SUBSIDIARY HOMOCLINIC ORBITS TO A SADDLE-FOCUS FOR REVERSIBLE-SYSTEMS

A dynamical system is said to be reversible if there is an involution of phase space that reverses the direction of the flow. Examples are classical Hamiltonian systems with quadratic kinetic energy. For reversible systems, homoclinic orbits that are invariant under the reversible transformation typically persist as parameters are varied. This paper concerns reversible systems for which a primary homoclinic orbit to a saddle-focus is assumed to exist. The problem under investigation is a characterisation of the subsidiary homoclinic orbits which then exist in a neighbourhood of the primary one. Such orbits have applications as solitary water waves and as buckling solutions of nonlinear struts. A Shil'nikov-type analysis is performed for four-dimensional linearly reversible systems. It is shown that each subsidiary homoclinic orbit can be labelled by a symmetric string of positive integers. All possible strings of length one, two or three correspond to the existence of a homoclinic orbit, whereas only certain of those of length four or greater do. This situation contrasts with known results if the reversible system is also Hamiltonian. The analysis is supported by performing careful numerical experiments on the equation u + Pu + u - (1 - alpha) u(2) - alpha u(2) = O, where P and are parameters; a good agreement with the theory is found.