An isolated closed periodic orbit in the phase portrait of an autonomous continuous-time dynamical system that is invariant under the flow and has a neighborhood free of other distinct periodic orbits; it may be attracting, repelling, or of saddle type with characteristic transverse stability properties.

Principle

A limit cycle is characterized by isolation (no nearby continuous family of periodic solutions) and orbital stability that determines whether nearby trajectories converge to, diverge from, or intersect the cycle transversely.

Demonstration

The van der Pol oscillator has a unique attracting limit cycle for appropriate parameter values: trajectories starting from different initial conditions converge in phase and amplitude to the same closed orbit.

Misapplication

Labeling any periodic solution a limit cycle even when it lies inside a continuous family of periodic orbits (a center) or when the system is non-autonomous and the orbit is not invariant under an autonomous flow.

Consequence

The existence of a stable limit cycle implies persistent self-sustained oscillations robust to small perturbations and determines long-term behavior independent of initial conditions within its basin of attraction.

Reversal

An equilibrium point (fixed point) or a quasiperiodic torus, which are invariant sets with fundamentally different local dynamics and stability characteristics from an isolated periodic orbit.

Boundary

Defined for finite-dimensional, autonomous continuous-time systems (ordinary differential equations); periodic orbits in forced/nonautonomous systems or discrete maps require adapted notions (forced response or periodic points).

Semantic Tension

Often contrasted with a center (non-isolated family of periodic orbits) or with neutrally stable cycles; the key tension is isolation and transverse stability versus marginal families of cycles.

Synthesis

A limit cycle is an isolated, invariant closed trajectory of an autonomous ODE whose transverse stability properties determine whether it organizes robust sustained oscillations in the system.