Piecewise Smooth Dynamical Systems Regularized by Convolution

TL;DR

This paper introduces a regularization method for piecewise smooth dynamical systems with complex discontinuity sets, transforming them into smooth systems on manifolds with corners to analyze their dynamics.

Contribution

It develops a regularization procedure using blowings-up for piecewise smooth vector fields with normal crossings discontinuities, enabling analysis of their smooth dynamics.

Findings

Regularization reduces complex discontinuity sets to smooth manifolds with corners.

Unexpected dynamical phenomena can occur even in simple piecewise constant systems.

Method applicable to systems in $\

Abstract

We present a general regularization procedure for piecewise smooth vector fields whose discontinuity locus is a variety of normal crossings type. We show that such regularization can be smoothed through a finite sequence of blowings-up, thereby reducing the problem to study of the dynamics of a smooth vector field in a manifold with corners. The procedure will be illustrated in the cases of piecewise smooth vector fields on $\mathbb{R}^2$ with discontinuity locus $x=0$ or $xy=0$, and on $\mathbb{R}^3$ with discontinuity locus $xyz=0$. We will see that some unexpected dynamical phenomena may arise even in the case of piecewise constant vector fields.