ...
Matthew's user avatar
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Best references for a more intuitive introduction to Dynamical Systems, Ergodic Theory and Quantum Chaos

As a graduate math and physics student, I am introducing myself to the study of Smooth Dynamical Systems and Ergodic Theory, with the aim of applying it to Quantum Chaos and Quantum Ergodic Theory. I ...
ayphyros's user avatar
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Are there continuous chaotic systems where nearby paths diverge at rates other then exponential?

I know that in general, nearby paths in a chaotic system tend to diverge exponentially, but are there continuous systems where paths diverge at other rates? For example, is there a system where nearby ...
Colonizor48's user avatar
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Finding and studying Poincaré map

I know what the definition of a Poincaré map is but I am struggling a little bit with the exercises and with computing it. I am trying to solve the following problem: Given the system (in polar ...
MathsLearner's user avatar
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Doubt in a paper on Devaney Chaos and dense periodic points

I am reading from the paper here. In Lemma 5, the authors make the following assertion: Let $f : [0, 1] \to [0, 1]$ be a continuous function with a dense set of periodic points. If $f$ has no proper ...
Ajin Shaji Jose's user avatar
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Does an infinite number of points in a closed interval imply density?

It is not very important, but I am asked to prove that the periodic points in the tent map are dense on the closed interval $[0,1]$. Anyway, I can prove that there are an infinite number of periodic ...
Matthew's user avatar
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Henon map's sink fixed point and saddle fixed point.

It is noticeable that the sinking and saddling points of the Henon map are appropriately satisfied in both $a = 0.27, -0.09$, and $b=0.4$. As written in this fig: My question is, why the Henson map ...
simple1's user avatar
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2 votes
1 answer
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Lyapunov Exponents for $n$-Dimensional Matrix $A$

I am wondering whether my solve is correct. I know how to solve the 2, or 3 dimension of the state matrix. But what if the state matrix goes to n-dim? Here is what I tried: To find the Lyapunov ...
lulu's user avatar
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Exploring Sensitivity Dependence in Chaotic Systems

After analyzing the sensitivity dependence of orbits under the map $f(x) = 2x \mod 1$, we have found that every initial point has nearby points that eventually diverge by at least $1/2$ unit after ...
simple1's user avatar
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On the mathematical expression for the logistic map

Mathematically, the logistic map is written as follows $$x_{n+1}=rx_{n}(1-x_{n})$$ Here, $x_n\in[0,1]$ represents the ratio of the existing population at each step ($n$) to the maximum possible ...
haiku's user avatar
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1 answer
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For any chaotic system such as a chaotic attractor does there exist a higher dimensional representation in which the system is no longer chaotic? [closed]

For a given n dimensional chaotic system such as a chaotic attractor or really a time dependent chaotic dynamic system does there exist a higher dimensional representation where the behavior is in ...
Matthew Wander's user avatar
1 vote
1 answer
25 views

Action of the adjoint operator in tangent space

I'm reading a paper on the computation of covariant Lyapunov vectors (https://arxiv.org/pdf/1212.3961.pdf) and, as I have a Machine Learning background, I have some gaps concerning dynamical systems. ...
Pepper08's user avatar
2 votes
1 answer
84 views

Is this chaotic behaviour? Weakly coupled Van der Pol oscillators

I was investigating the following system of two weakly coupled identical Van der Pol oscillators $$\left\{\begin{array}{@{}l@{}} \ddot{x}_1 + x_1 + \epsilon(x_1^2 - 1)\dot{x}_1 = \epsilon k(x_2 - x_1)...
Hervé Schmit-Veiler's user avatar
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60 views

Connection between ergodicity and Lyapunov exponents

This will be a soft reference question in a sense, as I will not state any rigorous theorems/results. Whenever I happen to read about ergodic systems, be it a specific book, article or a blog post, a ...
Epsilon Away's user avatar
15 votes
1 answer
355 views

Why does the Mandelbrot fractal appear when plotting $\underbrace{x\cos(x\cos( \cdots x\cos}_9(x))))$?

while plotting the function $x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x\cos(x)))))))))$ using matplotlib in python I found the mandelbrot fractal. What is the reason that the mandelbrot fractal ...
intro's user avatar
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How do I change this to integration over a trajectory?

I'm working through CHAOS: An Introduction to Dynamical Systems by Alligood et. al. and I'm on Challenge 7 step 5. At this point we have a system of ODEs. (Note: a dot over a variable is its ...
roundsquare's user avatar
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How do you show that a chaotic system such as the logistic map acts transitively upon an infinite set?

The logistic map with parameter $r=4$ and the tent map with parameter $\mu=2$ are topologically conjugate chaotic dynamical systems. Question Given an infinite set of points $S$ within the domain, how ...
it's a hire car baby's user avatar
2 votes
0 answers
45 views

Pre-images of the critical point of of $3.83 x (1-x)$

This question may be easy; however, I have been unable to locate any references regarding the specific scenario described below. Let $T:[0,1]\to [0,1]$ be the quadratic map $T(x) = 3.83 x (1-x)$. It ...
Matheus Manzatto's user avatar
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1 answer
65 views

How does it follow from $(|f(x)-f(p)|)/|x-p| <a$ that $|f^k(x)-p| \leq a^k|x-p|$

The exercise is asking me to show the latter theorem $|f^k(x)-p| \leq a^k|x-P|$ for $k=2$. But I am confused as to how they even derived that theorem. In the proof of a theorem, that had: $(1)$ There ...
Matthew's user avatar
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1 vote
0 answers
50 views

Precisely computing nth iterate of tent map

Suppose we have the tent map, $T_\mu(x) = \mu \min(x, 1-x)$ with parameter $\mu = 2$. Then note that if we pick an $x_0 \in [0, 1]\cap \mathbb{Q}$, so that if $x_0 = p/q$ with $p, q$ co-prime and $q$ ...
user918212's user avatar
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0 answers
21 views

How can non-chaotic curves fill a (hyper)torus and chaotic curves fill the entire energy hypersphere?

