The Poincaré-Bendixson Theorem is a fundamental result in the theory of dynamical systems, specifically for planar systems, stating that a non-empty compact limit set of a flow on the plane must be either a single equilibrium point, a periodic orbit, or consist of a finite number of periodic orbits. This theorem connects the behavior of nonlinear systems to phase portraits, providing insights into equilibrium points and phenomena like limit cycles and bifurcations.
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