Homoclinic Solutions of an Invertible ODE System

Near its statiinary point we study solutions of an invertible system of ordinary differential equations with a square nonlinearity and with parameters υ ∈ IR and σ = ±1. The system appeared from the water-wave problem after its reduction on the center manifold and a selection of the basic first approximation and a power transformation of coordinates. In a neighbourhood of a stationary point we study the system by means of its normal form for cases σ = ±1 and υ = 1, when there is a double zero eigenvalue. We have found local families of periodic solutions, of conditionally periodic solutions and of homoclinic solutions. We make a comparison with the Hamiltonian normal form.