Strogatz's textbook "Nonlinear Dynamics and Chaos", Chapter 7 presents the Poincare-Bendixson theorem, which gives conditions under which one can conclude the existence of a closed orbit within some compact subset $R$ of the phase plane.
As presented in Strogatz, one of the hypotheses of the theorem is the existence of a trajectory $C$ which remains always within $R$ (the Wikipedia presentation makes this less obvious, but is equivalent). Strogatz thus presents a trick to find such a curve: we construct $R$ to be an annular region upon whose boundaries the flow points everywhere inward. Such a region must obviously contain a trajectory with the desired properties.
This, however, seems to assume that the limit cycle is stable (if it were unstable the trajectories would point everywhere outward, at best). I was wondering whether we can adapt the theorem to find unstable limit cycles as well, by considering the time reversed system. Essentially: does the existence of a stable limit cycle in a system imply the existence of an unstable limit cycle in its time-reversed counterpart?