Melnikov vector in higher dimensions

Abstract

A HOMOCLINIC orbit to a hyperbolic singular point of a system of autonomous ordinary differential equations is an orbit that is contained in the intersection of the stable and unstable manifolds of that singular point. Along the homoclinic orbit, tangent bundles of the stable and unstable manifolds do not span the whole space, and the intersection of these tangent bundles, in general, is of higher dimension. When a time periodic perturbation is applied to this system, the original stable and unstable manifolds will be perturbed. These are the stable and unstable manifolds of a Poincare map associated with the perturbed system. Furthermore, if the perturbation depends on parameters, then we are led to a bifurcation problem and we are interested in how the intersection of the stable and unstable manifolds of the Poincare map depends on parameters. If these two manifolds intersect transversally, namely if tangent spaces of these two manifolds at a point of intersection span the whole space, then ‘horseshoes’ will be created by the Smale-Birkhoff theorem [ 1 l] and the system exhibits an extremely rich dynamical behavior. If the stable and unstable manifolds of the Poincare map have quadratic tangency, then a still more complicated dynamics would occur, see [8]. One of the analytical methods to deal with these problems is the Poincare-Melnikov-Arnold method, often simply called the Melnikov method [2, 7, lo]. This method is based on measuring the separation of the stable and unstable manifolds of (a Poincare map of) the system. This method has been extensively studied, see, e.g. [3, 4, 6, 131 and references therein. However for higher dimensional systems this method seems not to have been fully developed. For example it is often assumed that a given system is a completely integrable Hamiltonian system. In this case the stable and unstable manifolds of a hyperbolic singular point coincide along a homoclinic orbit and hence we have a homoclinic manifold. In this situation the Melnikov method can be used not only to verify the transversal intersection but also to verify the tangential intersection. The purpose of these notes is to establish the Melnikov method for systems of higher dimension from a unified and geometrical viewpoint and to examine the possibilities and limitations of this method. This paper is organized as follows. In Section 2 we derive the expressions of the stable and unstable manifolds of a Poincare map associated with a periodically perturbed system. Using these expressions we obtain the separation vector between the stable and unstable manifolds in Section 3. The Melnikov vector is obtained as the first order approximation of the separation vector. In Section 4 we derive sufficient conditions for the transversal intersection of the stable and unstable manifolds of the Poincare map. In general these conditions cannot be necessary and sufficient conditions. However in Section 5 we consider a special situation in which the