{"title":"Limit cycles of piecewise differential systems with linear Hamiltonian saddles and linear centres","authors":"J. Llibre, C. Valls","doi":"10.1080/14689367.2022.2037519","DOIUrl":null,"url":null,"abstract":"We study the continuous and discontinuous planar piecewise differential systems formed by linear centres together with linear Hamiltonian saddles separated by one or two parallel straight lines. When these piecewise differential systems are either continuous or discontinuous separated by one straight line, they have no limit cycles. When these piecewise differential systems are continuous and are separated by two parallel straight lines they do not have limit cycles. On the other hand, when these piecewise differential systems are discontinuous and separated by two parallel straight lines (either two centres and one saddle, or two saddles and one centre), we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle. If the piecewise differential systems separated by two parallel straight lines have three linear centres or three linear Hamiltonian saddles it is known that they have at most one limit cycle.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"262 - 279"},"PeriodicalIF":0.5000,"publicationDate":"2022-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Dynamical Systems-An International Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/14689367.2022.2037519","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 2
Abstract
We study the continuous and discontinuous planar piecewise differential systems formed by linear centres together with linear Hamiltonian saddles separated by one or two parallel straight lines. When these piecewise differential systems are either continuous or discontinuous separated by one straight line, they have no limit cycles. When these piecewise differential systems are continuous and are separated by two parallel straight lines they do not have limit cycles. On the other hand, when these piecewise differential systems are discontinuous and separated by two parallel straight lines (either two centres and one saddle, or two saddles and one centre), we show that they can have at most one limit cycle, and that there exist such systems with one limit cycle. If the piecewise differential systems separated by two parallel straight lines have three linear centres or three linear Hamiltonian saddles it is known that they have at most one limit cycle.
期刊介绍：
Dynamical Systems: An International Journal is a world-leading journal acting as a forum for communication across all branches of modern dynamical systems, and especially as a platform to facilitate interaction between theory and applications. This journal publishes high quality research articles in the theory and applications of dynamical systems, especially (but not exclusively) nonlinear systems. Advances in the following topics are addressed by the journal:
•Differential equations
•Bifurcation theory
•Hamiltonian and Lagrangian dynamics
•Hyperbolic dynamics
•Ergodic theory
•Topological and smooth dynamics
•Random dynamical systems
•Applications in technology, engineering and natural and life sciences
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