The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems

Lijun Wei, Yun Tian and Yancong Xu
Additional contact information
Lijun Wei: School of Mathematics, Hangzhou Normal University, Hangzhou 310036, China
Yun Tian: Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
Yancong Xu: School of Mathematics, Hangzhou Normal University, Hangzhou 310036, China

Mathematics, 2022, vol. 10, issue 9, 1-14

Abstract: This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function H = 1 2 y 2 + 1 2 x 2 − 2 3 x 3 + a 4 x 4 ( a ≠ 0 ) under two types of polynomial perturbations of degree m , respectively. It is proved that the Hamiltonian system perturbed in Liénard systems can have at least [ 3 m − 1 4 ] small limit cycles near the center, where m ≤ 101 , and that the related near-Hamiltonian system with general polynomial perturbations can have at least m + [ m + 1 2 ] − 2 small-amplitude limit cycles, where m ≤ 16 . Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, [ · ] represents the integer function.

Keywords: Liénard system; near-Hamiltonian system; Hopf bifurcation; elementary center (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/9/1483/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/9/1483/ (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:10:y:2022:i:9:p:1483-:d:805378

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 ().