Qualitative Theory of Dynamical Systems

, Volume 10, Issue 1, pp 51–64 | Cite as

Invariant Torus in 3D Lotka–Volterra Systems Appearing After Perturbation of Hopf Center

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Abstract

We study a three dimensional Lotka–Volterra systems. In the paper Bobieński and Żołądek (J Ergod Theory Dyn Syst 25:759–791, 2005) four cases of center (i.e. an invariant surface supporting a center) were found. In this paper, we study a codimension 2 component LV Hopf and its versal deformation. We prove that at most one invariant torus may appear. This invariant torus corresponds to the limit cycle bifurcating in the amplitude system.

Keywords

Lotka–Volterra system Abelian integrals Limit cycles 

Mathematics Subject Classification (2000)

Primary 34C07 37C27 Secondary 34C23 34C26 

Notes

Acknowledgments

The preliminary version of the paper was written while the author was visiting the Laboratorie Emile Picard, Université Paul Sabatier. The author thanks the University for hospitality. The author thanks H. Żołądek for helpful discussions. The author thanks the referee for his helpful suggestions concerning the presentation of this paper.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Institute of MathematicsWarsaw UniversityWarsawPoland

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