Stochastic Perturbations of Periodic Orbits with Sliding

摘要

Vector fields that are discontinuous on codimension-one surfaces 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 piecewise-smooth vector field 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 analysed by the use of a series expansion of the transitional probability density function. Stochastically perturbed sliding motion is analysed 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.