The The 1:-1:1 Resonance Integrable Problem for a Cubic Lotka-Volterra Systems.

Abstract

This paper is devoted to investigate the integrability and linearizability problems around a singular point at the origin of a cubic three-dimensional Lotka-Volterra differential system with -resonance. A complete set of necessary conditions for both integrability and linearizability are given. For sufficiency part, we show that the system has two analytic first integrals at the origin. In particular, we use Darboux method of integrability, linearizable node and transformation technique to show that the system admits two independent first integrals.

Keywords

• Integrability; Linearizability; First integral; Lotka-Volterra; Darboux method.