Stochastically perturbed sliding motion in piecewise-smooth systems

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November 2014, 19(9): 2889-2913. doi: 10.3934/dcdsb.2014.19.2889

Stochastically perturbed sliding motion in piecewise-smooth systems

D. J. W. Simpson ^1, and R. Kuske ^2,

1.	Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand
2.	Department of Mathematics, University of British Columbia, Vancouver, BC, Canada

Received April 2013 Revised February 2014 Published September 2014

Abstract
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Sliding motion is evolution on a switching manifold of a discontinuous, piecewise-smooth system of ordinary differential equations. In this paper we quantitatively study the effects of small-amplitude, additive, white Gaussian noise on stable sliding motion. For equations that are static in directions parallel to the switching manifold, the distance of orbits from the switching manifold approaches a quasi-steady-state density. From this density we calculate the mean and variance for the near sliding solution. Numerical results of a relay control system reveal that the noise may significantly affect the period and amplitude of periodic solutions with sliding segments.

Keywords: relay control, nonsmooth system, stochastic differential equation., noise, bifurcation, Filippov system.

Mathematics Subject Classification: Primary: 34F05, 37H9.

Citation: D. J. W. Simpson, R. Kuske. Stochastically perturbed sliding motion in piecewise-smooth systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2889-2913. doi: 10.3934/dcdsb.2014.19.2889

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