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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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Is the butterfly effect real?

Is the butterfly effect real? It is a well known statement that a butterfly can, by flapping her wings in a slightly different way, cause a hurricane somewhere else in the world that wouldn't occur if ...
6k views

How far ahead can we predict solar and lunar eclipses?

The solar system is non-integrable and has chaos. The sun-earth-moon three-body system might be chaotic. So, how far into the future can we predict solar eclipses and/or lunar eclipses? How about ...
3k views

Why are we sure that integrals of motion don't exist in a chaotic system?

The stadium billiard is known to be a chaotic system. This means that the only integral of motion (quantity which is conserved along any trajectory of motion) is the energy $E=(p_x^2+p_y^2)/2m$. Why ...
5k views

Are double pendulums eventually periodic?

I've often heard it said that the motion of a double pendulum is non-periodic. (This may be related to the fact that it's a chaotic system, but I'm not sure about that.) But this does not seem ...
3k views

How do computers “solve” the three-body-problem?

I've done a bit of research, and have learned that computers "solve" the three-body-problem by using "Numerical methods for ordinary differential equations", but I can't really find anything about it ...
9k views

Why do many people link entropy to chaos?

I understand that, in thermodynamics, entropy has a precise definition (the infinitesimal change of entropy being the infinitesimal heat transfer divided by the temperature), and that in statistical ...
1k views

Why is the computer useful if a chaotic system is sensitive to numeric error?

In every textbook on chaos, there are a lot of numerical simulations. A typical example is the Poincare section. But why is numerical simulation still meaningful if the system is very sensitive to ...
1k views

Staying in orbit - but doesn't any perturbation start a positive feedback?

I am not a physicist; I am a software engineer. While trying to fall asleep recently, I started thinking about the following. There are many explanations online of how any object stays in orbit. The ...
912 views

The natural metric of a phase space and the Lyapunov exponent

For me, it seems that there is no apparent metric on a phase space of a dynamical system. Of course one can naively define an Euclidean metric on it, but it seems that this metric has not much to do ...
1k views

What is a quantum scar?

This notion was proposed by Heller in 1984. But his paper is hard to follow (at least for me). Does anyone has a good understanding? Is it just judged by the naked eye?
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What creates the chaotic motion on a double pendulum?

As we know, The double pendulum has a chaotic motion. But, why is this? I mean, the mass of the two pendulums are the same and they have the same length. But, what makes its motion random? I'm just ...
1k views

A good, concrete example of using “chaos theory” to solve an easily understood engineering problem? [closed]

Can anyone suggest a good, concrete example of using "chaos theory" to solve an easily understood engineering problem? I'm wondering if there is a an answer of the following sort: "We have a high ...
1k views

Are all aperiodic systems chaotic?

So I understand that a chaotic system is a deterministic system, which produces aperiodic long-term behaviour and is hyper-sensitive to initial conditions. So are all aperiodic systems chaotic? Are ...
12k views

Poincaré maps and interpretation

What are Poincaré maps and how to understand them? Wikipedia says: In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the ...
304 views

Is there a connection between Bertrands theorem and Chaos theory?

Bertrand's theorem states Among central force potentials with bound orbits, there are only two types of central force potentials with the property that all bound orbits are also closed orbits, ...
2k views

Calculate/Estimate the fractal dimention of the logistic map

This is the logistic map:. It is a fractal, as some might know here. It has a Hausdorff fractal dimension of 0.538. Is it possible to calculate/measure its fractal dimension using the box counting ...
3k views

Does the “Andromeda Paradox” (Rietdijk–Putnam-Penrose) imply a completely deterministic universe?

Wikipedia article: http://en.wikipedia.org/wiki/Rietdijk–Putnam_argument Abstract of 1966 Rietdijk paper: A proof is given that there does not exist an event, that is not already in the past for ...
8k views

Chaos theory and determinism

My professor in class went a little over chaos theory, and basically said that Newtonian determinism no longer applies, since as time goes to infinity, no matter how close together two initial points ...
2k views

Will glass always break in the same way?

