The Solution of the Extended 16th Hilbert Problem for Some Classes of Piecewise Differential Systems

Louiza Baymout, Rebiha Benterki and Jaume Llibre ()
Additional contact information
Louiza Baymout: Mathematical Analysis and Applications Laboratory, Department of Mathematics, University Mohamed El Bachir El Ibrahimi of Bordj Bou Arréridj, El Anasser 34000, Algeria
Rebiha Benterki: Mathematical Analysis and Applications Laboratory, Department of Mathematics, University Mohamed El Bachir El Ibrahimi of Bordj Bou Arréridj, El Anasser 34000, Algeria
Jaume Llibre: Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Barcelona, Catalonia, Spain

Mathematics, 2024, vol. 12, issue 3, 1-29

Abstract: The limit cycles have a main role in understanding the dynamics of planar differential systems, but their study is generally challenging. In the last few years, there has been a growing interest in researching the limit cycles of certain classes of piecewise differential systems due to their wide uses in modeling many natural phenomena. In this paper, we provide the upper bounds for the maximum number of crossing limit cycles of certain classes of discontinuous piecewise differential systems (simply PDS) separated by a straight line and consequently formed by two differential systems. A linear plus cubic polynomial forms six families of Hamiltonian nilpotent centers. First, we study the crossing limit cycles of the PDS formed by a linear center and one arbitrary of the six Hamiltonian nilpotent centers. These six classes of PDS have at most one crossing limit cycle, and there are systems in each class with precisely one limit cycle. Second, we study the crossing limit cycles of the PDS formed by two of the six Hamiltonian nilpotent centers. There are systems in each of these 21 classes of PDS that have exactly four crossing limit cycles.

Keywords: discontinuous piecewise differential system; Hamiltonian nilpotent center; cubic polynomial differential system; limit cycle; vector field (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/12/3/464/pdf (application/pdf)
https://www.mdpi.com/2227-7390/12/3/464/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:12:y:2024:i:3:p:464-:d:1330700

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().