All Questions
Tagged with complex-systems chaos-theory
130 questions
5
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How can bounded orbits diverge arbitrarily far at late times?
While there are many definitions of chaotic dynamics, many commonly used ones require a positive Lyapunov exponent, which is defined as
$$\lambda := \lim_{t \to \infty} \lim_{\delta{\bf Z}_0 \to {\bf ...
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-3
votes
1
answer
123
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Lyapunov is wrong - got unstable on a stable system [closed]
I'm angry with the Lyapunov stability criteria. Consider this system:
Here, $u$ is the input and $x_1$, $x_2$ are my state variables. Now, solve for the transference of the system, defining my output ...
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1
vote
2
answers
128
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Is it possible to define a universal formula for chaos?
I've been working on a chaos project. I have noticed that there are several formulas to find the behavior of chaos, for example:
The logistic map is a simple equation that exhibits chaotic behavior ...
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1
vote
0
answers
78
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Symmetry and integrability in classical Hamiltonian
I am trying to understand the behaviour of an Hamiltonian system I'm simulating. I will give a quick context setting. The system is defined as
$$
\mathcal{H}(\mathbf{z};\mathbf{z}^*) = \sum_{i=1}^{M}...
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0
votes
0
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164
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Lyapunov Exponent for Double Pendulum
I want to calculate the Lyapunov Exponent for a double pendulum, with a small change in the initial angle. In this study, the authors used the formula $\frac{1}{t}{ln(\frac{d}{d_0})}$ as $t$ tends to ...
- 136
1
vote
0
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48
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Two-body problem + shield
Let us consider two point charges, one positive, one negative, interacting via Coulomb force.
In the absence of any other force, this system constitutes an elementary example of two-body problem, and ...
- 1,252
3
votes
1
answer
111
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What exactly is KAM stability and how can I determine if an orbit is KAM stable or not?
I have been working on the three-body problem lately and came across KAM stability. I read that KAM stability generally means that the solution is stable at different initial conditions (that of ...
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1
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0
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110
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Curvature and stability
In Topological methods in hydrodynamics 1 mentioned that "The Riemannian curvature of a manifold has a profound impact on the behavior of geodesics on it. If the Riemannian curvature of a ...
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1
vote
1
answer
128
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Calculating the Lyapunov exponents spectrum from particle trajectories
I am simulating a forced, compressible 2D flow, that is turbulent and statistically steady, but not stationary.
I want to calculate the Lyapunov exponents spectrum from the trajectories of Lagrangian ...
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2
votes
0
answers
94
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Does Poisson Distribution means the system is chaotic?
The Berry-Tabor Conjecture says that for classically integrable systems, the corresponding quantum systems obey the Poisson distribution for their energy-level spacing. But generally, the integrable ...
- 21
3
votes
2
answers
90
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Effect of reorthonormalisation step size when calculating Lyapunov exponents using the Gram–Schmidt reorthonormalisation (GSR) procedure
I am trying to determine the Lyapunov exponent using Gram–Schmidt reorthonormalisation (GSR), for a well-defined dynamical system (I know the differential equations etc). I believe I have implemented ...
- 31
1
vote
1
answer
169
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Stroboscopic map
I am trying to plot the stroboscopic map of the classical kicked rotor, which is characterized by the equations:
$$p_{n+1} = p_n - \frac{dV}{dx}|_{x=x_n}$$
$$x_{n+1} = x_n +p_{n+1}$$
where $x_n$ is on ...
- 359
0
votes
1
answer
131
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Why the 3d Lorenz attractor has a butterfly shape? Why isn't it 3 dimensional too? [closed]
The Lorenz attractor has a butterfly shaped a strange attractor, but we plot it in 3D. Why is not it has a 3D shape too? It has a strange shape? It is a non-integer dimensional attractor.
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8
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4
answers
2k
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Do all dynamical systems have attractors?
Do all dynamical systems have attractors?
Is there any chance that there are two or more absolutely the same sets of states in one attractor?
- 181
1
vote
1
answer
127
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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 $x(t) = \sum_{n=0} c_n \cos(n \omega_0 t)$, $\lim_{n \to \infty} c_n = 0$. The Fourier transform $\tilde{x}(\omega)$...
- 39
2
votes
1
answer
136
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Lyapunov exponent of "real life"
Today I simply forgot watching soccer WM on TV, and promptly my national team lost. Assume there is a meaningful alternative universe where I turned on the TV (quantum and relativity theorists already ...
- 233
1
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2
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89
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Literature reference: example of stable and unstable manifolds in Henon-Heiles system
There is a quite classical description of chaotic systems based on the behaviour of stable and unstable manifolds around a stationary point of the Poincaré section. It is presented, for example, [here,...
