Limit-cycle
From Free net encyclopedia
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 -infinity. Such behavior is exhibited in some nonlinear systems. In the case where all the neighboring trajectories approach the limit-cycle as time t <math>\rightarrow +\infty</math>, it is called a stable or attractive limit-cycle. If instead all neighboring trajectories approach it as time t <math>\rightarrow</math> <math>-\infty</math>, it is an unstable or non-attractive limit-cycle. In all other cases it is neither "stable" nor "unstable".
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 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.
and also:
- Philip Hartman, "Ordinary Differential Equation", Society for Industrial and Applied Mathematics, 2002.
- Witold Hurewicz, "Lectures on Ordinary Differential Equations", Dover, 2002.
- Solomon Lefschetz, "Differential Equations: Geometric Theory", Dover, 2005.
- Lawrence Perko, "Differential Equations and Dynamical Systems", Springer-Verlag, 2006.