Limit cycle

In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity.

Limit cycles have been used to model the behavior of many real-world oscillatory systems.

The study of limit cycles was initiated by Henri Poincaré (1854–1912).

Such a trajectory is called closed (or periodic) if it is not constant but returns to its starting point, i.e. if there exists some

By the Jordan curve theorem, every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.

, then there is a neighborhood around the limit cycle such that all trajectories in the interior that start in the neighborhood approach the limit cycle for time approaching

, and also for trajectories in the exterior approaching the limit cycle.

If instead, all neighboring trajectories approach it as time approaches negative infinity, then it is an unstable limit cycle (α-limit cycle).

If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity, and another one which spirals into it as time approaches negative infinity, then it is a semi-stable limit cycle.

Stable limit cycles are examples of attractors.

They imply self-sustained oscillations: the closed trajectory describes the perfect periodic behavior of the system, and any small perturbation from this closed trajectory causes the system to return to it, making the system stick to the limit cycle.

The Bendixson–Dulac theorem and the Poincaré–Bendixson theorem predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.

Finding limit cycles, in general, is a very difficult problem.

The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of Hilbert's sixteenth problem.

are quadratic polynomials of the two variables, such that the system has more than 4 limit cycles.

Limit cycles are important in many scientific applications where systems with self-sustained oscillations are modelled.

Stable limit cycle (shown in bold) and two other trajectories spiraling into it
Stable limit cycle (shown in bold) for the Van der Pol oscillator
Examples of limit cycles branching from fixed points near Hopf bifurcation . Trajectories in red, stable structures in dark blue, unstable structures in light blue. The parameter choice determines the occurrence and stability of limit cycles.