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I am working with a second-order non-linear differential equation and I have a very good guess that it should have a stable limit cycle (because of the physics involved in the phenomena which is described by these differential equation). I was wondering whether there is a general method for the above purpose!

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    $\begingroup$ Show me the equation. As I had a nonlinear second order ode which depending on the parameters yielded a limit cycle as determined by the theorem outlined by the answer below. $\endgroup$
    – Chinny84
    Feb 4, 2015 at 15:16

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I believe you can apply the Poincare-Bendixson Theorem to show there is a periodic orbit, or you can use Bendixsons Negative Criterion to show that no such periodic orbit exists.

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