On piecewise smooth vector fields tangent to nested tori
type: Item
instance of: scholarly article
title: On piecewise smooth vector fields tangent to nested tori
zbMATH Open document ID: 1350.34020
published in: Journal of Differential Equations
publication date: 2016-08-29T00:00:00Z
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C45
zbMATH DE Number: 6620354
zbMATH Keywords: piecewise smooth vector field
zbMATH Keywords: invariant tori
author: Tiago de Carvalho
author: Marco A. Teixeira
MaRDI profile type: Publication
full work available at URL: j.jde.2016.06.015
OpenAlex ID: W2472390662
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DOI: 10.1016/J.JDE.2016.06.015
review text: The authors start with a piecewise smooth vector field on \(\mathbb R^3\) of the form NEWLINE\[NEWLINEZ_0 = ((\mathrm{sgn} z)x, (\mathrm{sgn} z)y, f(x^2 + y^2))NEWLINE\]NEWLINE with switching set the \((x, y)\)-plane that is tangent to a foliation of \(\mathbb R^3\) whose leaves are topological 2-tori except for two topological circles, tangent to all planes that contain the \(z\)-axis, and all of whose orbits are topological circles. They prove the existence, for each \(k \in \mathbb Z^+ \cup \{ \infty \}\), of a one-parameter family of piecewise smooth vector fields \(Z_\varepsilon^k = Z_\varepsilon\) that satisfies a number of properties, among them: (a) \(Z_\varepsilon \to Z_0\) as \(\varepsilon \to 0\); (b) \(Z_\varepsilon\) has exactly \(k\) hyperbolic nested limit cycles in each connected component of a vertical plane through the \(z\)-axis, minus the \(z\)-axis; (c) \(Z_\varepsilon\) has exactly \(k\) hyperbolic nested invariant topological tori; and (d) a particular circle \(x^2 + y^2 = r^2\) in the plane \(z = 0\) is an attractor for all trajectories of \(Z_\varepsilon\).NEWLINENEWLINEThey prove a similar theorem for starting system \(Z_0^\eta\) in which a small rotation \(\eta\) has been introduced for \(z< 0\).
description: scientific article; zbMATH DE number 6620354