A limit-cycle is a closed trajectory in phase space exhibited by nonlinear systems. As a dynamical system evolves, its trajectory might tend to spiral in towards a closed loop in the phase space. The neighboring trajectories may either spiral towards the limit-cycle or move away from it. In the case where all the neighboring trajectories move towards the limit-cycle, it is called a stable limit-cycle. Otherwise it is an unstable limit-cycle.

Stable limit-cycles imply self-sustained oscillations. Any small perturbation from the closed trajectory would cause the system to return to the limit-cycle, making the system stick to the limit-cycle.

Figure illustrating a stable limit cycle for the Van der Pol oscillator . As seen in the figure, all the trajectories for various initial states of this system, make the system converge to the limit cycle. Hence, this system exhibits self-sustained oscillations.

Further Reading:

• Steven H. Strogatz, "Nonlinear Dynamics and Chaos", Addison Wesley publishing company, 1994.

• M. Vidyasagar, "Nonlinear Systems Analysis, second edition, Prentice Hall, Englewood Cliffs, New Jersey 07632.