  We study the Poincare-Bendixson theorem for two-dimensional continuous
dynamical systems in compact domains from the point of view of computation,
seeking algorithms for finding the limit cycle promised by this classical
result. We start by considering a discrete analogue of this theorem and show
that both finding a point on a limit cycle, and determining if a given point is
on one, are PSPACE-complete.
  For the continuous version, we show that both problems are uncomputable in
the real complexity sense; i.e., their complexity is arbitrarily high.
Subsequently, we introduce a notion of an "approximate cycle" and prove an
"approximate" Poincar\'e-Bendixson theorem guaranteeing that some orbits come
very close to forming a cycle in the absence of approximate fixpoints;
surprisingly, it holds for all dimensions. The corresponding computational
problem defined in terms of arithmetic circuits is PSPACE-complete.
