Hot answers tagged chaos-theory
20
An orbit is stable because of conservation of angular momentum.
Suppose we start with an object in an exactly circular orbit and slow it down slightly. That means it is moving at less than orbital velocity so it starts to fall inwards. However as its distance to the Sun decreases the tangential component of its velocity has to increase to conserve angular ...
8
Well, yes. In a purely mathematical world where you can specify initial conditions exactly, chaotic systems are fully deterministic. It's not like a quantum system with wavefunction collapse, whose evolution can never be specified exactly by the initial conditions.
But in practice, we can never specify (or know) the initial conditions exactly. So there will ...
8
The gravitational potential is what is known as a central force, which means that "how strong" the potential is only depends on how far away you are, and not on what angle you are relative to it.
Having said that, gravitational systems are often treated in terms of an effective potential (full explanation provided on the Wikipedia page) which look like this ...
7
If you take a well-behaved physical system and perturb it a little bit, then you expect the total behavior or your system to be changed only a little bit. You can quantify this by saying that if your initial perturbation is $\delta$, then the final perturbation can never exceed $\gamma \delta$ for some constant $\gamma$. In many cases, such perturbations ...
6
David Bar Moshe's answer is fine, but I wanted to go into more detail. The main reason that random matrices show up in dynamical systems is because they describe the level statistics of classically chaotic motions. In classically integrable systems, there is a semiclassical formula for the level-spacing, determined by the Bohr-Sommerfeld rule. If you know ...
6
The basic idea is that statistical properties of complex physical systems
fall into a small number of universal classes. A very known example of
this phenomenon is the universal law implied by the central limit theorem
where the sum of a large number of random variables belonging to a large
class of distrubutions converges to the normal distribution. Please ...
6
Classical mechanics is perfectly integrable for two bodies as a closed or isolated system. However, early on it was found that problems existed, where Newton found he could not find a solution for the motion of the planets in a complete form. He made his famous statement that God had to readjust the solar system now and them. Poincare solved the Sweden ...
6
I think the following from the wikipedia entry clears up well the terminology:
Chaos theory is a field of study in applied mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions; an effect which is ...
6
Good question. You are correct in that without any restoring force, an object balanced precariously in an equilibrium position will be unstable. In physics, we use the scalar quantity of "potential" to find the equilibrium positions. These will be the maxima and minima in the potential field. The negative gradient of the potential gives the force.
You've ...
6
Qualitative discussion
(almost math free)
The real key to understand orbits is the conservation of angular momentum. A two body orbit is nice this way insofar as it is a planar system and we get an easy expression for the angular momentum (we'll assume a satellite much, much less massive than the primary and not bother with the canonical transformation ...
5
Jack Wisdom at MIT has extensively studied the question of the stability of the solar system. He has a list of papers with links to freely-readable PDF files on his website:
http://groups.csail.mit.edu/mac/users/wisdom/
A good starting point might be "Is the Solar System Stable? and Can We Use Chaos to Make Measurements?" (PDF) (in Chaos, proceedings ...
5
It is true that the double pendulum exhibits integrable behavior, when the initial angles are very small, however, in general, it is very difficult to characterize the chaotic behavior of the double pendulum in terms of the initial angles. There are other representations which provide a clearer picture of its chaotic behavior.
The introductory section of ...
4
Strictly speaking, there is no quantum chaos. Time evolution is
unitary, which implies that small changes in state are not magnified
in size. Thus the sensitive dependence on initial conditions, the
prerequisite for chaos in classical mechanics, is absent in quantum
mechanics. Even stronger, in discrete quantum systems with a
finite-dimensional Hilbert ...
4
I'm guessing that when you talk about randomness you're thinking about the collapse of the wavefunction and that the the result of the collapse is apparently random. If so, most us currently believe that the randomness is only apparent and is the result of decoherence.
Decoherence describes the interaction of a quantum system with the environment around it. ...
4
In general, no. It is possible to recognise a (system of) differential equation(s) as being nonlinear purely by inspection, but there are plenty of non-chaotic nonlinear systems. Chaos is a stronger (and, unfortunately, not well-defined) condition.
Also, many (most?) systems, including the famed Lorenz attractor, only exhibit chaos under certain conditions. ...
4
Let's start out using notation similar to the example you linked to:
$$
\ddot{y}+b\dot{y}+\sin(y)=a\cos(ct)+d
$$
As in the example, we'll write this in autonomous form:
$$
\mathbf{x}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=\begin{pmatrix}y\\\dot{y}\\ct\end{pmatrix}
$$
the differential equation can now be written:
$$
...
