Questions tagged [chaos-theory]
Chaos theory is the study of systems that are highly sensitive to slight, even imperceptible changes in initial conditions. This is popularly known as the butterfly effect. Many natural systems exhibit chaotic behavior, including weather and electron orbitals.
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Assumptions in the bound on chaos
In the paper A bound on chaos, by Maldacena, Shenker and Stanford. They mention two assumptions to prove that the Lyapunov exponent in the OTOCs must be smaller than or equal to $2\pi T$.
One of the ...
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Is a free rigid body in 3D space an integrable system? [duplicate]
I am trying to find three integrable systems with 6 degrees of freedom using the Liouville–Arnold theorem. That means that a set of integrals of motion that correspond to a conserved quantity for ...
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Why Black Hole is maximally chaotic?
I understand intuitively that black holes are chaotic. However, people say black holes are not just chaotic, they are "maximally chaotic". What is the quantitative definition of "...
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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 ...
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Why there’s level repulsion in quantum chaotic systems?
What’s the physical intuition of level repulsion in thermalized quantum systems? And from the opposite, why integratable systems display a Poisson distribution of the spectrum?
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Does the value of Maximum Lyapunov exponent depend on the eigenvalues of the system?
I am currently reading this paper where on page 8, the authors say that:
Negative eigenvalues correspond to unstable systems.
This correlates with Figure 8 on page 12.
Does it mean that there is a ...
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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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Is it possible to realize a probabilistic Maxwell's demon using Tesla valve?
Imagine a container, with balls of diameter 1x and 10x, moving randomly in all directions. These balls are mixed, so it is a low order system.
Now this container is connected to another container via ...
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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?
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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 $\tilde{x}(\omega)$...
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Per Newtonian mechanics, a coin toss exhibits deterministic chaos theory, but could relativity cause a probabilistic outcome of a coin toss?
It took me a long time to accept that a coin toss boils down to deterministic chaos theory. For example, the typical near 50/50 odds for outcomes of heads or tails results from complex initial ...
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Boltzmann's equation for interacting case
Gravitation Foundations and Frontiers book by T. Padmanabhan tells in its first chapter that
This result allows us to introduce distribution functions in relativistic theory in exact analogy with non-...
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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 ...
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Can a saddle point corresponding to black hole horizon act as a source of chaos?
I was going through this paper where on page 5 they argue that in the given Poincare section:
Most of the orbits form regular tori, while we can also see that some tori near the saddle point of the ...
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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,...
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Kinetic term of the Hamiltonian constructed from the action of perturbative string motion is not positive definite
I am trying to reproduce the results from this paper. On page 10 of the paper, they have an equation: $$\frac{S}{T}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+11.3 c_0^3+21.5 c_0 c_1^2+10....
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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, ...
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What is needed to base physics on iterated functions? [closed]
Iterated systems are considered candidates for the foundation of physics. Iterated functions have an even stronger claim:
Arnold and Avez (1968)
Let $M$ be a smooth manifold, $\mu$ a measure on $M$ ...
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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'' ...
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How Far in Advance can Eclipses be Predicted? [duplicate]
How far into the future (or past) can we predict when eclipses will occur, given the chaotic nature of the Sun-Earth-Moon system?
I know that the location of an eclipse becomes subject to considerable ...
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Uncomputability of the $n$-body problem
The gravitational $n$-body problem is well known to be uncomputable; one can not find a general algorithm that works in all cases that can predict the trajectories of $n$n-bodies. However, in contrast ...
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Under dynamic scaling, Ising model emerge fractals, which implicate what？
In the lecture note of statistic field theory of David Tong,
Fluctuations occur on all length scales, big and small.
he post a link, a youtube video, which describe the Ising model in different ...
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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 ...
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Chaotic Pendulum - Strange Attractor, no zero lyapunov exponent in spectrum
I have been studying the chaotic behavior of pendulum systems inside a buoy on the ocean. I've simulated such a system using Simscape multibody, and I obtain all the usual characteristics of chaos. It ...
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How can non-chaotic curves fill a (hyper)torus and chaotic curves fill the entire energy hypersphere?
What I already know
Before I ask my question, I would prefer to briefly explain what I already know, so that any gap in my understanding could be rectified.
Note: I consider only bounded phase space ...
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How can we generalize the Poincaré recurrence time to other sets of events?
