Homoclinic orbits to small periodic orbits for a class of reversible systems

Abstract

In this paper a one-parameter class of four-dimensional, reversible vector fields is investigated near an equilibrium. We call the parameter μ and place the equilibrium at 0. The differential at 0 is supposed to have ±iq, q > 0, as simple eigenvalues and 0 as a double, nonsemisimple eigenvalue. Our ultimate goal is to construct homoclinic connections of periodic orbits of arbitrary small size, in fact we shall show that the oscillations of the homoclinic orbits at infinity are bounded by a flat function of μ. This result receives its significance from the still unsolved question as to whether solutions exist which are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. First we construct the periodic solutions. In contrast to previous work, we find these in a full rectangle [0, K0] × ]0,μ0], where K measures the amplitude of the periodic orbits. Then we show that for each n ∈ ℕ there is a μn and a family of periodic solutions X(μ), μ ∈]0,μn[, of Size μn. To each of these solutions, we can find two homoclinic orbits, which are distinguished by their phase shift at infinity. One example of such a vector field occurs when describing the flow of an inviscid irrotational fluid layer under the influence of gravity and small surface tension (Bond number b < ⅓), for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalised solitary wave. Our work shows that there exist solitary waves with oscillations at infinity of order less than |μ|n for every n.