Random perturbations of systems with periodic impulse effects

TL;DR

This paper investigates how small random perturbations affect systems with periodic impulses, proving convergence to deterministic behavior and describing the fluctuations, with applications demonstrated on a nonlinear pendulum example.

Contribution

It establishes limit theorems for stochastic impulsive systems, showing convergence to deterministic systems and characterizing fluctuations in the zero noise limit.

Findings

Convergence of stochastic impulsive systems to deterministic limits as noise vanishes.

Characterization of fluctuation processes via linear stochastic differential equations.

Numerical illustration on a periodically kicked nonlinear pendulum.

Abstract

The principal aim of the present work is to explore limit theorems for small random perturbations of dynamical systems with periodic impulse effects, in the limit of vanishing noise intensity. We start with a system whose time evolution is governed by a nonlinear ordinary differential equation in between impulses, and a nonlinear resetting map at impulses; the latter are assumed to arrive in a time-periodic manner. We next consider small state-dependent Brownian perturbations of this system and explore the zero noise limit on finite, but arbitrary, time horizons. For the resulting stochastic system with impulse effects, we prove convergence to the underlying deterministic impulsive system as the noise goes to zero. More importantly, we prove convergence of the rescaled fluctuation process about the deterministic limit in a strong pathwise sense on finite time intervals to a limiting fluctuation process governed by a linear time-dependent stochastic differential equation in between impulses and a linear time-dependent resetting map at impulses.The results are illustrated numerically for a periodically kicked nonlinear pendulum with state-dependent kick sizes.