Geometry of the Melnikov Vector

Abstract

Publisher Summary This chapter discusses the geometry of the Melnikov vector in higher dimensional cases and extends the theories for the two-dimensional case to higher dimensional cases. Palmer showed that the linear variational system along the homoclinic orbit of the unperturbed autonomous system has exponential dichotomies on half-lines. Using this fact, explicit expressions of the local stable and unstable manifolds of the perturbed system is derived. The case of Palmer in higher dimensional cases is used to derive the Melnikov vector. The chapter introduces a notion of an index of a homoclinic or heteroclinic orbit, which is useful to classify the cases that can occur in higher dimensional cases. It discusses a relation between the dimension of the Melnikov vector and the index of the homoclinic or heteroclinic orbit. Numerical aspect of the Melnikov vector is also discussed in the chapter. Several special cases in which the Melnikov vectors take simpler forms are considered and the tangency of the stable and unstable manifolds is discussed in the chapter. These general theories are applied to Hamiltonian systems. These theories are extended to the case of a heteroclinic orbit for invariant tori and as a by-product a formula is derived that guarantees the transversal intersection of the stable and unstable manifolds of a two-dimensional system with a quasi-periodic perturbation.