Stable periodic orbits for a class of three dimensional competitive systems

H. R. Zhu, Hal Smith

Research output: Contribution to journalArticlepeer-review

54 Scopus citations


It is shown that for a dissipative, three dimensional, competitive, and irreducible system of ordinary differential equations having a unique equilibrium point, at which point the Jacobian matrix has negative determinant, either the equilibrium point is stable or there exists an orbitally stable periodic orbit. If in addition, the system is analytic then there exists an orbitally asymptotically stable periodic orbit when the equilibrium is unstable. The additional assumption of analyticity can be replaced by the assumption that the equilibrium point and every periodic orbit are hyperbolic. In this case, the Morse-Smale conditions hold and the flow is structurally stable.

Original languageEnglish (US)
Pages (from-to)143-156
Number of pages14
JournalJournal of Differential Equations
Issue number1
StatePublished - May 1994

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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