On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in <inline-formula> <mml:math id="mm999" display="block"> <mml:semantics> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:semantics> </mml:math> </inline-formula>

Abstract

Here we study 3-dimensional Lotka–Volterra systems. 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. We remark that these systems with such six periodic orbits are non-competitive Lotka–Volterra systems. The proof is done using the algorithm that we provide for computing the periodic solutions that bifurcate from a zero-Hopf equilibrium based in the averaging theory of third order. This algorithm can be applied to any differential system having a zero-Hopf equilibrium.

Keywords

• Lotka–Volterra differential systems
• periodic orbit
• Hopf bifurcation
• averaging theory