A new method for measuring the splitting of invariant manifolds

We study the so-called Generalized Arnol'd Model (a weakly hyperbolic near- integrable Hamiltonian system), with d + 1 degrees of freedom (d greater th an or equal to 2), in the case where the perturbative term does not affect a fixed invariant d-dimensional torus. This torus is thus independent of th e two perturbation parameters which are denoted epsilon(epsilon > 0) and mu . We describe its stable and unstable manifolds by solutions of the Hamilton- Jacobi equation for which we obtain a large enough domain of analyticity. T he splitting of the manifolds is measured by the partial derivatives of the difference DeltaS of the solutions, for which we obtain upper bounds which are exponentially small with respect to epsilon. A crucial tool of the method is a characteristic vector field, which is def ined on a part of the configuration space, which acts by zero on the functi on DeltaS and which has constant coefficients in well-chosen coordinates. It is in the case where \mu is bounded by some positive power of epsilon th at the most precise results are obtained. In a particular case with three d egrees of freedom, the method leads also to lower bounds for the splitting. (C) 2001 Editions scientifiques et medicales Elsevier SAS.