0
votes
4
answers
458
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Chaos theory: What exactly drives the future outcome?
Chaos theory states that we can't predict future because we can't measure initial conditions of a system to infinite precision. I get that.
That alone doesn't mean that the future is not determined, ...
- 153
0
votes
2
answers
68
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Trying to prove chaotic motion from the equation of a nonlinear oscillation [closed]
So I'm given the equation of a nonlinear oscillation:
$x''+ω_0^2x=λx^3$
Assume that $x_1$ and $x_2$ are solutions to the differential equation above.
Therefore;
$x = αx_1+βx_2$
$x' = αx_1'+βx_2'$
$x'' ...
- 41
1
vote
3
answers
748
views
What is a phase space?
What is a phase space? And can the phase space be specified with x and y instead of with theta and omega?
I am currently working on a problem where I am graphing the trajectories of three masses (the ...
- 151
1
vote
1
answer
266
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Area of Phase Space and Dependence on Energy
The phase curve for a system is made for some configuration, for example - The Harmonic Oscillator. Now as we increase the energy, the phase curve enlarges i.e. area enclosed by the curve increases.
...
6
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3
answers
1k
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A rule for when phase-space orbits may cross
Note: in this question when I talk about "phase space," I will be refering to velocity vs. position space, which can also be correctly referred to as "state space." Many sources (...
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1
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1
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220
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How to understand the largest Lyapunov exponent?
Some more information and answers are here: https://math.stackexchange.com/q/4451013/577710 .
It is said that
..the largest Lyapunov exponent, which measures the average exponential rate of ...
- 207
0
votes
0
answers
219
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Calculating the Maximal Lyapunov Exponent for a Hamiltonian System
I am consider the following Hamiltonian: $$\mathcal{H} = \frac{1}{2}(\dot{x}^2 + \dot{y}^2) + \frac{1}{2}(x^2 + y^2) + x^2y - \frac{y^3}{3}.$$ The first step I took were to solve the equations of ...
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2
votes
1
answer
81
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Advection Term in the Lorenz 96 Model
The Lorenz 96 Model is defined as
$$\frac{dx_i}{dt}=\underbrace{(x_{i+1}-x_{i-2})x_{i-1}}_{advection}-x_i+F$$
with some forcing $F$ and periodic boundary conditions so that $x_{i+N}=x_i$ for some $N$.
...
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0
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0
answers
77
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Calculating Lyapunov exponent (LE) for pendulum using ellipsoid growth - code yields negative LEs
Per my advisor, I have read the textbook Chaos, an introduction to dynamical systems by Alligood, Sauer, and Yorke. (side not, I have really enjoyed this book). In chapter 5, the numerical calculation ...
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2
votes
1
answer
276
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Phase space portrait for dynamical system with Bifurcations
I have this dynamical system
$$x'=y, y'=-x^3-y+mx$$
and I want to draw the phace space diagram for $m=-1/8, m=1/4,$ the bifurcation points. 1st of all I cant find what kind of bifruction I have( I go ...
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1
vote
1
answer
68
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Mixing for Burgers equation in 2+1D
Let us consider the following (2+1)-dimensional Burgers-like equation:
$$
u_t + (u^2)_x + (u^3)_y=0.
$$
Here the unknown is a function $u= u(t,x,y):(0,\infty) \times \mathbb R^2 \to \mathbb R$.
Is ...
- 111
1
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1
answer
130
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Question about energy threshold for bounded and unbounded motion from a research paper
I was reading a research paper titled "Dynamical analysis of bounded and unbounded orbits in a generalized Hénon-Heiles system." The link to the paper is here and on arXiv. In this paper, ...
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2
votes
1
answer
133
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A coupled nonlinear dynamical system in four dimensional phase space
I have come across a coupled nonlinear dynamical system given below
$$ r\, \ddot{x} + \dot{x} = \sin y~,$$
$$ r\, \ddot{y} + \dot{y} = \sin x~,$$
where $r$ is some real number and $\dot{x}$ denotes $\...
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0
votes
1
answer
134
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Time taken for a system to return to it's original state
Consider the following system:
There are N particles (point-like particles) of $1$ Kg each in a Sphere of radius $R$ centered at origin in three dimensions. Randomly assign these N particles their ...
- 101
1
vote
1
answer
52
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Linearization of 1D maps about a fixed unstable point [closed]
Recently, I was going through the paper Controlling Chemical Chaos in a three variable autocatalator system, by Peng et al. Here are the references
Although I have been introduced to 1D maps and the ...
