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Questions tagged [chaos-theory]

For questions in chaos theory.

3
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1answer
37 views

Are there non-chaotic systems which exhibits topological mixing?

I was recently reading about chaos theory. Chaos is commonly defined to exhibit 3 properties: Sensitivity to initial conditions/Lyapunov coefficient is positive Exhibits topological mixing Dense ...
4
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1answer
210 views

Fractal Dream Attractor behavior

This question is similar to this one: Properties of King's Dream fractal The fractal is described here: Softology - Visions Of Chaos 2D Strange Attractor Tutorial I accidentally modefied the formula ...
3
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2answers
86 views

Can Brownian motion be regarded as chaos?

I want to know if it is included in chaos. Does it have boundedness, deterministic, initial value sensitivity that is characteristic of chaos?
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0answers
48 views

What is the name of this chaotic sequence?

A long time ago I came across a chaotic sequence of the form x[n+1] = f(x[n], y[n]) y[n+1] = g(x[n], y[n]) If I remember correctly, ...
0
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0answers
29 views

Examples of chaotic motion in non-deterministic dynamical systems?

I was reading this article on Hyperbolic dynamics, in which I came across the line "deterministic chaos - the appearance of chaotic motions in purely deterministic dynamical systems". Hence I wonder ...
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0answers
80 views

Proof check: Show that $x_{n + 1} = f(x_n) = 15 - x_n^2$ is a (Smale) horseshoe

A solution is given here: However, I believe it to be incorrect because $f(K_1) = f(K_2) \neq (-\sqrt{15}, \sqrt{15})$. Attempt at a solution: Referring to the diagram from the question... By the ...
1
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0answers
24 views

To classify bifurcations by calculating partial derivatives evaluated at the bifurcation point

Question: Given a one dimensional system $\dot{x} = f(x;\; \mu)$ for which a bifurcation of fixed points occurs at $(x^\star, \mu_c)$, where $x^\star$ is a fixed point and $\mu_c$ is the ...
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1answer
24 views

Determining Rate of Convergence for Chaotic Behavior

I've got a set of data that's been manipulated solely numerically. But anyway, for a certain part of that data set, I see that it's convergent (image). But I'm sort of at a loss for how depict that ...
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0answers
14 views

If a system is defined in $n$-dimension then it has $n$ Lyapunov exponents

I wanted to know if there is a formal proof to show that if a system is defined in $n$-dimension then it has $n$ Lyapunov exponents. Any links regarding information about this and Kaplan-Yorke ...
2
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1answer
45 views

To find the approximate period and approximate equation of the limit cycle for a system with a Hopf bifurcation

Question: The system $$ \dot{x} = 3y + 3x^3 + xy $$ $$\dot{y} = -3x + \mu y + 2xy^2 - y^3$$ undergoes a Hopf bifurcation at $(x, y) = (0, 0)$ as $\mu$ passes through 0. Calculate the approximated ...
5
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1answer
111 views

Do all systems of differential equations have a solution?

I'm not really sure that I have the required knowledge to pose this question properly so I'll just pose it poorly. The rise of numerics has allowed us to study systems of differential equations for ...
1
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2answers
120 views

Show that the origin of the following system is stable (non-linear vs linear stability)

$$\dot{x} = -2y -x^3 + x^2 y^2$$ $$\dot{y} = x - x^2 y$$ Jacobian evaluation at $(0, 0)$: $$\begin{pmatrix}0 & -2\\ 1 & 0\end{pmatrix}$$ Clearly the determinant is greater than zero but the ...
0
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1answer
47 views

What is the “bifurcation of a fixed point”?

Isn't the bifurcation of a system, say $\dot{x} = \mu \sin(x) - 2x$, related to the system rather than a fixed point, say $x^{\star} = 0$?. My understanding is that a bifurcation occurs when there's ...
1
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0answers
69 views

Understanding the definition of Sensitive dependence on initial conditions?

I was trying to understand the rigorous definition of sensitive dependence on initial conditions which is as follows - $f : X \mapsto X$ where $X$ is a metric space. If there exists $\epsilon > ...
10
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1answer
124 views

Lorenz Attractor, its Geometric Model, and 14th Smale's problem.

