#PhysicsFactlet
Does a chaotic system always behave chaotically?
Not really, as many chaotic systems have a subset of possible initial conditions that lead to a quasi-periodic motion.
As an example, below are two sets (black and orange) of 20 double pendula each, all with the same initial energy, and each group starting with very similar initial conditions.
The first group (black) spread out a little bit with time, but nearby initial conditions keeps evolving into nearby dynamics, which is typical of integrable systems.
On the other hand the pendula in the second group (orange) also starts with similar initial condition, but after a short transient evolve each very differently from each other, which is a mark of a chaotic system. #Physics#Visualization#Chaos
@j_bertolotti Neat! Is there a paper somewhere on regions of integrability in the phase space of the double pendulum? My cursory search didn't pull one up.
Released into the #PublicDomain, and uploaded to #WikimediaCommons together with the #Mathematica script used to generate it: https://commons.wikimedia.org/wiki/File:Non_chaotic_double_pendulum.gif
@jaztrophysicist I already have a lecture on it, so I am ready 😉
The problem with Hamiltonian chaos is the huge amount of pregress knowledge necessary to talk meaningfully about it, which makes any attempt at being precise in its popularization quite hard.
But I might try when I have a bit of time!
@j_bertolotti See the solar system - technically a chaotic system, but stable enough that the sun will probably go red giant before Earth gets kicked out of orbit. And even self-organising at times, trojan/greek asteroids at Jupiter's Lagrange points, tidal locking, orbital resonances etc. Not that any of it can be easily predicted in detail. "Chaos within reason" perhaps.
@_thegeoff I think the solar system case is a bit different. We know it is chaotic, but we also know it has a VERY small Lyapunov exponent (i.e. a VERY long time before chaos sets in).
But there are sets of initial conditions for the 3-body problem that are known to lead to a perfectly periodic motion (there are also for a system as big and complicated as the solar system, but nobody bothered to find them, as far as I know).
@j_bertolotti Yeah, I suppose it's trivial to find 3-body solutions that are stable for "a long time" (mostly placing at least one "far away"?), if not permanently, mathematically stable (which I'm not sure is possible in the real world, non rigid bodies etc?)
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