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Theorem: Suppose we have a region in the plane in which the following two conditions are satisfied (e.g. Figure 1):
Then, there exist at least one stable limit cycle inside .
- All trajectories passing through the boundary of the region point inwards.
- All equilibrium points inside the region are unstable
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).