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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)?
Qmechanic
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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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