Piecewise linear perturbations of a linear center

Abstract

This paper is mainly devoted to study 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 Σ and the singular point of the unperturbed system is in Σ. 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 confirm 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 these last 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.

The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems

“…There are analogous results for piecewise smooth systems, for the case of continuous systems see for example [6,7,26,27], and for the case of discontinuous systems see [1,8,11,12,14,18]. In the discontinuous ones we can have more than one limit cycle, either all crossing cycles or including one sliding cycle, and in fact, the determination of the number of limit cycle has been the subject of several recent papers, see [2,3,4,10,15,16,17,20,22,23,24].…”

Piecewise linear differential systems without equilibria produce limit cycles?

“…Up to now we know that there are discontinuous systems with at least three limit cycles, see for instance [2,4,3,6,8,9,10,11,12,13,14,15,22,17,18,19,20,22,24].…”

“…But now we assume that this piecewise differential systems are discontinuous, i.e. we do not consider the conditions (4).…”

Center boundaries for planar piecewise-smooth differential equations with two zones

“…In the literature we can find a lot of works that deal with limit cycles of discontinuous piecewise linear differential systems, see for instance [5,6,9,10,11,12]. Han and Zang, in [10], provide discontinuous systems with two limit cycles, and they conjecture that the maximum number of limit cycles for this class is exactly two.…”

“…Later on, Llibre and Ponce provide in [12] a proof of the existence of such three limit cycles. In [5] the authors obtain three limit cycles from a piecewise perturbation of a linear center, and they can choose from which periodic orbits of the linear center the limit cycles bifurcate. To the best of our knowledge, we do not know an example of planar piecewise linear systems separated by a straight line W with four or more limit cycles.…”