We consider a mechanical system consisting of n-penduli and a d-degree-of-freedom rotator. The phase space of the rotator defines a normally hyperbolic invariant manifold ( _0 ). We apply a time-dependent perturbation, which is not assumed to be either Hamiltonian, or periodic, or quasi-periodic, as we allow for rather general time dependence. The strength of the perturbation is given by a parameter ( R ). For all (| | ) sufficiently small, the augmented flow—obtained by making the time into a new variable—has a normally hyperbolic locally invariant manifold ( _ ). For ( =0 ), ( _0= _0 R ). We define a Melnikov-type vector, which gives the first-order expansion of the displacement of the stable and unstable manifolds of ( _0 ) under the perturbation. We provide an explicit formula for the Melnikov vector in terms of convergent improper integrals of the perturbation along homoclinic orbits of the unperturbed system. We show that if the perturbation satisfies some explicit non-degeneracy conditions, then the stable and unstable manifolds of ( _ ), (W^ s ( _ ) ) and (W^ u ( _ ) ), respectively, intersect along a transverse homoclinic manifold, and, moreover, the splitting of (W^ s ( _ ) ) and (W^ u ( _ ) ) can be explicitly computed, up to the first order, in terms of the Melnikov-type vector. This implies that the excursions along some homoclinic trajectories yield a non-trivial increase of order (O( ) ) in the action variables of the rotator, for all sufficiently small perturbations. The formulas that we obtain are independent of the unperturbed motions in ( _0 ), and give, at the same time, the effects on periodic, quasi-periodic, or general-type orbits. When the perturbation is Hamiltonian, we express the effects of the perturbation, up to the first order, in terms of a Melnikov potential. In addition, if the perturbation is periodic, we obtain that the non-degeneracy conditions on the Melnikov potential are generic.
A Melnikov potential whose critical points give rise to transverse homoclinic intersections in perturbation of periodic points is introduced in @cite_26 @cite_19 . In @cite_41 @cite_37 the Melnikov potential is established in a situation where the hyperbolic part consists of a single pendulum. A geometric version of the Melnikov integrals for periodic points with codimension one manifolds is studied in @cite_3 . The paper @cite_3 assumes explicitly that the perturbations vanish on the manifold.
@cite_37 @cite_26 @cite_41 @cite_3 @cite_19
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We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n+1 degrees of freedom. The integrable system is assumed to have n-dimensional hyperbolic invariant tori with coincident whiskers (separatrices). Following Eliasson, we use a geometric approach closely related to the Lagrangian properties of the whiskers, to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar function of n phases, which we call splitting potential. This geometric approach works for both the singular (or weakly hyperbolic) case and the regular (or strongly hyperbolic) case, and provides the existence of at least n+1 homoclinic intersections between the perturbed whiskers. In the regular case, we also obtain a first order approximation for the splitting potential, that we call Melnikov potential. Its gradient, the (vector) Melnikov function, provides a first order approximation for the splitting distance. Then the nondegenerate critical points of the Melnikov potential give rise to transverse homoclinic intersections between the whiskers. Generically, when the Melnikov potential is a Morse function, there exist at least 2^n critical points. The first order approximation relies on the n-dimensional Poincare-Melnikov method, to which an important part of the paper is devoted. We develop the method in a general setting, giving the Melnikov potential and the Melnikov function in terms of absolutely convergent integrals, which take into account the phase drift along the separatrix and the first order deformation of the perturbed hyperbolic tori. We provide formulas useful in several cases, and carry out explicit computations that We show that certain mechanical systems, including a geodesic ow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy. The assumptions we make in the case of geodesic ows are: a) The metric and the external perturbation are smooth enough. b) The geodesic ow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection. c) The frequency of the external perturbation is Diophantine. d) The external potential satisOes a generic condition depending on the periodic orbit considered in b). The assumptions on the metric are C2 open and are known to be dense on many manifolds. The assumptions on the potential fail only in inOnite codimension spaces of potentials. The proof is based on geometric considerations of invariant manifolds and their intersections. The main tools include the scattering map of normally hyperbolic invariant manifolds, as well as standard perturbation theories (averaging, KAM and Melnikov techniques). We do not need to assume that the metric is Riemannian and we obtain results for Finsler or Lorentz metrics. Indeed, there is a formulation for Hamiltonian systems satisfying scaling hypotheses. We do not need to make assumptions on the global topology of the manifold nor on its dimension. We study families of volume preserving diffeomorphisms in R3 that have a pair of hyperbolic fixed points with intersecting codimension one stable and unstable manifolds. Our goal is to elucidate the topology of the intersections and how it changes with the parameters of the system. We show that the primary intersection of the stable and unstable manifolds is generically a neat submanifold of a fundamental domain. We compute the intersections perturbatively using a codimension one Melnikov function. Numerical experiments show various bifurcations in the homotopy class of the primary intersections. We study perturbations of diffeomorphisms that have a saddle connection between a pair of normally hyperbolic invariant manifolds. We develop a first-order deformation calculus for invariant manifolds and show that a generalized Melnikov function or Melnikov displacement can be written in a canonical way. This function is defined to be a section of the normal bundle of the saddle connection.We show how our definition reproduces the classical methods of Poincare and Melnikov and specializes in methods previously used for exact symplectic and volume-preserving maps. We use the method to detect the transverse intersection of stable and unstable manifolds and relate this intersection to the set of zeros of the Melnikov displacement. The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called the Melnikov potential) are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, if the unperturbed invariant manifolds are completely doubled, it is shown that there exist, in general, at least @math primary homoclinic orbits (4n in antisymmetric maps). Both lower bounds are optimal.