The three-dimensional generalized Lotka–Volterra systems

Abstract

This work is devoted to the study of periodic solutions of generalized Lotka–Volterra systems in three dimensions. The group A₃ of isometries of the regular tetrahedron acts on the twelve-dimensional space of such systems. We find four components (modulo the action of A₃) of the center variety, i.e. the semi-algebraic subset consisting of systems with a family of periodic solutions (which degenerate at the center). We focus attention on the component which consists of systems X₀ with an invariant plane L₀ transversal to the coordinate axes and with a (partial) first integral on L₀. Under the assumption of normal hyperbolicity of X₀ along L₀ we begin study of limit cycles appearing for a perturbed system $X_{0}+\varepsilon X_{1}$ (in the Lotka–Volterra class). The linearization of the problem leads to the problem of zeroes of certain Poincaré–Melnikov integrals of a new type. We determine the asymptotic behavior of these integrals near two critical values of the partial first integral of the restricted system X₀|_L0. We also study the situation near the intersection of different components of the center variety.