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).


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.".

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  • $\begingroup$ And what about a double pendulum or a waterfall? $\endgroup$ – Deschele Schilder Jun 18 '19 at 11:49
  • $\begingroup$ @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. $\endgroup$ – stafusa Jun 18 '19 at 12:12
  • $\begingroup$ 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. $\endgroup$ – Deschele Schilder Jun 18 '19 at 13:07
  • $\begingroup$ 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. $\endgroup$ – stafusa Jun 18 '19 at 14:52
  • $\begingroup$ 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? $\endgroup$ – Deschele Schilder Jun 18 '19 at 16:09

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