Persistence of Dynamical Systems under Random Perturbations

Abstract

Random perturbations may decisively affect the long-term behavior of dynamical systems. Random effects are modeled by the addition of Gaussian white noise to the system. The resulting diffusion equation is solved asymptotically, when the strength of the noise is small. Such solutions can be found by a ray method. The rays, in turn, can be interpreted as paths of maximum likelihood. The lowest eigenvalue for the system can then be approximated by means of the asymptotic solution of the diffusion equation. The reciprocal of this eigenvalue gives the persistence of the system.