On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations

David J. W. Simpson

doi:10.3934/dcds.2014.34.3803

Article Contents

2014, Volume 34, Issue 9: 3803-3830. Doi: 10.3934/dcds.2014.34.3803

This issue Previous Article Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms Next Article Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables

On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations

David J. W. Simpson^1,

1.
Institute of Fundamental Sciences, Massey University, Palmerston North

Received: March 31, 2013

Revised: November 30, 2013

Published: August 2014

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Abstract

A two-fold singularity is a point on a discontinuity surface of a piecewise-smooth vector field at which the vector field is tangent to the surface on both sides. Due to the double tangency, forward evolution from a two-fold is typically ambiguous. This is an especially serious issue for two-folds that are reached by the forward orbits of a non-zero measure set of initial points. However, arbitrarily small perturbations of the vector field can make forward evolution well-defined, and from an applied perspective, such perturbations may represent additional model features that enhance the realism of a piecewise-smooth mathematical model. Three physically motivated forms of perturbation: hysteresis, time-delay, and noise, are analysed individually. The purpose of this paper is to characterise the perturbed dynamics in the limit that the size of the perturbation tends to zero. This concept is applied to a two-fold in two dimensions. In each case the limit leads to a novel probabilistic notion of forward evolution from the two-fold.

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Mathematics Subject Classification: Primary: 34E10, 37E99; Secondary: 34F05.

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