A 3D competitive Lotka–Volterra system with three limit cycles: A falsification of a conjecture by Hofbauer and So

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• 作者：Mats Gyllenberg ; Ping Yan and Yi Wang
• 关键词：Competitive ; Lotka– ; Volterra system ; Limit cycles ; Carrying simplex ; Hopf bifurcation
• 刊名：Applied Mathematics Letters
• 出版年：2006
• 期刊代码：5_08939659
• 类别：mt
• 出版时间：2006
• 卷：19
• 期：1
• 页码：1-7
• 文件大小：172 K

摘要

For three-dimensional competitive Lotka–Volterra systems, Zeeman [M.L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems, Dyn. Stab. Syst. 8 (1993) 189–217] identified 33 stable nullcline equivalence classes. Among these, only classes 26–31 may have limit cycles. Hofbauer and So [J. Hofbauer, J.W.-H. So, Multiple limit cycles for three dimensional Lotka–Volterra equations, Appl. Math. Lett. 7 (1994) 65–70] conjectured that the number of limit cycles is at most two for these systems. In this paper, we construct three limit cycles for class 29 without a heteroclinic polycycle in Zeeman’s classification.