I've found a post with a beautifully animated video that states the following: In 2001 mathematician Warwick Tucker proved that the paper model accurately describes the motion on the Lorenz ...
0
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0answers
77 views

Understanding fixed point of Arnold's Cat Map

I'm currently performing a small investigation into the properties of Arnold's Cat Map, a function which is defined as: C:[0,1) -> [0,1) C(x,y) = (2x+y, x+y) (mod 1) This function has exactly one ...
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0answers
47 views

A question on Smale-Birkhoff homoclinic theorem

Does Smale-Birkhoff homoclinic theorem ("given a transverse homoclinic intersection for a diffeomorphism $f$, there exists an integer $n \geq 1$ such that $f^{n}$ has an invariant Cantor set on which ...
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0answers
41 views

Measure of the periodic orbits of this simple dynamical system

Consider the dynamical system $\phi$ on a phase space $M =\{(x,y): x \in [0, 2\pi), y \in [0,\pi)\}$ where we take $x(mod 2\pi)$, i.e. $M$ is a cylinder. Define $(x_{1}, y_{1}) = \phi(x_{0}, y_{0})$ ...
1
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1answer
36 views

Does positive topological entropy imply horseshoes?

I am aware that positive topological entropy implies chaos in the sense of Li-Yorke. I want to get an idea of what consequences positive topological entropy has for the the presence (or lack of) ...
4
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0answers
29 views

Peaks in the circle chaos game

The chaos game is a random walk that steps halfway between your current position and a set of predefined points. If the points are on an equilateral triangle, the resulting set is Sierpinski's ...
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0answers
147 views

What can linear stability analysis say about trajectories starting in the neighbourhood of a fixed point? The role of Arnold diffusion

Consider a Hamiltonian dynamical system having more degrees of freedom than invariant quantities (Suppose that the number of degrees of freedom is $D>1$ and that the only invariant quantity is the ...
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0answers
28 views

Differences in the eigenvalue distribution between “regular” and chaotic domains

Weyl's conjecture gives us the asymptotic distribution of the eigenvalues $\lambda$ of the Laplace operator in a domain $\Omega$ with homogeneous Dirichlet boundary condition on $\partial \Omega$: $$ ...
2
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1answer
46 views

A question on Lyapunov exponents associated with a fixed point of a vector field.

The Lyapunov exponent $\chi(x_{0}, e)$ in a direction $e \in \mathbb{R}^{n}$ along the trajectory $x(t,x_{0})$ of a vector field $\dot{x} = f(x)$ through a point $x_{0}$ is defined to be $$\chi(x_{0},...
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0answers
139 views

The surface of Aizawa attractor

Aizawa attractor creates a closed surface. Let's ignore the transient state in the beginning. I am wondering how to calculate its surface? I even don't know where to start. The dynamic model of ...
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0answers
67 views

Inverse of Baker's map

Consider a matrix $A$ whose size is $n\times n$, and the following Baker's map, that maps $A$ to $B$, whose size is $\frac{n}{2}\times2n$: $$ B(i,j) = \begin{cases} \frac{1}{2}\left(A(2i,\lceil\frac{...
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0answers
18 views

measure of the chaotic region of the logistic map

The logistic map has infinitely many regions with periodic behavior after the onset of chaos, called "islands of stability". I'm interested how large the chaotic region actually is, after you subtract ...
-1
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1answer
27 views

logistic map with lamba greater than 4

I was doing some recreational math about the logistic map. (If you're not familiar with what the logistic map is, here are some links you can check out) https://www.youtube.com/watch?v=ETrYE4MdoLQ ...
3
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1answer
89 views

Chaos of Irrational Numbers

I just finished a course on chaos and fractals, and it got me thinking about irrational numbers, and I was thinking about whether or not the distribution of the digits are chaotic. Consider a map $\...
0
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0answers
30 views

How are Chaotic systems different from Ill-posed systems?

I got puzzled when I thought of two concepts that is 1) Chaotic Systems 2)Well posed and ill-posed systems. Chaotic systems are those dynamical systems which have the sensitive dependence to the ...
1
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1answer
45 views

How do I find the fixed points of this 2-d map?

Consider the two-dimensional map $$ \begin{aligned} X_{n+1} &= X_n + Y_n + r\left(X_n - X^3_n\right)\\ Y_{n+1} &= Y_n + r\left(X_n - X^3_n\right) \end{aligned} $$ where $r$ is a control ...
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0answers
60 views

Poincaré-Bendixson criterion applied to non-autonomous dynamical systems

I'm fairly new to dynamical systems and chaos. I'm dealing with a second order non-autonomous system of differential equations: $$\dot{x} = y $$ $$\dot{y} = f(x,y,t)$$ I can further split $f$ as ...
1
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0answers
26 views

Newhouse Ruelle Takens theorem and three frequency quasiperiodicty

NRT theorem suggests that there exist arbitrarily small perturbations of a vector field on an $m$-torus for $m\geq3$ leading to strange axiom A attractors. Although, three frequency quasi periodic ...
1
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1answer
42 views

Periodic points of $F(x)=\lambda x + \beta x^3$ where $\lambda >1$ and $\beta <0$

So, when the coefficients are somewhat arbitary, I'm quite confused. Obviously there are 3 fixed points. There are other quite obvious ones, by letting $\lambda x+ \beta x^3 = -x$. However when we ...
2
votes
2answers
134 views

Solving $F(x) = F(e^x) - F(-e^x)$

What are non-trivial functions such that for all $x \in \mathbb{R}$, $$F(x) = F(e^x) - F(-e^x)?$$ I obtained this equations after seeking for random variable $X$ with CDF $F$ such that $\log |X|$ is ...
2
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2answers
76 views

Are multiple kinds of attractors (chaotic and otherwise) possible within a single system of differential equations?

