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1.2.5 The Poincaré-Bendixson Theorem

Theorem: Suppose we have a region $ R$ in the plane in which the following two conditions are satisfied (e.g. Figure 1):
  1. All trajectories passing through the boundary of the region point inwards.
  2. All equilibrium points inside the region are unstable
Then, there exist at least one stable limit cycle inside $ R$.
Figure 1: An example for a region R in which the Poincaré-Bendixon conditions are satisfied (there is a single unstable equilibrium at the origin (0,0).
Image R_Poincare_bendixson

Eran Borenstein