This question has had me thinking for a while. If I have two large panes of glass and a rock or similar item is thrown in exactly the same place on the glass, would the two panes break in the same way....
599 views

Chladni figures: avoided crossings of nodal lines

As khown, Chladni figures display nodal lines of eigenfunctions satisfying the equation $\Delta^2\psi=k^4\psi$ with appropriate boundary conditions. One can note these lines don't like to cross each ...
361 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant in essential ...
1k views

Ljapunov exponent of driven damped pendulum

I have written a computer simulation of the driven damped pendulum, pretty much as the one shown here, only that I did it Python. Next, I have found some parameters for which the pendulum behaves ...
717 views

Is that true that real quantum chaos doesn't exist?

I read several books and papers on quantum chaos, to my understanding they all emphases that the quantum chaos does not really exist because the linearity of the Schrodinger equation. Some works were ...
726 views

Is it really impossible to calculate in advance the result of throwing dice?

Is it really impossible to calculate in advance the result of throwing dice? After all, the physics of dice throwing is in the world of classical mechanics, rather than quantum mechanics.
156 views

Are there any known models with limit cycles in their RG flow?

The text-book presentation of the renormalization group (RG) leaves one with the impression that all systems will eventually flow to a fixed point. This is somewhat enforced by the phenomenological ...
446 views

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Physics-oriented books on fractals

I'm looking for some good books on fractals, with a spin to applications in physics. Specifically, applications of fractal geometry to differential equations and dynamical systems, but with emphasis ...
1k views

How and why can random matrices answer physical problems?

Random matrix theory pops up regularly in the context of dynamical systems. I was, however, so far not able to grasp the basic idea of this formalism. Could someone please provide an instructive ...
2k views

What is the highest energy position for a double pendulum? And for which energy positions is it chaotic?

Math/physics teachers love to break out the double pendulum as an example of chaotic motion that is very sensitive to initial conditions. I have some questions about specific properties: For a ...
817 views

Can a small change in the Earth temperature give rise to large-scale climate changes?

Earth's atmosphere is a chaotic system. In such systems arbitrarily small changes the conditions can give rise to very large effects. There are many rumors about the physical and large scale ...
2k views

Randomness, Chaos, Quantum mechanical probability functions

Can someone explain these 3 concepts into a unified framework. Randomness : Randomness as seen in a coin toss, where the system follows known and deterministic (at the length and scale and precision ...
305 views

What advantages have a symplectic or geometric integrator over a simple one, say, RK4?

I heard that a symplectic integration algorithm has a property related to the phase space of a system, but i don't understand much further than that. I'm interested in applying that method to a non-...
213 views

What causes the chaotic trajectory of a balloon going flat? [duplicate]

I think this is not a duplicate of the question John mentions. That question asks specifically why the balloon makes a spiraling movement, which in general is not the case (the answers pay only ...
2k views

Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics?

Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics? Apart from perhaps thermodynamics?
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Is long-term weather forecast impossible in principle?

This question can be asked about any chaotic dynamical system, but hydrodynamics of the atmosphere makes it more concrete. Arnold describes his 1966 result as follows: I have calculated the ...
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Non-integrability of the 2D double pendulum

Context: For a system with $n$ degrees of freedom (DOF), one has to deal with $2n$ independent coordinates ($2n$ dimensional phase space), of position $q$ and $\dot{q}$ in Lagrangian formulation, or ...
719 views

Will double or triple pendulums sync when they are on a platform they can affect?

what i am talking about is something along the lines of this video, http://www.youtube.com/watch?v=5v5eBf2KwF8 where 30 metronomes sync themselves on a table. Will the same happen with double or ...
872 views

Does a simple double pendulum have transients?

Suppose, we have the most simple double pendulum: Both masses are equal. Both limbs are equal. No friction. No driver. Arbitrary initial conditions (no restriction to low energies) Does this ...
3k views

What is a physical example of a Saddle-Node Bifurcation?

I am doing a presentation on bifurcations and would like physical examples to go along with each type of bifurcation but I am unable to find or think of any good example of a simple Saddle Node ...
98 views

Simplest model of chaos with time-independent smooth Hamiltonian and trivial topology?

What is the simplest model of chaos governed by a time-independent smooth Hamiltonian on a phase-space with trivial topology? We know that... With trivial topology, the minimal number of dimension ...
875 views

What is discrete phase space?

I've been reading a little about the usual, continuous Wigner functions and phase space quasi-distributions in general, and I believe I understand the idea behind them. The Wigner function arises when ...
The Out-of-time-ordered correlator (OTOC) is defined as $\langle[W(t),V(0)]^2\rangle$, and can be considered as a new way to extract quantum chaos. However, the understanding of this special ...