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The world around us abounds with chaotic systems: dripping taps (when a certain dripping rate is reached the dripping becomes irregular, which can be seen in this old but very entertaining video, around 30 minutes; the straw that breaks the camel's back [around 6 minutes] is, by the way, not an example of chaos but of criticality), waterfalls, double pendulums, bouncing billiard balls on a pool table, the onset of turbulence and of course the weather system.

Now assume these systems don't contain potential energy sources to be released (like dams with a huge amount of water behind them, a fully blown ballon, the already mentioned camel, or humans; it's obvious that in examples containing cases like this very different histories develop if, for example, an explosive device that is coupled to these dams which will explode (thereby releasing the potential energy of the water) if some conditions vary very little from the non-explosive state then clearly the histories are very different; or if a person sees the doors of the train he had to catch in time close before his eyes then his day will completely change; I think these are both examples of criticality, like the straw that...; so here's no chaos involved).

If we consider a double pendulum, we can't say (to determine if it's subsequent movement is chaotic) to vary a tiny part of the double pendulum, like changing the movements of a number of atoms somewhere in the pendulum (in proportion with changing a small part of the global weather system).

I can't see that there are potential energy sources in the weather system to be released by a tiny change of a puff of air. So isn't it fair to say that in determining if the weather system is chaotic we have to change the whole weather system by a small amount (give all the air molecules the same tiny change in phase space)?

Deschele Schilder
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Some points should be make clear:

  • There's no need for a system to have "potential energy sources" in order for it to display chaotic behavior, as the existence of Hamiltonian chaos shows.

  • There are no atoms in the double pendulum - its equations only account for two angles and speeds, the model is therefore agnostic to the bobs or arms being continuous or discrete. So, changing "atoms somewhere in the pendulum" can only be represented in the equations as a change in one or more of its 4 variables.

  • Lastly, none of the above has to do with determining whether the system is chaotic or not, only with the possibility that it is. Proving that a system is chaotic, in general, is not an easy problem.

As for the last question:

No, you definitely don't have to change all the variables of the system to see its evolution diverging exponentially from the unperturbed one. In a chaotic region of the phase space, a change in any direction, including one that corresponds to changing a single variable, will typically originate a divergent trajectory.

stafusa
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  • I meant potential energies that can be released are absent (such criticalities are not present in the global weather system). Of course, there are atoms in the double pendulum, just as there are air molecules in the weather, whose phase space positions you can change by tiny, coherent amounts. By a tiny change of the initial configuration of the double pendulum, you change all the constituents in a tiny coherent way. Do you think that a change of one air molecule in the weather phase space gives rise to totally different weather after a long time? – Deschele Schilder Apr 25 '18 at 12:38
  • In the same way the positions of atoms are not present in the double pendulum model, weather models don't include individual molecules. But, yes, any arbitrarily small change, even, metaphorically, changing "an air molecule", is going to result in an exponentially diverging orbit, if the system is chaotic. – stafusa Apr 25 '18 at 17:25
  • As for "potential energies that can be released [...] in the global weather system", doesn't the huge amount of heat stored in the oceans count? – stafusa Apr 25 '18 at 21:46
  • Of course, but that can't be released by the tiny change of some little piece of the huge weather system, like the energy of the water behind a dam can, for example, be released if you put an explosive in the dam, which is very sensitive to some variable. The device will explode if you make a minuscule change in this variable. This isn't happening in the weather system. I wasn't asking for weather models but real weather systems, which obviously consist out of molecules. If you talk about one molecule then it's trajectory will become very different indeed. But does that also is the case for – Deschele Schilder May 01 '18 at 07:23
  • the whole weather system (which is what we are talking about)? The movement of this one molecule between the other air molecules mayb chaotic, but from this you can't conclude that the weather system itself is chaotic (of course I believe it's chaotic, but not on the basis of one air molecule). – Deschele Schilder May 01 '18 at 07:28
  • @descheleschilder If the system is chaotic, then a change, no matter how small, might generate a divergent trajectory. Blowing up a dam with explosives has no relation to chaos. – stafusa May 01 '18 at 08:51
  • But you must consider the whole system. If a little part of the weather system (say the air around the butterfly's wings) is changed, like the billiard balls on an open table, it's clear that the balls on this little table trace out entirely different paths in phase space. But the further you are away from this changed (very, very) little piece of the whole, there will be no significant change in the system as a whole (the global weather), just like changing the movements of some molecules in a double pendulum won't change the pendulum's behavior. Explosives have indeed no relation to – Deschele Schilder May 01 '18 at 09:27
  • chaos, but they is often used in examples to show chaotic behavior (which they are not*). For example (again) the explosive in the dam. If a device, connected to the trigger of the explosive, undergoes a tiny change, the explosion takes place. So you would say, a very tiny change leads to a totally different outcome (the dam explodes, the water floods big areas, etc. But this is not an example of chaos, but of criticallity. – Deschele Schilder May 01 '18 at 09:36