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Maoan Han,Jaume Llibre,Yun Tian

doi:10.3390/math8071137

On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in     R 3   

Maoan Han, Jaume Llibre + Show 1 more

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https://doi.org/10.3390/math8071137

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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.

Highlights

Introduction and Statement of ResultsAn equilibrium point of a 3-dimensional autonomous differential system having a pair of purely imaginary eigenvalues and a zero eigenvalue is a zero-Hopf equilibrium.A 2-parameter unfolding of a 3-dimensional autonomous differential system with a zero-Hopf equilibrium is a zero-Hopf bifurcation
As far as we know the number of periodic orbits which can bifurcate from a zero-Hopf equilibrium point when this is perturbed inside the class of all Lotka–Volterra systems (1) only has been studied partially in the paper [23] using averaging theory of second order
We shall use the averaging theory of third order for studying the number of periodic orbits which can bifurcate from a zero-Hopf equilibrium point of a Lotka–Volterra system (1)

Summary

Introduction and Statement of Results

An equilibrium point of a 3-dimensional autonomous differential system having a pair of purely imaginary eigenvalues and a zero eigenvalue is a zero-Hopf equilibrium. As far as we know the number of periodic orbits which can bifurcate from a zero-Hopf equilibrium point when this is perturbed inside the class of all Lotka–Volterra systems (1) only has been studied partially in the paper [23] using averaging theory of second order. We shall use the averaging theory of third order for studying the number of periodic orbits which can bifurcate from a zero-Hopf equilibrium point of a Lotka–Volterra system (1). Previous results in this direction are the following.

The Algorithm for Computing the Periodic Solutions Bifurcating from a

Proof of Theorem 1