A limit cycle attractor is a closed-loop trajectory in phase space corresponding to a stable periodic oscillation. Unlike a fixed equilibrium, a limit cycle draws nearby trajectories into a repeating cycle. For example, a heartbeat or the Van der Pol oscillator can exhibit a limit cycle: the system's state converges to a stable loop representing a sustained oscillation (no decay or unbounded growth).
Mathematically, limit cycles occur in nonlinear systems when a balance of driving and dissipative forces produces self-sustained oscillations. Visually, in a phase portrait, a limit cycle appears as a closed curve that neighboring trajectories spiral towards or around. The interior of the loop may contain an unstable fixed point (repeller) while the loop itself is asymptotically stable – any perturbed trajectory in the vicinity is “attracted” back to this closed orbit.
Properties: Limit cycles have at least one zero Lyapunov exponent (reflecting neutral motion along the cycle) and negative exponents for transverse directions, indicating stability of the cycle. They often arise via Hopf bifurcation when a stable fixed point becomes unstable and gives birth to a periodic orbit. Behaviorally, a limit cycle implies the system will settle into cyclic behavior (a stable rhythm or oscillation) rather than a static state.