Stochastic sensitivity of cycles in periodic dynamical systems

Abstract

A non-linear dynamical system with periodic parameters is considered in presence of random noise. A dispersion of stochastic trajectories around the deterministic cycle is studied on the base of the stochastic sensitivity analysis. For weak noise, the asymptotics of this dispersion is found in a form of periodic matrix function named by the stochastic sensitivity matrix. This matrix is a solution of the boundary value problem for some matrix linear differential equation. A mathematical analysis of this problem is carried out, and an explicit solution is presented for one-dimensional case. The elaborated mathematical method is applied to the analysis of the stochastic population model with Allee effect and periodic modulation. A dependence of the stochastic sensitivity of oscillations on the amplitude and frequency of periodic forcing is investigated. A phenomenon of the noise-induced transition from persistence to extinction is studied by confidence domains constructed on the base of the stochastic sensitivity function technique.