Let's assume one asserts the existence of a closed orbit by applyling the Poincaré-Bendixson theorem to a trapping region $R$ that is constructed such that all phase vectors on its boundary point inwards and all fixed points are excluded from $R$ itself.
Furthermore, assume that there was precisely one unstable fixed point that one excluded in such a manner. Let's also assume that that/these closed orbits truly correspond to limit cycles (The Poincaré-Bendixson theorem stricly speaking only implies the existence of a closed orbit which is not necessarily a limit cycle.)
Then what can one conclude about its/their stability? If it were only one, I suppose it has to be stable such that the flow of the system is continuous? Can there even be multiple limit cycles around a single fixed point, and if so can they be nested into each other or can they lie "beside" each other? What can we then conclude about their stability, e.g., based on the continuity of the flow in $R$?
I'm thinking of these questions since in the book "Nonlinear Dynamics and Chaos (Studies in Nonlinearity)" by Strogatz he treats the glycolytic oscillator as an example on page 205 - 209. There we precisely have the situation I just described and on page 209, Strogatz then states that in that particular system "numerical integration shows that it is actually a stable limit cycle".
I though maybe one could find out something about the stability without resorting to numerics?
