TY - JOUR

T1 - Existence of at most two limit cycles for some non-autonomous differential equations

AU - Gasull, Armengol

AU - Zhao, Yulin

N1 - Publisher Copyright:
© 2023 American Institute of Mathematical Sciences. All rights reserved.

PY - 2023/3

Y1 - 2023/3

N2 - It is know that the non-autonomous differential equations dx/dt = a(t) + b(t)|x|, where a(t) and b(t) are 1-periodic maps of class C1, have no upper bound for their number of limit cycles (isolated solutions satisfying x(0) = x(1)). We prove that if either a(t) or b(t) does not change sign, then their maximum number of limit cycles is two, taking into account their multiplicities, and that this upper bound is sharp. We also study all possible configurations of limit cycles. Our result is similar to other ones known for Abel type periodic differential equations although the proofs are quite different.

AB - It is know that the non-autonomous differential equations dx/dt = a(t) + b(t)|x|, where a(t) and b(t) are 1-periodic maps of class C1, have no upper bound for their number of limit cycles (isolated solutions satisfying x(0) = x(1)). We prove that if either a(t) or b(t) does not change sign, then their maximum number of limit cycles is two, taking into account their multiplicities, and that this upper bound is sharp. We also study all possible configurations of limit cycles. Our result is similar to other ones known for Abel type periodic differential equations although the proofs are quite different.

KW - Non-autonomous differential equation

KW - Limit cycle

KW - Periodic orbit

UR - http://www.scopus.com/inward/record.url?scp=85163374378&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/22c9e746-30bd-3333-8744-2e1b1bedb5d5/

U2 - 10.3934/cpaa.2023016

DO - 10.3934/cpaa.2023016

M3 - Article

SN - 1534-0392

VL - 22

SP - 970

EP - 982

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

IS - 3

ER -