Limitzyklen

Limitzyklen, also known as limit cycles, are closed invariant trajectories of continuous-time dynamical systems in phase space. In particular, they are periodic orbits that are isolated from other periodic orbits, and every trajectory starting sufficiently close to the cycle stays near it for all future times.

A limit cycle is classified by its stability. If nearby trajectories converge to the cycle as time grows, it is stable (attracting). If nearby trajectories diverge away, it is unstable. Some cycles can be semi-stable, attracting from one side and repelling from the other. The stability is typically analyzed using return maps (Poincaré maps) or Floquet multipliers; for a planar system, one multiplier is always 1 due to time translation, while the others determine stability.

Limit cycles commonly occur in two-dimensional systems and can arise through parameter changes, notably via Hopf bifurcations, where a fixed point loses stability and a small-amplitude limit cycle is created. Classical examples include the van der Pol oscillator and various Liénard-type systems. In planar dynamics, limit cycles can be unique or part of a finite or infinite family, depending on the system.

Key results guide their existence and properties. The Poincaré-Bendixson theorem describes possible omega-limit sets in the plane, limiting them to equilibria or limit cycles under certain conditions. The Bendixson-Dulac criterion provides nonexistence results in regions where the divergence of the vector field does not change sign. Limit cycles are central to the study of nonlinear oscillations in science and engineering.

Applications span biology, chemistry, physics, and electronics, where rhythmic or periodic behavior is modeled by systems exhibiting limit cycles. A well-known example is the self-sustained oscillation of the van der Pol oscillator.