What I already know Before I ask my question, I would prefer to briefly explain what I already know, so that any gap in my understanding could be rectified. Note: I consider only bounded phase space ...
Souparna Nath's user avatar
1 vote
1 answer
77 views

Find the other fixed point of this 2-adic isometry

Let the Collatz function go $(3x+1)/2$ for odd numbers and $x/2$ for even. Now running the function, write a $1$ every time you hit an odd number and a $0$ every time you hit an even number. By this ...
it's a hire car baby's user avatar
3 votes
1 answer
131 views

Reference request: Why is an integrable system called an integrable system and why is the dynamical billiard on a disk completely integrable?

I am seeking detailed reference or references to help me understand the following: Relevant history and motivation behind the term "integrable system" with appropriate primers The meaning ...
Cartesian Bear's user avatar
2 votes
0 answers
70 views

Procedure of largest lyapunov exponent calculation

I am currently learning about chaos theory and lyapunov exponents. Specifically I am looking at a double pendulum and I am trying to calculate its largest lyapunov exponent. For that I am using the ...
Davide Farassino's user avatar
2 votes
1 answer
103 views

Deriving the Lyapunov function of synchronization error of Lorenz systems

I am currenty studying the synchronization of Lorenz systems, in which one system transmits one of its coordinates to the other and this drives the other system to converge towards it exponentially, ...
Georgios Paraskevopoulos's user avatar
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1 answer
42 views

What is the unstable or stable space of the Smale-Williams solenoid? [closed]

Consider the set $$ N = S^1 \times \{ (x,y)\in \mathbb{R} : x^2+y^2 \leq 1 \} $$ and the $\operatorname{map} f: N \rightarrow N$ defined by $$ f(\theta, x, y)=\left(2 \theta, \lambda x+\frac{1}{2} \...
Tiago Verissimo's user avatar
1 vote
0 answers
61 views

On physical measure on post-critical set for unimodal map

Let $f: [0,1] \to [0,1]$ be an infinitely renormalizable unimodal map, let $c$ denote its critical point. It is well-known that the post-critical set $\omega(c)$ of $f$ (which is the omega limit set ...
Soren Gee's user avatar
2 votes
0 answers
42 views

In the definition for sensitivity to initial conditions what exactly does the distance between trajectories mean?

I've seen this definition for sensitive dependance in Modeling Life (Garfinkel et al, 2010): $$d(M_{t} - N_{t}) = e^{\lambda * t} * d(M_0 - N_0)$$ or alternatively from Wikipedia: $$ {|\delta \mathbf {...
Hazumi's user avatar
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1 vote
0 answers
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Properties of the bifurcation diagram for the logistic function

Once the bifurcation diagram has been plotted ($x_{n+1}=rx_n(1-x_n)$), there are 3 elements or properties that I don't know haw to explain, and I have not found any article where they are explored. ...
Minerva González García's user avatar
1 vote
1 answer
65 views

Linearity of system of differential equations?

I am learning how to solve differential equations (ordinary and partial)and why they are so important for physics.One thing I have noticed so far is that we know so little on the nature of the ...
Volpina's user avatar
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1 vote
2 answers
324 views

Period $3$ orbit of the logistic map $x_{n+1}=r \cdot x_n(1-x_n)$

One can proof, that the logistic map has an stable orbit of period three for $r=1+2\sqrt{2}$. This can be done by looking at the third iterated of $f$ and investigate it for stable fixed points. For ...
David's user avatar
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2 votes
2 answers
385 views

Chaotic behavior of the logistic map at $r=4$.

With Sarkovskii's theorem I want to conclude the chaotic behavior of the logistic map $f(x)=r \cdot x(1-x)$. I can't find a value of $x$ which leads to a periodic three orbit. Does anyone know a value ...
David's user avatar
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2 votes
1 answer
292 views

Periodic orbits of the logistic map

I have a question about the period orbits of the logistic map $f(x)=r \cdot x(1-x), r \in [0,4], x \in [0,1]$. The bifurcation-diagram own for $r<3.5699$ only periodic orbits of period $ p=2^k,k \...
David's user avatar
  • 77
2 votes
1 answer
136 views

Sharkovskii`s theorem and a period three orbit of the logistic map at e.g. $r=3.83$

Let $f:[0,1]\to [0,1]$ be continuous with a period three orbit. We know with Sharkovskii`s theorem that f owns period orbits of any natural number. If we take a look to the bifurcation-diagram of the ...
David's user avatar
  • 77
1 vote
1 answer
58 views

Finding Critical Points of the Model $x=\exp(-rx)$ [closed]

Given the model $x_{t+1} = e^{-rx_t}$, how do I find the fixed/critical points? I know that critical points are $x = f(x)$ but how would you go about that here? Either as originally put or $\frac{\ln(...
L B's user avatar
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5 votes
0 answers
329 views

What is correlation dimension, actually?

I'm taking a course on chaotic dynamical systems, and we're talking about attractors with non-integer correlation dimensions, but I can't seem to find a satisfactory definition for this concept. ...
Frank Seidl's user avatar
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