Melnikov functions and limit cycles in piecewise smooth perturbations of a linear center using regularization method
type: Item
instance of: scholarly article
title: Melnikov functions and limit cycles in piecewise smooth perturbations of a linear center using regularization method
zbMATH Open document ID: 1393.34030
author: Denis De Carvalho Braga
author: Alexander Fernandes da Fonseca
author: Luis Fernando Mello
published in: Nonlinear Analysis. Real World Applications
publication date: 2017-03-30T00:00:00Z
review text: Consider a perturbed planar system of the form \[ dX/dt=AX+\varepsilon G(X), \] where \(X=(x,y)\in\mathbb{R}^2\), \(\varepsilon\geq0\), \(A\) is a matrix such that the equilibrium \((0,0)\) is a center of the unperturbed system and \(G(X)\) is a piecewise smooth vector field, \[ G(X)=G^-(X)\text{ for }y<0, \;G(X)=G^+(X)\text{ for }y>0 \] with \(G^{\pm}\) being smooth. The main results are concerned with the existence and the stability of a limit cycle of the system. The results are obtained using the Melnikov function method. Two approaches are used: (i) the Taylor expansion for the Poincaré first return map of the discontinuous system, (ii) the classical Melnikov method for a regularized approximation of the system. The link between these two approaches is investigated.
reviewed by: Tihomir Gyulov
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6698075
zbMATH Keywords: Melnikov function
zbMATH Keywords: piecewise smooth system
zbMATH Keywords: limit cycle
zbMATH Keywords: periodic orbit
zbMATH Keywords: regularization method
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2017.01.003
OpenAlex ID: W2581971491
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DOI: 10.1016/J.NONRWA.2017.01.003
description: scientific article; zbMATH DE number 6698075