Vector fields with homogeneous nonlinearities and many limit cycles

Armengol Gasull, Jiang Yu, Xiang Zhang

Research output: Contribution to journalArticleResearchpeer-review

8 Citations (Scopus)

Abstract

© 2015 Elsevier Inc. Consider planar real polynomial differential equations of the form x ˙=Lx+Xn(x), where x=(x,y)∈R2, L is a 2×2 matrix and Xn is a homogeneous vector field of degree n>1. Most known results about these equations, valid for infinitely many n, deal with the case where the origin is a focus or a node and give either non-existence of limit cycles or upper bounds of one or two limit cycles surrounding the origin. In this paper we improve some of these results and moreover we show that for n≥3 odd there are equations of this form having at least (n+1)/2 limit cycles surrounding the origin. Our results include cases where the origin is a focus, a node, a saddle or a nilpotent singularity. We also discuss a mechanism for the bifurcation of limit cycles from infinity.
Original languageEnglish
Pages (from-to)3286-3303
JournalJournal of Differential Equations
Volume258
Issue number9
DOIs
Publication statusPublished - 1 Jan 2015

Keywords

  • Focus
  • Homogeneous nonlinearities
  • Limit cycle
  • Nilpotent singularity
  • Node
  • Polynomial differential equations

Fingerprint Dive into the research topics of 'Vector fields with homogeneous nonlinearities and many limit cycles'. Together they form a unique fingerprint.

Cite this