Integrable and non-integrable Lotka-Volterra systems

Abstract

In a recent paper [1], completely integrable cases were discovered of the Lotka Volterra Hamiltonian (LVH) system without linear terms, dxidt=x˙i=∑j=1naijxixj,i=1,…,n, aij=−aji, satisfying the condition H=∑i=1nxi=h=const. In this paper, we first generalize this system to one that includes an arbitrary set of linear terms that preserve the Hamiltonian integral. We thus discover a wide class of LVH systems which we claim to be integrable, since their equations possess the Painlevé property, i.e. their solutions have only poles as movable singularities in the complex t-plane. Next, we focus on the case n=3 and vary some of the parameters, including additional nonlinearities to look for nonintegrable extensions with interesting dynamical properties. Our results suggest that, in this class of systems, non - integrability generally yields simple dynamics far removed from the type of complexity one expects from non - integrable 3 - dimensional nonlinear systems.