Piecewise linear perturbations  of a linear center

Claudio Buzzi; Claudio Pessoa; Joan Torregrosa

doi:10.3934/dcds.2013.33.3915

Article Contents

2013, Volume 33, Issue 9: 3915-3936. Doi: 10.3934/dcds.2013.33.3915

This issue Previous Article Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane Next Article Liouville type theorems for poly-harmonic Navier problems

Piecewise linear perturbations of a linear center

1.
Departamento de Matemática, Universidade Estadual Paulista, 15054-000, São José do Rio Preto
2.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona

Received: February 29, 2012

Revised: January 31, 2013

Published: August 2013

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Abstract

This paper is mainly devoted to the study of the limit cycles that can bifurcate from a linear center using a piecewise linear perturbation in two zones. We consider the case when the two zones are separated by a straight line $\Sigma$ and the singular point of the unperturbed system is in $\Sigma$. It is proved that the maximum number of limit cycles that can appear up to a seventh order perturbation is three. Moreover this upper bound is reached. This result confirms that these systems have more limit cycles than it was expected. Finally, center and isochronicity problems are also studied in systems which include a first order perturbation. For the latter systems it is also proved that, when the period function, defined in the period annulus of the center, is not monotone, then it has at most one critical period. Moreover this upper bound is also reached.

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Mathematics Subject Classification: Primary: 37G15, 34C05, 34C99.

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