SummaryWe 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 w e call splitting potential. This geometric approach w orks 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 rst order approximation for the splitting potential, that we call Melnikov potential. Its gradient, the (vector) Melnikov function, provides a rst 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, w h e n the Melnikov p o t e n tial is a Morse function, there exist at least 2 n critical points.The rst order approximation relies on the n-dimensional Poincar e{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 i n tegrals, which t a k e into account the phase drift along the separatrix and the rst 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 di erent kinds of examples.MSC numbers.58F05, 34C37, 58F36, 34C30, 70F15 PAC numbers. 45.20.Jj, 02.40.Vh, 95.10.Fh, 45.50.Pk 1 Setup and introduction Perturbation of a hyperbolic integrable HamiltonianIt is well-known that the problem of giving conditions for the splitting of the whiskers of hyperbolic invariant tori is one of the main di culties related with the Arnold di usion, a phenomenon of instability in perturbations of integrable Hamiltonian systems with more than 2 degrees of freedom. The present paper is concerned with the study of the existence of homoclinic orbits and splitting in a wide class of Hamiltonians. The tools used are a geometric approach based on Eliasson's work Eli94], and the Poincar e{Melnikov method. 1

We give a precise statement for the KAM theorem in a neighbourhood of an elliptic equilibrium point of a Hamiltonian system. If the frequencies of the elliptic point are nonresonant up to a certain order K 4, and a nondegeneracy condition is fulfilled, we get an estimate for the measure of the complement of the KAM tori in a neighbourhood of given radius. Moreover, if the frequencies satisfy a Diophantine condition, with exponent {, we show that in a neighbourhood of radius r the measure of the complement is exponentially small in (1Âr) 1Â({+1) . We also give a related result for quasi-Diophantine frequencies, which is more useful for practical purposes. The results are obtained by putting the system in Birkhoff normal form up to an appropiate order, and the key point relies on giving accurate bounds for its terms.

The two main stability results for nearly-integrable Hamiltonian systems are revisited: Nekhoroshev theorem, concerning exponential lower bounds for the stability time (effective stability), and KAM theorem, concerning the preservation of a majority of the nonresonant invariant tori (perpetual stability). To stress the relationship between both theorems, a common approach is given to their proof, consisting of bringing the system to a normal form constructed through the Lie series method. The estimates obtained for the size of the remainder rely on bounds of the associated vectorfields, allowing one to get the``optimal'' stability exponent in Nekhoroshev theorem for quasiconvex systems. On the other hand, a direct and complete proof of the isoenergetic KAM theorem is obtained. Moreover, a modification of the proof leads to the notion of nearly-invariant torus, which constitutes a bridge between KAM and Nekhoroshev theorems. AcademicPress, Inc.

Abstract. We study the existence of transverse homoclinic orbits in a singular or weakly hyperbolic Hamiltonian, with 3 degrees of freedom, as a model for the behaviour of a nearly-integrable Hamiltonian near a simple resonance. The example considered consists of an integrable Hamiltonian possessing a 2-dimensional hyperbolic invariant torus with fast frequencies != p " and coincident whiskers or separatrices, plus a perturbation of order = " p

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