All Questions
Tagged with nonlinear-dynamics chaos-theory
37
questions
1
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18
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Can probe trajectories to compute Lyapunov exponents get "stuck in more regular orbits" after rescaling?
I am computing Lyapunov exponents, and there is something that I do not understand about the data.
The model has a regime for $\delta \approx 1$ (in some units) where it is fully chaotic, and the ...
- 31
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0
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25
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Logistic map: bifurcation and domain of attraction
Let $f(x) = \mu x(1-x)$ be the logistic map, the question is divided into 3 parts:
Part (1): what can you say about the domain of attraction of the 2-cycle in $3<\mu<1+\sqrt 6$?
My attempt: let $...
- 305
2
votes
1
answer
94
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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
$$\left\{\begin{array}{@{}l@{}}
\ddot{x}_1 + x_1 + \epsilon(x_1^2 - 1)\dot{x}_1 = \epsilon k(x_2 - x_1)...
5
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0
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444
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What is correlation dimension, actually?
I'm taking a course on chaotic dynamical systems, and we're talking about attractors with non-integer correlation dimensions, but I can't seem to find a satisfactory definition for this concept. ...
- 1,016
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38
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Distinguishing between chaos and multiperiodic oscillations from the Fourier spectrum
Consider a system which exhibits multiperiodicity, say with oscillations of the form
$$x(t) = \sum_{n=0} c_n \cos(n \omega_0 t), \qquad \lim_{n \to \infty} c_n = 0$$
The Fourier transform $\tilde{x}(\...
- 121
1
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1
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113
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Show the system has one equilibrium point
I was wondering how we would show that the system:
$$\frac{dx}{dt}=-x^3+2x-4y \\
\frac{dy}{dt}=-y^3+2y+4x$$
has only one equilibrium point.
I have seen cases where the system is, for example:
$$\...
- 69
1
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0
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31
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Lyapunov dimension
I have a nonlinear differential equation system composed of 4 equations. I calculated Lyapunov's dimension of each of the states to be a little bit over 3 (say 3.11, 3.1, 3.13, 3.14). How can I ...
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2
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116
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No Chaos in $\mathbb{R}^2$
While reading some basic introductory texts in nonlinear dynamics, it was asserted that no chaotic behaviour for flows can occur in $\mathbb{R}^2$ because of the Poincare-Bendixson Theorem. ...
- 1,020
1
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0
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91
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What is the difference between Lyapunov Exponent vs time and Lyapunov Exponent vs parameter?
What is the difference between Lyapunov Exponent vs time and Lyapunov Exponent vs parameter?
I noticed that some plot graph of Lyapunov Exponent but are not usually the same x-axis. Some plot graph ...
- 466
0
votes
1
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88
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Consider the set, recursively built, starting from the unit interval and removing the first $\frac{1}{3}$ at each step. Find the similarity dimension. [closed]
My thinking for this question is that it is just a slight variation of the standard Cantor set and will therefore have the same similarity dimension. My logic is that at each new step, the interval ...
- 105
1
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1
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97
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Use polar co-ordinates to show the fixed point of the dynamical system $x^\prime = x-y-x^{3}$ and $y^\prime = x+y-y^{3}$ has distance <2 from (0,0)
Using polar co-ordinates to solve the dynamical system:
$$x^\prime = x-y-x^{3}$$
$$y^\prime = x+y-y^{3}$$
I have arrived at: \begin{align}
r' &= r-r^3(cos^4\theta+sin^4\theta)\\
\theta' &= ...
- 105
0
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0
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103
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For the Hénon map $y_{n+1} = 1 - a{y}_n^{2}+bx_n$ and $x_{n+1} = y_n$, Assess the stability of the period-2 orbit when $b=0.3$ and $a=3.675$.
For the Hénon map $y_{n+1} = 1 - a{y}_n^{2}+bx_n$ and $x_{n+1} = y_n$, Assess the stability of the period-2 orbit when $b=0.3$ and $a=3.675$.
I understand that the two points of the orbit must satisfy ...
- 105
1
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1
answer
168
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Using polar co-ordinates to find the fixed point of the dynamical system $x^\prime = x-y-x^{3}$ and $y^\prime = x+y-y^{3}$.
Using polar co-ordinates to solve the dynamical system:
$$x^\prime = x-y-x^{3}$$
$$y^\prime = x+y-y^{3}$$
I would say that I mostly understand this question and need to manipulate the equations using ...
- 105
0
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0
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30
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Assume $f(x)$ is a 1D map exhibiting a period-3 orbit P. Then is P a period 3 orbit of $f(f(x))$?
Assume $f(x)$ is a 1D map exhibiting a period-3 orbit P. Then P is a period 3 orbit of $f(f(x))$?
I know this is a somewhat elementary question but I'm only asking it to confirm if my thought process ...
- 105
1
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0
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46
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Can $x(t) = Acos(at^2+b)$ be interpreted as a generic chaotic trajectory of some dynamical system?
Can $x(t) = A\cos(at^2+b)$ be interpreted as a generic chaotic trajectory of some dynamical system?
We have just been introduced to the background theory behind chaotic systems but haven't worked ...
- 105