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In the study of dynamical systems, one often talks about solutions that repeat themselves after a certain time, hence their name of "periodic orbits". Then one moves to the distinction of "stable" (e.g. a harmonic 2D pendulum) or "unstable" periodic orbits.

First question:

  • Do all dynamical systems (integrable or not) exhibit stable and unstable periodic orbits and if yes, why are periodic orbits inherent to all systems? (e.g. does the non-chaotic 2D simple pendulum ever exhibit unstable periodic orbits?)

Second question:

  • In the study of dynamical systems (namely also chaotic ones), why is there so much emphasis on the study of periodic orbits (solutions) of a system (be it stable or unstable) instead of the other types of solutions that can be found (e.g. any closed orbit not being periodic, or other types)?
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user929304
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1 Answers1

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1st question:

Yes all systems have areas of periodic (or almost-periodic, or isolated periodic) motion. For integrable systems this is the expected. But in fact the existence of isolated periodic solutions is a form of non-integrability a well (related to 2nd question as well). For the exact meaning of isolated periodic solution check the link on integrability above.

Considering stability, yes all systems (integrable or not) can have stable or unstable periodic solutions, of course for non-linear chaotic systems this is the norm.

The simple non-chaotic 2D pendulum, does not exhibit unstable periodic orbits (it does exhibit unstable equilibrium points, though)

2nd question:

You might want to check the KAM theory, which simply states that a no-integrable system which is close to an integrable system will have areas where the orbits are sustained (or in other words any sufficiently small perturbation away from an integrable system, can be studied by KAM theory using the periodic orbits, or invariant tori in phase space)

UPDATE:

Stable periodic solutions (as per the links in question) are periodic solutions which are dynamically stable, in other words if slightly perturbed the orbit will remain close to the original,

Unstable periodic solutions are those periodic solutions which are not stable in the previous sense.

Isolated periodic solutions are special kinds of solutions which relate to tests of non-integrability for some special cases of hamiltonians

Almost periodic solutions are solutions which are periodic in a probability sense. Meaning the system does not return exactly to the same point but close to it with probability 1 at certain periods.

Another link about (measuring) chaos in hamiltonian systems http://www.staff.science.uu.nl/~verhu101/SAMOSpap.pdf

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Nikos M.
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    NOte, however, that the only central forces of the form $F = -kr^{n}$ that have stable orbits are those that have $n = 1 $ or $-2$ – Zo the Relativist Oct 29 '14 at 13:36
  • @user929304, as per the link isolated periodic motion is a test of non-integrability (for some hamiltonian systems) – Nikos M. Oct 29 '14 at 13:46
  • @user929304, the tests or definition of chaos relates to laypunov exponents (which themselves relate to unstable periodic solutions), the isolated periodic solutions in the answer refer to tests of non-integrability, are you asking for the connection between chaotic systems and non-integrable systems? – Nikos M. Oct 29 '14 at 13:53
  • @user929304: a proper derivation of that result is a whole chapter of the Landau/Lifschitz classical mechanics book. I refer you to there if you're interested. And really, if you're interested in questions like this, you should already have the Landau/Lifschitz book. – Zo the Relativist Oct 29 '14 at 13:55
  • @user929304, i'm not sure, billiards are main examples of ergodic systems which can exhibit complex behaviour (but which also include simpler orbits), i would have to look more thoroughly, if i can update the question to make sth more clear (per my understanding) i can do that – Nikos M. Oct 29 '14 at 14:24
  • @user929304, if you need a rough estimate (from the figures) about the 8 eliptical billiards, i would say the first 2 exhibit almost-periodic motion, the stable periodic solutions are those with polygonal envelopes, number 4 is unstable – Nikos M. Oct 29 '14 at 14:36
  • @user929304, 1) well yes one may say that in some sense, 2) a poincare map is a slice of the phase space for specific values of parameters (in order to visualise in a sense the orbits in complex systems), no ellipses do not correspond to quasi-periodic orbits (ellipses are just deformed circles), a curve in a poincare map corresponds to a stable orbit, chaotic regions appear as dense points (quote "If the initial point is inside an elliptic island, its trajectory is restricted on a 1-d curve") etc.. – Nikos M. Oct 30 '14 at 15:22
  • @user929304, well these observations are rough and general, one cannot explain all the patterns and intricasies (which might require extensive calculations) in a question or a comment, maybe getting some references on non-linear dynamics (the landau/lipshitz book is a good introduction, there are other more specific of course), hope this helpful – Nikos M. Oct 30 '14 at 15:35
  • “Yes all systems have areas of periodic (or almost-periodic, or isolated periodic) motion.” Not true — how many periodic orbits (or almost-periodic orbits, etc.) does $\dot{x}=1$ have? – Matthew Kvalheim Jul 16 '19 at 08:08