Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two
type: Item
instance of: scholarly article
title: Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two
zbMATH Open document ID: 1491.34031
author: Dingheng Pi
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2022-01-21T00:00:00Z
review text: In this paper, the author investigates the dynamical behaviors of a class of piecewise smooth dynamical systems defined in \(\mathbb{R}^n\). Let \(x=(x_1, \cdots, x_n)\in \mathbb{R}^n\). The system has a codimension 2 switching manifold \(\Sigma\) which is an intersection of two hyperplanes \(\Sigma_1:=\{x\in \mathbb{R}^n: \ x_1=0\}\) and \(\Sigma_2:=\{x\in \mathbb{R}^n: \ x_2=0\}\). The author first introduces the double regularized system with two small parameters \(\varepsilon\) and \(\eta\) of the original piecewise smooth system. Then the relation between periodic orbits of the piecewise smooth system and periodic orbits of its double regularized system is studied. By applying the implicit function theorem and geometric singular perturbation theory, the author proves that if the piecewise smooth system has an asymptotically stable sliding periodic orbit \(\gamma\), then under some conditions, the regularized system has also a unique, asymptotically stable periodic orbit \(\gamma_{\varepsilon\eta}\) in a neighborhood of \(\gamma\). Moreover, \(\gamma_{\varepsilon\eta}\) converges to \(\gamma\) as both \(\varepsilon\) and \(\eta\) converge to 0.
reviewed by: Zhengdong Du
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 7461168
zbMATH Keywords: piecewise smooth systems
zbMATH Keywords: regularization
zbMATH Keywords: limit cycle
zbMATH Keywords: codimension two switching manifold
zbMATH Keywords: asymptotic stability
MaRDI profile type: Publication
full work available at URL: dcdsb.2021080
OpenAlex ID: W3134934504
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DOI: 10.3934/DCDSB.2021080
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description: scientific article; zbMATH DE number 7461168