Questions tagged [chaos-theory]
For questions in chaos theory.
3
votes
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
votes
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
votes
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?
0
votes
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
votes
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 ...
0
votes
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
vote
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 ...
-1
votes
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 ...
0
votes
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
votes
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
votes
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
vote
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
votes
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
vote
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
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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
vote
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
votes
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 ...
1
vote
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 ...
0
votes
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 $\partial \Omega$:
$$
...
2
votes
1answer
46 views
A question on Lyapunov exponents associated with a fixed point of a vector field.
The Lyapunov exponent $\chi(x_{0}, e)$ in a direction $e \in \mathbb{R}^{n}$ along the trajectory $x(t,x_{0})$ of a vector field $\dot{x} = f(x)$ through a point $x_{0}$ is defined to be
$$\chi(x_{0},...
1
vote
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 ...
1
vote
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 $\frac{n}{2}\times2n$:
$$ B(i,j) = \begin{cases} \frac{1}{2}\left(A(2i,\lceil\frac{...
0
votes
0answers
18 views
measure of the chaotic region of the logistic map
The logistic map has infinitely many regions with periodic behavior after the onset of chaos, called "islands of stability". I'm interested how large the chaotic region actually is, after you subtract ...
-1
votes
1answer
27 views
logistic map with lamba greater than 4
I was doing some recreational math about the logistic map. (If you're not familiar with what the logistic map is, here are some links you can check out)
https://www.youtube.com/watch?v=ETrYE4MdoLQ
...
3
votes
1answer
89 views
Chaos of Irrational Numbers
I just finished a course on chaos and fractals, and it got me thinking about irrational numbers, and I was thinking about whether or not the distribution of the digits are chaotic.
Consider a map $\...
0
votes
0answers
30 views
How are Chaotic systems different from Ill-posed systems?
I got puzzled when I thought of two concepts that is 1) Chaotic Systems 2)Well posed and ill-posed systems.
Chaotic systems are those dynamical systems which have the sensitive dependence to the ...
1
vote
1answer
45 views
How do I find the fixed points of this 2-d map?
Consider the two-dimensional map
$$
\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}
$$
where $r$ is a control ...
0
votes
0answers
60 views
Poincaré-Bendixson criterion applied to non-autonomous dynamical systems
I'm fairly new to dynamical systems and chaos. I'm dealing with a second order non-autonomous system of differential equations:
$$\dot{x} = y $$
$$\dot{y} = f(x,y,t)$$
I can further split $f$ as ...
1
vote
0answers
26 views
Newhouse Ruelle Takens theorem and three frequency quasiperiodicty
NRT theorem suggests that there exist arbitrarily small perturbations of a vector field on an $m$-torus for $m\geq3$ leading to strange axiom A attractors. Although, three frequency quasi periodic ...
1
vote
1answer
42 views
Periodic points of $F(x)=\lambda x + \beta x^3$ where $\lambda >1$ and $\beta <0$
So, when the coefficients are somewhat arbitary, I'm quite confused. Obviously there are 3 fixed points. There are other quite obvious ones, by letting $\lambda x+ \beta x^3 = -x$.
However when we ...
2
votes
2answers
134 views
Solving $F(x) = F(e^x) - F(-e^x)$
What are non-trivial functions such that for all $x \in \mathbb{R}$,
$$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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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^{...
8
votes
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}) =
\...
1
vote
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-...
0
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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 ...
