TY - JOUR
T1 - On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium
AU - Li, Shimin
AU - Llibre, Jaume
PY - 2019/11/1
Y1 - 2019/11/1
N2 - © 2019 American Institute of Mathematical Sciences. All rights reserved. This paper deals with planar discontinuous piecewise linear differential systems with two zones separated by a vertical straight line x = k. We assume that the left linear differential system (x < k) and the right linear differential system (x > k) share the same equilibrium, which is located at the origin O(0, 0) without loss of generality. Our results show that if k = 0, that is when the unique equilibrium O(0, 0) is located on the line of discontinuity, then the discontinuous piecewise linear differential systems have no crossing limit cycles. While for the case k ≠ 0 we provide lower and upper bounds for the number of limit cycles of these planar discontinuous piecewise linear differential systems depending on the type of their linear differential systems, i.e. if those systems have foci, centers, saddles or nodes, see Table 2.
AB - © 2019 American Institute of Mathematical Sciences. All rights reserved. This paper deals with planar discontinuous piecewise linear differential systems with two zones separated by a vertical straight line x = k. We assume that the left linear differential system (x < k) and the right linear differential system (x > k) share the same equilibrium, which is located at the origin O(0, 0) without loss of generality. Our results show that if k = 0, that is when the unique equilibrium O(0, 0) is located on the line of discontinuity, then the discontinuous piecewise linear differential systems have no crossing limit cycles. While for the case k ≠ 0 we provide lower and upper bounds for the number of limit cycles of these planar discontinuous piecewise linear differential systems depending on the type of their linear differential systems, i.e. if those systems have foci, centers, saddles or nodes, see Table 2.
KW - Limit cycle
KW - Piecewise discontinuous linear systems
UR - https://ddd.uab.cat/record/221276
U2 - https://doi.org/10.3934/dcdsb.2019111
DO - https://doi.org/10.3934/dcdsb.2019111
M3 - Article
VL - 24
SP - 5885
EP - 5901
JO - Discrete and Continuous Dynamical Systems - Series B
JF - Discrete and Continuous Dynamical Systems - Series B
SN - 1531-3492
ER -