Piecewise linear perturbations of a linear center

Abstract

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