Existence of generalized homoclinic solutions of Lotka-Volterra system under a small perturbation (Q530252)

The author considers the perturbed Lotka-Volterra system \[ v_{xx}=-\mu(1-a_2 u-v)v+\epsilon f(\epsilon,v,v_x,u,u_x),\;\;u_{xx}=-(1-u-a_1v)u+\epsilon g(\epsilon,v,v_x,u,u_x), \] for \(x\in \mathbb{R}\), with constants \(\mu>0\), \(a_1<1<a_2\), where \(f,g\) are smooth and \(\epsilon>0\) is a small parameter. Moreover, it is assumed that the functions \(f,g\) have certain reflection symmetries which yield a reversible system. For \(\mu\) sufficiently small, they establish the existence of a solution that is homoclinic to a small amplitude periodic orbit that emerges from a Hopf bifurcation.