Melnikov Functions in Quadratic Perturbations of Generalized Lotka–Volterra Systems

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• 作者：Henryk ?o?a?dek
• 关键词：Quadratic plane vector fields ; Lotka–Volterra systems ; Limit cycles ; Melnikov functions ; Primary 05C38 ; 15A15 ; Secondary 05A15 ; 15A18
• 刊名：Journal of Dynamical and Control Systems
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：21
• 期：4
• 页码：573-603
• 全文大小：635 KB
• 参考文献：1.Arnold VI, (ed). 2004. Arnold’s problems. New York: Springer.
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• 作者单位：Henryk ?o?a?dek (1)
  
  1. Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Calculus of Variations and Optimal Control
  Analysis
  Applications of Mathematics
  Systems Theory and Control
  
• 出版者：Springer Netherlands
• ISSN：1573-8698

文摘

We present a detailed analysis of Melnikov functions which arise in quadratic perturbations of generalized Lotka–Volterra vector fields with the first integral x α y β (1 ?x ?y). That analysis was sketched in ?o?a?dek (J Differ Equ 109:223-73, 1994). In particular, we prove that the maximal number of limit cycles in the generic case equals 2 and in the Hamiltonian triangle case, this number is 3. Keywords Quadratic plane vector fields Lotka–Volterra systems Limit cycles Melnikov functions