Multibump homoclinic solutions to a centre equilibrium in a class of autonomous Hamiltonian systems

We consider an autonomous Hamiltonian system of fourth-order that has a cen tre with non-semi-simple imaginary eigenvalues and such that some coefficie nt of the corresponding normal form at the centre is (strictly) negative. Firstly, we prove the existence of two-dimensional stable and unstable mani folds to the centre, made of orbits converging polynomially to the equilibr ium. Then we show that the existence of a homoclinic orbit that is the tran sverse intersection of the stable and unstable manifolds, implies the exist ence of an infinite number of 'multibump' homoclinic solutions. In particul ar the topological entropy of the system is positive. Our approach relies partially on the calculus of variations.