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I'm just having a small trouble with the nomenclature. So basically we spent a lot of time in class discussing limit cycles in the phase plane. However it never really became 100% clear to me whether all periodic solutions of a nonlinear ODE represent a limit cycle.

Or in other words, when we are searching for a periodic solution, then is it the same as searching for limit cycles? And in that case the different theorems regarding limit cycles can be applied (taking into account their conditions of course), right?

Thanks in advance!

Jim
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    "limit cycle" usually refers to an attracting or repelling isolated periodic solution. So while all limit cycles are periodic solutions, all periodic solutions are not limit cycles. E.g. in simple pendulum, there are no limit cycles. However, there are infinitely many (non-isolated) periodic solutions – nonlinearism Dec 12 '16 at 23:23
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    A limit cycle is a periodic orbit which is isolated: it means that there is some neighbourhood of this trajectory which doesn't contain any other periodic orbits. @nonlinearism mentioned that limit cycles could be attracting or repelling, but actually saddle limit cycles are very, very important too. So the key is that if periodic orbit is isolated, then it is a limit cycle. – Evgeny Dec 13 '16 at 19:19

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