TY - JOUR

T1 - Limit cycles created by piecewise linear centers

AU - Llibre, Jaume

AU - Zhang, Xiang

PY - 2019/5/1

Y1 - 2019/5/1

N2 - © 2019 Author(s). In the last few years, the interest for studying the piecewise linear differential systems has increased strongly, mainly due to their applications to many physical phenomena. In the study of these differential systems, the limit cycles play a main role. Up to now, the major part of papers which study the limit cycles of the piecewise linear differential systems consider only two pieces. Here, we consider piecewise linear differential systems with three pieces. In this paper, we study the limit cycles of the discontinuous piecewise linear differential systems in the plane R 2 formed by three arbitrary linear centers separated by the set ς = {(x, y) ∈ R 2: y = 0 {or} x = 0 {and} y ≥ 0 }. We prove that such discontinuous piecewise linear differential systems can have 1, 2, or 3 limit cycles, with 3 the maximum number of limit cycles that such systems can have. Moreover, the limit cycles are nested and must intersect ς in three or four points. The limit cycles having three intersection points with ς can reach the maximum number 3. The limit cycles having four intersection points with ς are at most 1, and if it exists, the systems could simultaneously have 1 or 2 limit cycles intersecting ς in three points.

AB - © 2019 Author(s). In the last few years, the interest for studying the piecewise linear differential systems has increased strongly, mainly due to their applications to many physical phenomena. In the study of these differential systems, the limit cycles play a main role. Up to now, the major part of papers which study the limit cycles of the piecewise linear differential systems consider only two pieces. Here, we consider piecewise linear differential systems with three pieces. In this paper, we study the limit cycles of the discontinuous piecewise linear differential systems in the plane R 2 formed by three arbitrary linear centers separated by the set ς = {(x, y) ∈ R 2: y = 0 {or} x = 0 {and} y ≥ 0 }. We prove that such discontinuous piecewise linear differential systems can have 1, 2, or 3 limit cycles, with 3 the maximum number of limit cycles that such systems can have. Moreover, the limit cycles are nested and must intersect ς in three or four points. The limit cycles having three intersection points with ς can reach the maximum number 3. The limit cycles having four intersection points with ς are at most 1, and if it exists, the systems could simultaneously have 1 or 2 limit cycles intersecting ς in three points.

U2 - https://doi.org/10.1063/1.5086018

DO - https://doi.org/10.1063/1.5086018

M3 - Article

C2 - 31154799

VL - 29

JO - Chaos

JF - Chaos

SN - 1054-1500

M1 - 053116

ER -