TY - JOUR

T1 - Limit cycles of generic piecewise center-type vector fields in R3 separated by either one plane or by two parallel planes

AU - Villanueva, Yovani

AU - Llibre, Jaume

AU - Euzébio, Rodrigo

N1 - Funding Information:
The first author is partially supported by PDSE-CAPES grant 88881.624523/2021-01 and DS-CAPES 88882.386238/2019-01. The second author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The third author is partially supported by Pronex/FAPEG/CNPq grant 2012 10 26 7000 803 and grant 2017 10 26 7000 508, CAPES grant 88881.068462/2014-01 and Universal/CNPq grant 420858/2016-4.
Funding Information:
The first author is partially supported by PDSE-CAPES grant 88881.624523/2021-01 and DS- CAPES 88882.386238/2019-01 .
Funding Information:
The second author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00 , and the H2020 European Research Council grant MSCA-RISE-2017-777911 .
Funding Information:
The third author is partially supported by Pronex/ FAPEG /CNPq grant 2012 10 26 7000 803 and grant 2017 10 26 7000 508 , CAPES grant 88881.068462/2014-01 and Universal/ CNPq grant 420858/2016-4 .
Publisher Copyright:
© 2022 Elsevier Masson SAS

PY - 2022/10

Y1 - 2022/10

N2 - While the limit cycles of the piecewise differential systems in the plane R2 have been studied intensively during these last twenty years, this is not the case for the limit cycles of the piecewise differential systems in the space R3. The goal of this article is to study the continuous and discontinuous piecewise differential systems in R3, formed by linear vector fields similar to planar centers separated by one or two parallel planes. We call those “center-type” differential systems, which have two pure imaginary numbers and zero as eigenvalues. When these kinds of piecewise differential systems are continuous or discontinuous separated by one plane, then they have no limit cycles. Also, if they are continuous separated by two planes, then generically they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel planes, we show that generically they can have at most four limit cycles, and that there exist such systems with four limit cycles. The genericity here means that the statements hold in a residual set of the space of parameters associated to the differential system. We recall that the same problem but for discontinuous piecewise differential systems in R2 formed by linear differential centers separated by two parallel straight lines has at most one limit cycle.

AB - While the limit cycles of the piecewise differential systems in the plane R2 have been studied intensively during these last twenty years, this is not the case for the limit cycles of the piecewise differential systems in the space R3. The goal of this article is to study the continuous and discontinuous piecewise differential systems in R3, formed by linear vector fields similar to planar centers separated by one or two parallel planes. We call those “center-type” differential systems, which have two pure imaginary numbers and zero as eigenvalues. When these kinds of piecewise differential systems are continuous or discontinuous separated by one plane, then they have no limit cycles. Also, if they are continuous separated by two planes, then generically they do not have limit cycles. But when the piecewise differential systems are discontinuous separated two parallel planes, we show that generically they can have at most four limit cycles, and that there exist such systems with four limit cycles. The genericity here means that the statements hold in a residual set of the space of parameters associated to the differential system. We recall that the same problem but for discontinuous piecewise differential systems in R2 formed by linear differential centers separated by two parallel straight lines has at most one limit cycle.

KW - Continuous piecewise vector fields

KW - Discontinuous piecewise vector fields

KW - Filippov vector fields

KW - Limit cycles

UR - http://www.scopus.com/inward/record.url?scp=85135056422&partnerID=8YFLogxK

U2 - 10.1016/j.bulsci.2022.103173

DO - 10.1016/j.bulsci.2022.103173

M3 - Article

AN - SCOPUS:85135056422

SN - 0007-4497

VL - 179

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

M1 - 103173

ER -