I’m looking for any $n$-dimensional system of first order differential equations where depending on the initial conditions you can end up in a number of attractors, for example multiple chaotic orbits,...
2
votes
1answer
112 views

How to prove a fixed point is stable?

\begin{align*} \dot{x} &= 2 x - \frac{8}{5} x^2 - xy\\ \dot{y} &= \frac{5}{2} y - y^2 - 2 xy \end{align*} So I have this dynamical system. I linearized it and found that the fixed point (at $...
0
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0answers
27 views

Computation of largest Lyapunov exponent from the Variational equation.

Is there any computational method or analytical method to compute the largest Lyapunov exponent of the variational equation $\dot{\eta_{i}} = A\eta_{i} - \sigma\lambda_{i} H \eta_{i}$. Actually, ...
0
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0answers
26 views

Query on Master Stability Function applied in case of Diffusively coupled Chua circuits.

Let there be $N$ nodes and $\dot{x^{i}} = F(x^{i}) + \sum_{j=1}^{N} G_{ij} H(x_{j})$. Here $\dot{x^{i}}$ is the $m$ dimensional vector of dynamical variables of the $i$th node. The Dynamics of the $...
5
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1answer
81 views

Does Day and Night on a Klein bottle have a steady state?

Place a $m \times n$ ($m,n \ge 3$) square grid on a Klein bottle. On each square, we select an arbitrary non-mirror symmetric marker, and arrange them on the Klein bottle in some way. This arrangement ...
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0answers
17 views

construction of henon map

I am trying to construct a suspension that will model the Henon map. So far I have the suspension $\dot\phi=lnA\phi$ which is a linear suspension of a mapping A. The solution to this is $\phi=e^{...
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0answers
90 views

Can I define the Julia and Fatou sets of a complex non rational function? If not, how can I characterize this fractal?

Context: I have a family of discrete-time, nonlinear, dynamical system (the same one described on this preceding question) whose expression is as follows: $$S_{D,t}=\left\{(x,y): (x_{n+1},y_{n+1}) = \...
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0answers
26 views

non-deterministic basin of attraction plot

How to plot non-deterministic basin of attraction for given values? Let us consider a system is with two attractor states 0 and 15 and following table shows the probability for each attractor state-...
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0answers
25 views

Simple Billiard Table with Positive Lyapunov exponent

I can understand the Lyapunov exponent for a circular billiard table is zero. However, if I were right, I read the Lyapunov exponent for a stadium billiard table is non-zero which was surprising for ...
24
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1answer
2k views

Identifying this chaotic (?) recurrence relation

Update: I'm working on an interactive bifurcation diagram: http://matt-diamond.com/sineMap.html Here's the image when the starting coordinates are [0.5, 0.5] The bifurcation diagrams differ ...
3
votes
1answer
61 views

ireducible polynomials with coefficients in $\{0,-1\}$

I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - \cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-...
2
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1answer
53 views

Intuition behind dense orbits

I'm having some difficulties discerning the difference between attracting sets an attractors in my nonlinear systems course. The definition we've been given is that attractors are attracting sets that ...
5
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1answer
109 views

Is this chaotic map known?

Have you ever encountered the following map before? $$x_{n+1}=T(x_n), x\in[0,1]$$ where $$T(x)= \begin{cases} \frac{x}{1-x} & x\leq 1/2 \\ 1-\frac{1-x}{x} & x> 1/2 \end{...
3
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1answer
92 views

Does this dynamical system show an “absorbing area” or a “chaotic area”?

I am following the technical report by C.Mira: "Noninvertible maps: notion of chaotic area vs that of strange attractor" in order to characterize the behavior some dynamical systems of my own. In the ...
5
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2answers
79 views

Why do we need folding and a finite domain for chaos?

One of the generic ways to obtain chaos in phase space is when the system causes trajectories to stretch and fold. I understand that the stretching will cause neighboring initial conditions to ...
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0answers
62 views

Moser invariant curves in discrete dynamical systems and how they give stability

I have seen a theorem which says that under certain hypothesis, given an elliptic fixed point $P$ for particular discrete dynamical systems ($X_{k+1}=S(X_k)$ with $S$ a conservative diffeomorphism not ...