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

Abstract

We study a three dimensional Lotka–Volterra systems. In the paper Bobienski and Żolądek (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.