Questions tagged [chaos-theory]
For questions in chaos theory.
0
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0answers
29 views
Hypercyclic operators in $L_p (0,\infty)$
I'm pretty new to this area of study so if there are logical lacune in my proof (I'm sure there are many) please let me know.
This is material I'm self studying. I'm trying to adapt the methods used ...
2
votes
0answers
39 views
Although Lorenz system is a deterministic system, can it shows locally stochastic behavior?
The Lorenz system is a system of ordinary differential equations
$$\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x),\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y,\\{...
0
votes
0answers
30 views
Perron-Frobenius operator definition
According to
http://mathworld.wolfram.com/Perron-FrobeniusOperator.html, the P-F operator describes the time evolution of densities in phase space:
$$\rho_{n+1} = P (\rho_n).$$
But for the ...
0
votes
1answer
50 views
Hypercyclicity examples
Does anyone have simple practical examples of hypercyclicity they use in explaining the concept (graphically or numerically)? This appears often in texts about chaos in infinite dimensional linear ...
1
vote
1answer
33 views
Does a greater Lyapunov exponent result in a more chaotic system?
Let $\sigma_1$ be the Lyapunov exponent of a one-dimensional system $I_1$ and $\sigma_2$ be Lyapunov exponent of one-dimensional system $I_2$. Can I say that if $\sigma_1 > \sigma_2$, then $I_1$ is ...
4
votes
1answer
32 views
Periodic solutions of the double pendulum
I'm stuck: Are there periodic solutions of the double pendulum, or not?
The question is four-fold:
Does every double pendulum - defined by two masses $m_1$, $m_2$ and lengths $l_1$, $l_2$ - have ...
6
votes
1answer
60 views
How to calculate the envelope of the trajectory of a double pendulum?
Consider a double pendulum:
Background
For the angles $\varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$):
$\dot{\varphi_1} = 6\frac{...
2
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0answers
49 views
Project ideas on Chaos theory, Cellular Automata, Fractals, Games, IA [closed]
I'm a computer science student and I need to find a final year project. What interests me the most is Chaos, IA, Games, Fractals, CA..
Something I liked was the chaos theory within sudoku.
The ...
0
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0answers
6 views
How to calculte the Fourier Transform of a sovable chaos waveform?
Recently I am stucking in frequency estimation of a solvable chaos waveform.
Its analytic expression in time domain is
$$
z(t)=s_m(u_m-s_m)e^{\beta(t-mT)}\cos(\omega_0 t+\varphi)
$$
where $u_m \sim U(-...
0
votes
1answer
35 views
Let $M$ be a shift space over a finite alphabet $\mathcal{A}$. Prove that $M$ is compact in the metric topology $\tau_{\rho}$.
GIVEN
Define a map
\begin{equation*} \label{eq1}
\begin{split}
\rho(x,y) =
\begin{cases}
2^{-k} \ \ \ &\text{if } x \neq y, \text{ and } k \text{ is maximal so that } x_{[-k.k]} = y_{[-k,k]...
1
vote
0answers
35 views
Is $\tan(x)$ chaotic on the entire real line?
In Robert L. Devaney's "An Introduction to Chaotic Dynamical Systems" it defines a function $f:J\longrightarrow J$ to be expansive if there exists $\nu>0$ such that, for any $x,y\in J$, there ...
5
votes
2answers
164 views
Existence of infinite iteration of functions $f_\infty$?
Given a sequence of functions $\{f_n\}$ satisifying an iterated relation such as
$f_n(x)=g(x+f_{n-1}(x))$
$f_n(x)=g(xf_{n-1}(x))$
$f_n(x)=g(x/f_{n-1}(x))$
Where $g:=f_1$ is ...
1
vote
1answer
91 views
Diffeomorphism with only hyperbolic periodic points has finitely many periodic points (Morse-Smale)
Came across the following question during a course Chaotic Dynamical Systems:
If a diffeomorphism $f:I\to I$ is Morse-Smale (i.e. has only hyperbolic periodic points), then it has finitely many ...
1
vote
1answer
101 views
How reliable a measure of chaos is the largest Lyapunov exponent?
I’m dealing with a system of ODEs where I think I might have found chaotic solutions. I’ve calculated the largest Lyapunov exponent, and found it to be approximately 0.02. It’s positive, which ...
3
votes
1answer
48 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
286 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 ...
2
votes
2answers
122 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
52 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
41 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
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0answers
101 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
26 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
25 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
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0answers
15 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
51 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$$
undergoes a Hopf bifurcation at $(x, y) = (0, 0)$ as $\mu$ passes through 0. Calculate the approximated ...
5
votes
1answer
126 views
Do all systems of differential equations have a solution?
I'm not really sure that I have the required knowledge to pose this question properly so I'll just pose it poorly.
The rise of numerics has allowed us to study systems of differential equations for ...
1
vote
2answers
126 views
Show that the origin of the following system is stable (non-linear vs linear stability)
$$\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
49 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
97 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
138 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
91 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
53 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
43 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})$ ...
0
votes
1answer
53 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
40 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
157 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
31 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
48 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
$$\chi(x_{0},...
1
vote
0answers
182 views
The surface of Aizawa attractor
Aizawa attractor creates a closed surface. Let's ignore the transient state in the beginning. I am wondering how to calculate its surface? I even don't know where to start.
The dynamic model of ...
1
vote
0answers
78 views
Inverse of Baker's map
Consider a matrix $A$ whose size is $n\times n$, and the following Baker's map, that maps $A$ to $B$, whose size is $\frac{n}{2}\times2n$:
$$ B(i,j) = \begin{cases} \frac{1}{2}\left(A(2i,\lceil\frac{...
0
votes
0answers
28 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
34 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
105 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
1answer
66 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
61 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
70 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
28 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
47 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
135 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
85 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
141 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 $...
