Piecewise linear differential systems with only centers can create limit cycles?

Abstract

In this article we study the continuous and discontinuous planar piecewise differential systems formed only by linear centers separated by one or two parallel straight lines. When these piecewise differential systems are continuous they have no limit cycles. Also if they are discontinuous separated by a unique straight line they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel straight lines, we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle.

Limit cycles of 3-dimensional discontinuous piecewise differential systems formed by linear centers

“…In [11] it was proved that discontinuous piecewise linear differential systems in the plane separated by one straight line and formed by two linear centers have no limit cycles, besides discontinuous piecewise linear differential systems in the plane separated by two parallel straight lines and formed by three linear centers have at most one limit cycle, and that there are systems having such a limit cycle.…”

Limit cycles of discontinuous piecewise linear differential systems formed by centers or Hamiltonian without equilibria separated by irreducible cubics

“…In [14] authors proved that if Σ is a straight line then discontinuous piecewise linear differential centers have no crossing limit cycle. There are many other papers devoted to study the existence and the number of limit cycles of these systems when the curve of separation is a straight line, see for instance [2,8,9,10,11,12,15,18].…”

“…For the second family we will use the following lemma which provides a normal form for an arbitrary linear differential center. See a proof in [18].…”

Limit cycles created by piecewise linear centers

“…But in [12] it is proved the existence of discontinuous piecewise linear differential system separated by two parallel straight lines and formed by three linear centers, which exhibit one limit cycle. As far as we know this discontinuous system was the first example that only centers in a piecewise linear differential system can create limit cycles.…”

“…Subcase 1.1: K = 0. Under this assumption we can solve the first three equations of system (12) with respect the variables A, B and C obtaining…”