Uniqueness of Filippov sliding vector field on the intersection of two surfaces in \(\mathbb R^3\) and implications for stability of periodic orbits
type: Item
instance of: scholarly article
title: Uniqueness of Filippov sliding vector field on the intersection of two surfaces in \(\mathbb R^3\) and implications for stability of periodic orbits
zbMATH Open document ID: 1333.34022
published in: Journal of Nonlinear Science
publication date: 2015-12-17T00:00:00Z
review text: The following piecewise smooth system in \(\mathbb{R}^{3}\) is considered \[ \frac{dx}{dt} = f(x),\quad f(x) = f_{i}(x) \text{ for } x\in \mathbb{R}_{i}, \tag{1} \] where for \(i=1,2,3,4, \mathbb{R}_{i}\) are open, disjoint and connected sets, and (locally) \(\mathbb{R}^{3}=\mathbb{R}_{1}\cup\cdots \cup \mathbb{R}_{4}\). Moreover, each \(f_{i}\) is smooth (at least \(C^{1}\)) in an open neighborhood of each \(\mathbb{R}_{i}\) and \(\mathbb{R}^{3}\setminus \mathbb{R}_{1}\cup\cdots \cup \mathbb{R}_{4}\) has zero (Lebesgue) measure. The surface \(\Sigma =\Sigma_{1}\cup\Sigma_{2}\), \(\Sigma_{j}=\{x\in \mathbb{R}^{3}: h_j(x) =0\}, j=1,2\), with \(\nabla h_{j}\neq 0, h_{j}\in C^k, k\geq 2\) and \(\nabla h_1,\nabla h_2\) are linearly independent for \(x\in\Sigma\). Although, in general, there is no unique Filippov sliding vector field on \(\Sigma\) under natural conditions it is proved that all Filippov sliding vector fields are orbitally equivalent to \(\Sigma\). The implication of this result in the important case of a periodic orbit is examined.
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34A60
zbMATH DE Number: 6521646
zbMATH Keywords: Filippov convexification
zbMATH Keywords: orbital equivalence
zbMATH Keywords: periodic orbit
zbMATH Keywords: stability
author: Luca Dieci
author: Cinzia Elia
author: Luciano Lopez
MaRDI profile type: Publication
full work available at URL: s00332-015-9265-6
OpenAlex ID: W1133320253
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DOI: 10.1007/S00332-015-9265-6
description: scientific article; zbMATH DE number 6521646