3
The best reference for this is Feigenbaum's original article, reprinted in "Universality in Chaos" by Cvitanovic. The point is that when you iterate a map, every time you period double, you fold up the function one more time. The behavior is dominated by the solution to the following equation:
$$ \alpha g(g(x/\alpha)) = g(x)$$
Which says that g iterated ...
3
A stepping mortor or magnetic field lines in a Tokamak when two magnetic islands overlap are simple examples of chaotic behaviour (really different from turbulent) that can be easily controlled.
http://www.epj.org/_pdf/HP_EPJB_slowly_rocking.pdf
http://www-student.elec.qmul.ac.uk/people/josh/documents/ReissAlinSandlerRobert-ICIT2002.pdf
Lyshevski S., ...
3
I am assuming that the idea of fractal cosmology expresses that the universe has some sort of fractional pattern to it, which is the nature of fractal construct, and implies homogeneity.
No, this is basically wrong. Homogeneity is mutually exclusive with fractal. This isn't my area either, but I will try to describe the state of our understanding ...
2
The weather is a chaotic system and has the characteristic of having critical dependence on initial conditions. This in practice means that any forecast over 2 or 3 days starts to be flawed.
Chaotic systems are a feature of mathematics, and as such are present in classical physics:
Magnetic pendulum
Double pendulum
Dripping
In practice, when people use ...
2
The butterfly effect is to be taken metaphorically. The crux of it really is long-term unpredictability due to sensitivity to initial conditions. No serious meteorologist believes that the formation of a tornado could depend on whether a butterfly flaps its wings or not. Reliable short-term predictions of weather (over a few days) can be and are being made ...
2
You pretty much know it already. "Random" is a broad word that we use to mean that we can't predict behavior. Each of the three cases of randomness that you cite is unpredictable for a different reason, though - that's the difference.
Dice are random because they are complicated, chaotic pendulums are random because we aren't good enough to measure their ...
2
I want to examine whether the elucidated question:
the randomness and probabilities observed in quantum mechanics is really just the result of a chaotic (yet deterministic) system.
can be answered in the positive or the negative.
From the wikipedia entry one has an adequate definition of deterministic chaos :
Small differences in initial ...
2
Yes, there is investigation.
Some random names on the field (more on the physics side, NO specific order): Carl Dettmann, Tamás Tél, Ott, Ying-Cheng Lai, Adilson Motter, Celso Grebogi, Holger Kantz, Alessandro Moura, Eduardo G. Altmann, etc, etc, etc.
A quick search on some of these names should help you to find some recent papers on what is being done ...
2
In this PhD thesis (unfortunately in German :-/, a follow up paper can be found here), it is shown how for an action of a field theory containing even and Grassman fields, the renormalization group equation can be solved numerically (after expanding the action in derivatives and the fields). To investigate the corresponding renormalization group flow in the ...
1
Since time evolution is linear for a quantum system, this rules out classical chaos in terms of hypersensivity to initial conditions. The question of how quantum theory could explain classical chaos is adressed from the perspective of decoherence. This interesting document may be of some help: http://www.iqc.ca/publications/tutorials/chaos.pdf
1
No, or at least not in the sense the phrase "butterfly effect" is normally used. Well, possibly, but only if the critical system is chaotic.
The phrase is normally applied to systems that show chaotic behaviour. In such systems the trajectory of the system is very sensitive to the starting point i.e. if you take two points very close together in phase space ...
1
This is a famous period doubling experiment which is reprinted in Cvitanovic's "Universality in Chaos", along with other foundational papers on the period doubling route to chaos.
The paper you read is probably due to Libchaber and Maurer "A Rayleigh Bernard Experiment: Helium in a small box" Proceedings of the NATO Advanced Studies Institute on Nonlinear ...
1
I think it depends on the meaning of "application". In the wikipedia entry there are a lot of applications of chaos theory listed:
Chaos theory is applied in many scientific disciplines, including: geology, mathematics, microbiology, biology, computer science, economics, engineering, finance, meteorology, philosophy, physics, politics, population ...
1
Einstein train thought experiment
I understand this by analogy with Einsteins train thought experiment. Let's say I am on a sidewalk facing over the street, where two people pass each other in opposite directions (one of them facing Andromeda), while they just pass each other captain Zorg of Andromeda Galaxy decides to attack the earth and launches the ...
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