Qualitative Theory of Dynamical Systems

, Volume 10, Issue 1, pp 51-64

First online:

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

  • Marcin BobieńskiAffiliated withInstitute of Mathematics, Warsaw University Email author 


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.


Lotka–Volterra system Abelian integrals Limit cycles

Mathematics Subject Classification (2000)

Primary 34C07 37C27 Secondary 34C23 34C26