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13
votes
2
answers
1k
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Are chaotic systems the same as dissipative systems in inverse time?
Lyapunov exponents define whether a system expands or contracts in phase space and can be used to determine whether a dynamical system is chaotic, conservative, or dissipative. If the volume expands ...
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4
answers
1k
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How can one distinguish between a random process and a chaotic process? [duplicate]
Chaos is not a random process, although it may look like one. If I am given a set of observations, is it possible to determine if the observations are generated by a random process or if they are ...
2
votes
2
answers
155
views
What are the implications of deterministic chaos: useful or detrimental? [closed]
I am new to the concept of chaos theory and as a layman I am struggling to understand what is the significance and implication of chaos in ecological systems such as the chaotic predator prey model. I ...
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4
votes
3
answers
737
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Are chaotic systems examples of complex systems?
I am struggling to find a proper source or reference where examples of complex systems which are chaotic are given. Based on my understanding, complex systems consist of interacting components, each ...
- 235
0
votes
1
answer
369
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Lyapunov Exponent of the Logistic map [closed]
My dynamical system professor (and the textbooks we use) all claim that the Lyapunov exponent for the Logistic map with $r=4$ ($x_{n+1} = 4x_n(1-x_n)$) is $\log(2)$. Would someone be able to sketch ...
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2
votes
2
answers
195
views
When is molecular chaos dynamical chaos?
It is very common to have uncorrelated velocities in chaotic dynamical systems. Yet, we should be wary in equating the two quite different meanings of chaos.
Instead of matching dynamical chaos to ...
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4
votes
3
answers
193
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Non-Analytic Equations and Chaos
Could anyone please tell me an example of an equation with no analytic solution(s) that is not a chaotic one? And what is the physical meaning of having analytic solution? For instance, the three body ...
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2
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107
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Is there a mathematical way to determine if a force, phenomena or physical entity is in a state of chaos?
We often talk about chaos, but is chaos an objective term or a subjective term? If it's an objective term, is there a mathematical way to determine it? Is it possible there's a threshold where ...
user236241
1
vote
0
answers
73
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Is a satellite orbit around the Earth Lyapunov stable?
Presume there is a satellite orbiting the Earth in an orbit that follows a closed path around the planet (that is, escape orbits are not permitted here). As I understand it, there are two ...
3
votes
1
answer
99
views
One third of Lyapunov exponents are zero? What does it mean?
This may be quite a straightforward question, but I have a dynamical system with a high dimensional phase-space. I calculated the Lyapunov spectrum for it and saw that one third of my Lyapunov ...
- 598
0
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1
answer
39
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About natural frequencies in non-excited pendulums and Poincaré sections
How can a Poincaré map be defined for a double pendulum (or Furuta pendulum) when these systems don't have external excitations?
2
votes
0
answers
51
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Destruction of integrals of motion in chaotic systems: Fermi-Pasta-Ulam (FPU) paradox
I am trying to understand behavior of system studied by Fermi, Pasta and Ulam i.e. chain of oscillators interacting via nonlinear forces. I am generally not very familiar with chaos theory and ...
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2
votes
1
answer
153
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Is instability + sensitivity to initial conditions = Chaos?
Please correct me where wrong. I am having trouble finding answers to these specific questions.
(1) In chaotic systems, does the presence of chaos and a strange attractor indicate that there is no ...
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4
votes
2
answers
213
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Why would we want to calculate the Lyapunov exponent for experimental data?
Searching Google Scholar for "Lyapunov exponent from time series" turns up multiple papers (some of them highly cited) suggesting methods for estimating the largest Lyapunov exponent or sometimes even ...
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0
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74
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What can one conclude about the stability of limit cycles without the use of numerical methods?
Let's assume one asserts the existence of a closed orbit by applyling the Poincaré-Bendixson theorem to a trapping region $R$ that is constructed such that all phase vectors on its boundary point ...
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0
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1
answer
61
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What is a "Doppler instability"?
In the paper "Flow-induced control of chemical turbulence" by Berenstein and Beta, the term "Doppler instability" is mentioned in the context of the Belousov-Zhabotinsky reaction.
I am looking for a ...
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3
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3k
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What do physicists mean by an "integrable system"?
The notion of "integrability" is everywhere in physics these days. It's a hot topic in high energy theory, atomic physics, and condensed matter. I hear the word at least once a week, and every time, I ...
- 105k
2
votes
2
answers
576
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Arnold's Mathematical Methods of Classical Mechanics and Lyapunov stability
In Arnold's Classical Mechanics of Classical Mechanics, he refers to Lyapunov stability in many of the problems in the second chapter.
E.g. on page 20: "Problem: Consider a periodic motion along the ...
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