I know that the Poincaré recurrence tells us that every given physical arrangement in a finite physical space will eventually recur given enough time.
However, as I was thinking about it, even if my ...
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What are some chaotic long lasting initial conditions for the three-body problem?
I have been trying to find some good long lasting chaotic trajectories for the three-body problem, but it seems to be very hard to find.
All I find on the internet is a lot of periodic solutions for ...
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How to Interpret Solutions to Simple Chaotic Systems
Note: I'm a non-physicist so please take any misunderstandings/poor notation on my end with patience.
I remember watching a pop-physics video around 5 years ago which discussed the solution space of ...
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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.
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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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Unitarity and small perturbations
Consider a system whose initial state is written $|{\Psi(0)\rangle}=|{\chi(0)}\rangle+|{\epsilon(0)}\rangle$. Here $|{\epsilon(0)}\rangle$ represents a small perturbation i.e. $\lVert|{\epsilon}\...
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(Discrepancy in the) Statement of Eigenstate thermalization hypothesis
I am trying to understand ETH and unfortunately came across a seemingly contradicting definition by the same author (Mark Srednicki). I don't know which definition is correct.
At this instance in this ...
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What are some sources to study about chaos in dynamic systems? [duplicate]
I want to study about dynamic systems sensitive to initial conditions and chaotic motion.
Please recommend some lectures or notes for it.
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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 ...
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Are all turbulent flows chaotic?
I have often read that all turbulent flows are chaotic. I have come across a few explanations on web forums, but are there any authoritative references/textbooks that explicitly cover this question?
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The unlikely marriage of orbiting bodies
O.k. so we have an orbiting body such as the Earth around the sun or the moon around the Earth. The fact that they are orbiting does not fascinate me, it is my intuitive sense that there is a much ...
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Are there any symmetries in the Mixmaster model?
The Mixmaster model is usually presented as one example of chaotic models in physics and cosmology.
Usually, "chaos" or "randomness" may be related to a complete lack of symmetries ...
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Estimation of number of states to be used to obtain Wigner-Dyson distribution in a chaotic coupled oscilllator
A coupled harmonic oscillator with quadratic coupling -
$$
H = \frac{1}{2}(p_x^2 + p_y^2) + \frac{1}{2}(x^2 + y^2) + g x^2y^2,
$$
is known to be non-integrable, hence chaotic (for reference look at
...
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Does Wick theorem hold for ensembles of random unitary matrices?
Consider ensembles of random unitary matrices with weight function $w(\phi)$, which have partition functions of the form
$$
Z= \int_{U(N)} w(U)dU = \int \prod_{i=1}^N \frac{d \phi_i}{2\pi } w(\phi_i)\...
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How to understand Heisenberg time in random matrix theory?
Recently, from few papers, I have encountered the word 'Heisenberg time' $t_{\text{H}}$ which is an inverse of a mean level spacing $\Delta(\hat{\mathcal{H}})$ of a finite system Hamiltonian $\hat{\...
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Graphing the motion of a driven damped pendulum
$ \ddot{\phi}+2 \beta \dot{\phi}+\omega_{0}^{2} \sin \phi=\gamma \omega_{0}^{2} \cos \omega t$
The equation of movement is written above. I´m trying to plot the value of $\phi$ for different values of ...
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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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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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Can the phase portrait of SHO rotate counter-clockwise? or is it the case that there can be no physical motion corresponding to that?
Framing the question
In the case for Simple Harmonic Oscillation, we have the equation:
$$\ddot{x}+x=0 \tag{1} \label{1}$$
(say, we put all the coefficients to be 1)
Now, if we try to solve it in ...
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Is a coin toss pseudo-random or truly random? [duplicate]
I wonder if a coin toss is pseudo-random or truly random. Sure, you could say that a coin toss is pseudo-random because you don't know the speed of the coin or its rotation, but if you were to include ...
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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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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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Is pendulum only chaotic when its force is a single sinusoidal frequency?
Chaotic systems will be chaotic under certain parameters, but when the forced, damped pendulum is discussed, the initial conditions of its position and velocity are only mentioned. Is it assumed that ...
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How many flips does a tossed macroscopic coin need to go through until the coinflip's result becomes indeterministic?
A coinflip is a macroscopic event and is deterministic in nature. A coin-flipping machine that operates at the greatest physical precision possible would be able to predict the coinflip's result (...
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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 ...
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