More Than Three Limit Cycles in Discontinuous Piecewise Linear Differential Systems with Two Zones in the Plane

Abstract

In this paper, we study the existence of limit cycles for piecewise linear differential systems with two zones in the plane. More precisely, we prove the existence of piecewise linear differential systems with two zones in the plane with four, five, six and seven limit cycles. From our results we conjecture the existence of piecewise linear differential systems with two zones in the plane having exactly n limit cycles for all n ∈ ℕ.

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

“…After using some broken line as the boundary between linear zones, Braga and Mello in [3] put in evidence the important role of the separation boundary in determining the number of limit cycles. In [3] the authors exhibit an example with seven limit cycles having W as a polygonal curve and they state the conjecture: "Given n ∈ N there is a piecewise linear system with two zones in the plane with exactly n limit cycles". This conjecture was proved by the same authors in the paper [4].…”

“…In [3] the authors exhibit an example with seven limit cycles having W as a polygonal curve and they state the conjecture: "Given n ∈ N there is a piecewise linear system with two zones in the plane with exactly n limit cycles". This conjecture was proved by the same authors in the paper [4]. The main idea of [13] for three zones is still valid when only two zones exist.…”

“…In the paper [4], the authors consider piecewise linear systems sharing a singular point of focus type, both in the Jordan Normal Form. In Lemma 2 of [4] is proved that there exists a piecewise linear separation boundary such that the discontinuous system has a center.…”

“…In Lemma 2 of [4] is proved that there exists a piecewise linear separation boundary such that the discontinuous system has a center. Our work is inspired in this result.…”

“…In the present paper we treat only cases that f n 1 (X)f n 2 (X) > 0. Now we define the meaning of periodic orbit for Z W , given by (3). Assume that there are points p 1 , p 2 ∈ W , positive times t 1 , t 2 ∈ R and solutions γ 1 of (1), satisfying p 1 = γ 1 (0, p 1 ), and γ 2 of (2), satisfying p 2 = γ 2 (0, p 2 ).…”