Limit-cycle
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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.