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Piecewise Smooth Dynamical Systems Regularized by Convolution

Claudio A. Buzzi, Daniel Panazzolo, Paulo R. da Silva

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.

Piecewise Smooth Dynamical Systems Regularized by Convolution

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 with discontinuity locus or , and on with discontinuity locus . We will see that some unexpected dynamical phenomena may arise even in the case of piecewise constant vector fields.
Paper Structure (27 sections, 14 theorems, 187 equations, 25 figures, 1 table)

This paper contains 27 sections, 14 theorems, 187 equations, 25 figures, 1 table.

Table of Contents

  1. Introduction
  2. Overview of the paper
  3. Some examples in dimension $n=2$
  4. Regularization of the Sewing
  5. Regularization of other generic singularities
  6. A family of piecewise smooth vector field with poly-trajectories.
  7. Regularization by convolution
  8. Piecewise smooth spaces and functions
  9. Regularization of piecewise-smooth functions
  10. Smoothing $m_\varepsilon \ast f$ by a sequence of blowing-ups
  11. Blowing-up map
  12. Directional blowing-ups in piecewise smooth spaces
  13. Blowing-up in piecewise smooth regularized spaces
  14. Phase-directional blowing-up
  15. Family-directional blowing-up
  16. ...and 12 more sections

Key Result

Theorem 3.2

Given a mollifier $m$, the convolution operator defines a linear map from $L^1_{\operatorname{loc}} (\mathbbm{R}^n)$ to $C^{\infty} (\mathbbm{R}^n)$. If $\{ m_k \}$ is a regularizing sequence, then for each compact set $K$, the restriction of $m_k \ast f$ to $K$ converges to $f$ in the $L^1$-norm. Moreover, if $K \subset \mathbbm{R}^n \setminus \

Figures (25)

  • Figure 1: The mollifier $\mathbf{m}$.
  • Figure 2: Regularization around escaping and sliding points.
  • Figure 3: Regularization around saddle and fold-regular points.
  • Figure 4: Regularization around saddle-node and elliptical fold points.
  • Figure 5: Regularization around hyperbolic and parabolic folds points.
  • ...and 20 more figures

Theorems & Definitions (37)

  • Remark 2.1
  • Remark 3.1
  • Theorem 3.2
  • Definition 3.3
  • Remark 3.4
  • Remark 4.1
  • Example 4.2
  • Lemma 4.3
  • Remark 4.4
  • proof
  • ...and 27 more