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
2143611328 1872063233 2117676940 2091592576
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. 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 show that the Melnikov potential is a Morse function, in different kinds of examples. 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. Abstract We make explicit the geometric content of Mel’nikov’s method for detecting heteroclinic points between transversally hyperbolic periodic orbits. After developing the general theory of intersections for pairs of families of Lagrangian submanifolds N e ± , with N 0 + = N 0 − and constrained to live in an auxiliary family of submanifolds, we explain how the heteroclinic orbits of a given Hamiltonian system are detected by the zeros of the Mel’nikov 1-form. This 1-form admits an integral expression which is non-convergent in general. We discuss different solutions to this convergence problem.