Poincare-Bendixson theorem: bounded 2D autonomous ODE trajectories have periodic or fixed-point omega-limit

Poincare-Bendixson dichotomy framework. Setup: planar (2D) autonomous smooth ODE dx/dt = f(x, y), dy/dt = g(x, y) with forward orbit gamma(t) confined to a compact positively invariant set K not containing any fixed points. Theorem…