Limit cycle

A limit cycle is an isolated closed trajectory of a continuous dynamical system. Nearby trajectories may approach it or move away, making it an attracting, repelling, or semistable periodic motion.

Poincaré map: A map that takes points on a transverse section of a flow to their next return. Its fixed points represent periodic orbits, and their stability reveals whether a cycle attracts or repels.
Dynamical system: A mathematical model describing how a system's state changes over time. A limit cycle is an invariant trajectory within a continuous-time dynamical system.
Van der Pol oscillator: A nonlinear oscillator whose damping depends on amplitude and which has a stable periodic orbit. Its attracting oscillation is the textbook example of a stable limit cycle.
Center (dynamical systems): An equilibrium surrounded by closed periodic trajectories in a planar system. A center has a continuum of closed orbits, so none is an isolated limit cycle.
Floquet theory: A method for analyzing the stability of periodic solutions through the evolution of small perturbations over one period. Its multipliers quantify whether perturbations transverse to a limit cycle grow or decay.
Phase space: A space whose points represent the possible states of a dynamical system. A limit cycle is a closed curve in phase space, not necessarily a repeating path in physical space.
Hodgkin–Huxley model: A conductance-based model of how ionic currents generate electrical activity in neurons. Under suitable conditions, its repetitive firing can arise through a limit-cycle oscillation.
Homoclinic orbit: A trajectory that leaves a saddle equilibrium and returns to the same equilibrium. It is a loop in phase space but not a periodic orbit, because it approaches the saddle asymptotically.
Transverse stability: The growth or decay of perturbations perpendicular to an invariant trajectory. A cycle's transverse stability distinguishes attracting, repelling, and semistable behavior.
Periodic orbit: A trajectory that returns to its initial state after a fixed positive time. Every limit cycle is periodic, but only an isolated periodic orbit is a limit cycle.

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