Stochastic sensitivity of cycles in periodic dynamical systems. (English) Zbl 1515.37047
Summary: 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.
MSC:
| 37H10 | Generation, random and stochastic difference and differential equations |
| 37H30 | Stability theory for random and stochastic dynamical systems |
| 37C27 | Periodic orbits of vector fields and flows |
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