Limit Cycles

Mathematical Modeling · Axiom Academy

LESSON Limit Cycles - Mathematical Modeling Unit 3: Continuous Models - Mathematical Modeling A limit cycle is an isolated closed trajectory in the phase plane of a two-dimensional autonomous system. Unlike centers (where every nearby trajectory is also closed), a limit cycle is the only closed orbit in its neighborhood. "An isolated periodic orbit that nearby trajectories spiral toward or away from" Isolated: No other closed orbits arbitrarily close to it Periodic: Solutions on the limit cycle repeat with period T Attracting or repelling: Nearby trajectories approach or recede from it Self-sustained: Oscillations maintain constant amplitude without external forcing Stable vs Unstable Limit Cycles Stable (Attracting) Limit Cycle Trajectories starting near the limit cycle spiral toward it as . where is the limit cycle. All trajectories in a neighborhood eventually approach the periodic orbit. Physical interpretation: Self-sustained oscillations that persist despite small perturbations. Unstable (Repelling) Limit Cycle Trajectories starting near the limit cycle spiral away from it as . The limit cycle repels nearby trajectories. It acts as a separatrix between different behaviors. Physical interpretation: A threshold separating different dynamical regimes. Attracting from one side, repelling from the other. Trajectories inside spiral one way while trajectories outside spiral the opposite way. The Poincare-Bendixson Theorem

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