ANM Seminar: Stochastic perturbations of sliding motion and of periodic orbits with sliding segments
The problem of quantitatively determining the effects of small noise on a periodic orbit of a nonsmooth system with sliding segments is surprisingly complex. Understanding stochastic dynamics for sliding segments, non-sliding segments, and transitions from sliding to non-sliding segments, requires completely different mathematical approaches, and much of the theory of stochastic differential equations is simply not applicable to systems with discontinuities. In this talk I will describe various basic results for stochastic perturbations of nonsmooth systems with sliding. For short time-scales stochastically perturbed sliding motion can be understood by analysing occupation times through the Feynman-Kac formula, and for long time-scales the motion can be understood via stochastic averaging and an asymptotic solution to the Fokker-Planck equation. In the presence of noise the transition from sliding to non-sliding may be explained by a novel blow-up of phase space centred at the transition point. Non-sliding segments and a return to the sliding surface can be understood by boundary layer methods. The results are applied to an example in relay control for which small noise induces an unexpectedly large increase in the frequency of oscillations.
An Applied Nonlinear Mathematics seminar for the Summer Term.
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