Poincaré–Bendixson theorem

The Poincaré–Bendixson theorem states that a nonempty compact limit set of a planar continuous-time dynamical system, containing no equilibria, must be a periodic orbit.

Limit set: A set containing the accumulation points of a trajectory as time tends toward a limit. The theorem classifies compact limit sets when they contain no equilibrium.
Planar dynamical system: A continuous-time dynamical system whose state space is two-dimensional. The restriction to two dimensions is essential to the theorem's conclusion.
Chaos theory: The study of deterministic systems with sensitive dependence and complex long-term behavior. The theorem rules out chaotic compact limit sets without equilibria in planar flows.
Liénard system: A family of planar differential equations used to model nonlinear oscillators. The theorem helps establish periodic oscillations in suitable Liénard systems.
Omega-limit set: The set of accumulation points approached by a trajectory as time tends to positive infinity. The theorem is commonly applied to forward-time accumulation sets.
Continuous-time dynamical system: A system whose state evolves continuously in time according to a flow or differential equation. The theorem concerns trajectories evolving continuously rather than discrete iterates.
Lorenz system: A three-dimensional system of differential equations with a strange chaotic attractor. Its chaotic attractor illustrates behavior possible beyond the planar setting.
Van der Pol oscillator: A nonlinear oscillator whose trajectories approach a stable periodic orbit for positive damping parameter. Planar limit-set arguments help establish its sustained oscillation.
Equilibrium point: A state of a dynamical system that remains unchanged under its evolution. The theorem's periodic-orbit conclusion requires the limit set to contain no equilibria.
Compact space: A topological space in which every open cover has a finite subcover. Compactness prevents the limit set from escaping to infinity or losing accumulation points.

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