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Consider a deterministic system (a gas, a liquid, or a solid, each of which can have an arbitrary form; for example, the atmosphere, a waterfall, or a double pendulum) which consists of a huge number of constituents like atoms or molecules, which have a certain distribution of their momenta.

To see if the system behaves chaotically do we have to vary the momenta of all its constituents in a tiny (and in the same) way to see if the system behavior is chaotic, or can we just vary the momenta of a tiny portion of the system?

I ask this because in an answer to a question I read that varying a little piece of the weather system would imply that the weather system is a chaotic phenomenon (which it obviously is).

stafusa
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Deschele Schilder
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1 Answers1

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Nope. One certainly doesn't have to vary a huge number of system variables to test for divergence of trajectories (i.e., the sensitivity to initial conditions characteristic of chaos) - changing a single one is sufficient.

That's what justifies the famous hyperbole "Does the flap of a butterfly's wing in Brazil set off a tornado in Texas?", already covered in Physics SE here - whose answer, by the way, is "Well, yes, but it can also prevent the tornado or have a completely different effect (including nothing remarkable), just like every one of the innumerable arbitrarily small perturbations the system is constantly subjected to.".

stafusa
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  • And what about a double pendulum or a waterfall? – Deschele Schilder Jun 18 '19 at 11:49
  • @descheleschilder It's the same for the double pendulum. A chaotic system is sensitive to arbitrary perturbations in the state space, which typically includes perturbations along a single of its axes (corresponding to a given system variable). As for the waterfall, do you mean a chaotic waterwheel? Otherwise I don't know any chaotic waterfall model. – stafusa Jun 18 '19 at 12:12
  • So in the double pendulum, you let a very small part of the momenta of the atoms which make the DP up (with respect to the weather system, a very, very, very small part of all atoms which make up the pendulum) vary in a tiny way (atoms which, as said only form a very tiny part of the DP)? By a waterfall I mean a stream of water that starts somewhere on a high level and streams down to a lower level, meeting stones, obstacles, etc. on its way. I know it's not a real waterfall, but more a stream of water going downwards on a rough surface. – Deschele Schilder Jun 18 '19 at 13:07
  • Atoms aren't part of the usual model for the double pendulum, but yes, nearby trajectories diverge exponentially fast, so no matter how tiny the initial separation - including one of, say, $10^{-22}$ - one will observe them diverge eventually. I don't know any specific model for a waterfall off the top of my head, but what you describe should admit some chaotic models, perhaps even some displaying spatial-temporal chaos, pattern formation, etc. – stafusa Jun 18 '19 at 14:52
  • Do you think that that the motion of a double pendulum will change if you make a little variation in the momentum of one atom out of all the atoms that constitute the double pendulum? – Deschele Schilder Jun 18 '19 at 16:09
  • Likewise, do you really think that the total weather system (in which there are no critical regions) will change if one varies a virtually infinite part of it? Look at films of tornadoes and the enormous destruction they cause. How can this be connected to a butterfly that varies this infinite part in a slightly different way (w.r.t. the total weather system)? If so, how? – Deschele Schilder Jun 18 '19 at 16:19
  • "Do you think [...]" @descheleschilder It's not a matter of opinion. Some systems can be shown to be chaotic, which means that typical, arbitrarily small perturbations put them on new trajectories that are eventually totally different from the unperturbed one. And 'totally different' doesn't mean with more or fewer extreme events - they might just happen at different times, for instance. As for the butterfly wing flap, I believe David Hammen has answered you rather well already, back in 2016. :) – stafusa Jun 18 '19 at 22:04
  • I don't mean with "do you think..." that it is a matter of opinion. The small perturbations are put on the whole system, after which the whole system might diverge or not. In trying to show if a double pendulum behaves chaotically one does not put a small change on just one atom but on all atoms of the dp at once, after which the dp will after a certain time behave in a completely different way than if the small perturbations weren't put on the all atoms. Macroscopically this translates itself in small variations of the arms of the dp (the whole dp) and see what happens. – Deschele Schilder Jun 19 '19 at 00:17
  • @descheleschilder I mentioned it before, but let me emphasize: the usual equations for the double pendulum don't refer to atoms at all. It's a rigid-body or point-mass model so, strictly speaking, asking about changes on one atom is simply an ill-posed question. I've interpreted it so far as meaning an extremely minute change - if you do literally mean an atom, then you need to use a different description for the pendulum, one that includes its atoms. – stafusa Jun 19 '19 at 07:29