New global bifurcation diagrams for piecewise smooth systems: transversality of homoclinic points does not imply chaos
type: Item
instance of: scholarly article
title: New global bifurcation diagrams for piecewise smooth systems: transversality of homoclinic points does not imply chaos
zbMATH Open document ID: 1414.34013
published in: Journal of Differential Equations
publication date: 2018-12-03T00:00:00Z
review text: It is well-known that chaos arises if a smooth dynamical system which admits a transversal homoclinic point is perturbed. This paper presents a new result that this phenomenon may not exist for some piecewise smooth two-dimensional systems subjected to a periodic or almost periodic perturbation. The authors assume that the fixed point lies on the discontinuity surface and investigate two scenarios where the unperturbed homoclinic orbit crosses transversally the discontinuity surface. For the piecewise smooth systems where there is sliding close to the fixed point, they prove that chaos does not exist even though there is a transversal homoclinic point and the perturbation is periodic. On the other hand, for piecewise smooth systems where there is no sliding close to the fixed point, they conjecture that chaos is possible if there is a transversal homoclinic point and the perturbation is periodic.
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37G20
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C45
Mathematics Subject Classification ID: 34C28
zbMATH DE Number: 6987666
zbMATH Keywords: homoclinic orbit
zbMATH Keywords: Melnikov theory
zbMATH Keywords: piecewise smooth systems
zbMATH Keywords: sliding
zbMATH Keywords: chaos
zbMATH Keywords: transversal homoclinic points
author: Matteo Franca
author: Michal Pospíšil
reviewed by: Kwok-wai Chung
MaRDI profile type: Publication
full work available at URL: j.jde.2018.07.078
OpenAlex ID: W2886542398
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Wikidata QID: Q129416408
DOI: 10.1016/J.JDE.2018.07.078
description: scientific article; zbMATH DE number 6987666