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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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
$$
\mathcal{H}(\mathbf{z};\mathbf{z}^*) = \sum_{i=1}^{M}...
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When a non-linear dynamical system is put under stress, is the tendency towards infinite attractors a phase transition?
An attractor is a region of the state space which, when a system enters that state, keeps the system from evolving out of it.
Given certain parameters, such as energy, a system will settle into a set ...
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What are some highly unstable phenomena you can observe from lightyears away?
I'm currently drafting some scifi story where the protagonist is in a long timeloop and wants to figure out when the loop starts exactly based on observing some unstable phenomena, like atomic decay ...
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Could chaos theory and butterfly theory prove that time only exists in the present? [closed]
I am sorry if this were to sound dumb but, chaos theory is about reconizing patterns and the universe itself can only repeat it self
, would that mean that time only exist in the present cause if time ...
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Why the kinetic term of the Hamiltonian has to be positive definite for well-posed time evolution?
I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12:
$$\frac{\mathcal{S}}{\mathcal{T}}= \int \mathrm{d}t\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)...
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Why does chaos preclude exact solutions?
It is sometimes said that the n-body problem (using the initial positions and velocities of n point masses to calculate their future paths) has no general closed-form solution because the system is ...
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The establishment of Out-of-Time-Ordered Correlators (OTOCs) from Lyapunov Exponents (LEs)?
the OTOC in quantum system is
$$F(t) = \langle \hat{W}^\dagger(t) \hat{V}^\dagger(0) \hat{W}(t) \hat{V}(0) \rangle_{\beta}
$$
the Lyapunov exponent is
$$\lambda = \lim_{{t \to \infty}} \lim_{{d(0) \to ...
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Is there a formalization of the butterfly effect?
According to my understanding, the butterfly effect says, very informally, that even a tiny perturbation in a physical system can lead to significant alterations in future states of the physical ...
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Lyapunov exponents as a measure of chaos and how to report them
How are Lyapunov exponents of dynamical systems used as a measure of chaos? If I have the double pendulum, a chaotic system, then using the equation for the Lyapunov exponent (Wikipedia's article &...
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Information Loss in Chaos Theory
Is it accurate to suggest that in chaos theory, information is in practice lost due to the impossibility of characterizing the system's state with infinite precision, making it unfeasible to run the ...
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Entropy in chaos dynamics
I'm curious about how entropy is defined within chaos theory. Are there analogous laws similar to the second law of thermodynamics? How do we define steady-state or equilibrium within the state space ...
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Can protrusions on the smooth surface of a floating (or flying) body not slow it down, but accelerate it?
These protrusions are sure to create turbulent vortices. But what if these additional vortices can somehow lead to acceleration?
Additional clarification
It is clear that moving protrusions such as ...
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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 ...
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What kind of attractor have I obtained?
This is some sort of attractor I have obtained in the digital oscilloscope. The oscilloscope is connected to a chua circuit of my own design. My real question is whether that the phase plot I have ...
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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 ...
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Is there a generic behavior of Spectral Form Factor for Integrable models?
The spectral form factor is defined as (usually taken at $\beta = 0$ by definition along with disorder average)
\begin{equation}\label{eq:SFF1}
g(\beta,t) = \left| \frac{Z(\beta,t)}{Z(\beta)}\...
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Mutual information between energy levels
I have the following partition function
$$Z(\beta_1,\beta_2) = Z(\beta_1)Z(\beta_2) +Z(\beta_1,\beta_2)_c$$
Thus, I have a non-zero mutual information $I(Z(\beta_1);Z(\beta_2)) = S(Z(\beta_1))+S(Z(\...
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Which symmetry is broken during an AdS black hole phase transition and does it lead to chaos?
I was going through this paper in which the authors state that :
In Landau theory, a continuous phase transition is associated with a broken symmetry.
The phase transition in a black hole system can ...
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Intution for the physical meaning of high energy limit of a quantum states and uniform distribution in phase spacehow of a particle
Zeev Rudnick state in his talk Quantum Ergodicity for the Uninitiated (around 12 minute 40 second mark at the last text section of the slide) that a "a possible interpretation of the statement ...
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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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Is the universe deterministic, random, or both? [duplicate]
It occurred to me that the limits of possibility to the nature of the universe is it is either deterministic ie we are all at the will of natural laws that determine the outcome of events from the ...
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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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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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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 ...
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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 ...
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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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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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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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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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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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