On the zero-Hopf bifurcation of the Lotka-Volterra systems in R3

Abstract

Here we study the Lotka-Volterra systems in R3, i.e. the differential systems of the form dxi/dt = xi(ri - Σ3j=1 aijxj), i = 1, 2, 3. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least six periodic orbits bifurcating from one of their equilibrium points. The tool for proving this result is the averaging theory of third order.