Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems

The authors consider a class of fourth-order, reversible, autonomous Hamiltonian systems for which an orbit homoclinic to a node on the stable manifold is known to exist. If parameters are varied so that on the stable manifold the node become a focus (i.e. when 4 real eigenvalues become complex) it is shown that generic homoclinic orbits of systems in this class bifurcate into a countably infinite set of distinct symmetric homoclinic orbits. As an application they show that the problem du(t)Pu(t)+u(t)-u(t)²=0, u(0)=du(0)=0, u not=0, u(+or- infinity )=u(+or- infinity )=u(+or- infinity )=du(+or- infinity )=0 which has a unique (up to translations) solution, of large amplitude, for P<or=-2 has a countably infinite set of distinct large amplitude solutions for each P in (-2, -2+ in ) for some in >0. For P in (2- in ,2) it has at least two small amplitude solutions and for all P<2 it has at least one solution.