Stochastic Perturbations of Periodic Orbits with Sliding

TL;DR

This paper investigates how small stochastic perturbations affect the dynamics of Filippov systems with attracting periodic orbits on discontinuity surfaces, using specialized analysis methods to quantify changes in oscillation times.

Contribution

It introduces a novel approach combining series expansions and stochastic averaging to analyze noise effects on Filippov systems with sliding motion.

Findings

Small noise can significantly reduce oscillation time in relay control models.

The analysis identifies key geometric features influencing noise impact.

Quantitative results are obtained for a three-dimensional relay control system.

Abstract

Vector fields that are discontinuous on codimension-one surfaces are known as Filippov systems and can have attracting periodic orbits involving segments that are contained on a discontinuity surface of the vector field. In this paper we consider the addition of small noise to a general Filippov system and study the resulting stochastic dynamics near such a periodic orbit. Since a straight-forward asymptotic expansion in terms of the noise amplitude is not possible due to the presence of discontinuity surfaces, in order to quantitatively determine the basic statistical properties of the dynamics, we treat different parts of the periodic orbit separately. Dynamics distant from discontinuity surfaces is analyzed by the use of a series expansion of the transitional probability density function. Stochastically perturbed sliding motion is analyzed through stochastic averaging methods. The influence of noise on points at which the periodic orbit escapes a discontinuity surface is determined by zooming into the transition point. We combine the results to quantitatively determine the effect of noise on the oscillation time for a three-dimensional canonical model of relay control. For some parameter values of this model, small noise induces a significantly large reduction in the average oscillation time. By interpreting our results geometrically, we are able to identify four features of the relay control system that contribute to this phenomenon.