Exponentially small splitting of separatrices under fast quasiperiodic forcing

We consider fast quasiperiodic perturbations with two frequencies of a pendulum where is the golden mean number The complete system has a two dimensional invariant torus in a neighbourhood of the saddle point We study the splitting of the three dimensional invariant manifolds associated to this torus Provided that the perturbation amplitude is small enough with respect to and some of its Fourier coe cients the ones associated to Fibonacci numbers are separated from zero it is proved that the invariant manifolds split and that the value of the splitting which turns out to be exponentially small with respect to is correctly predicted by the Melnikov function AMS classi cation scheme numbers C F F J A Delshams V Gelfreich A Jorba and T M Seara Introduction At the end of the last century H Poincar e Poi discovered the phenomenon of the splitting of separatrices which seems to be the main cause of the stochastic behaviour in Hamiltonian systems He formulated the general problem of dynamics as a perturbation of an integrable Hamiltonian H I H I H I where is a small parameter I I I In n The values of the actions I such that the unperturbed frequencies k I H Ik are rationally dependent are called resonances As a model for the motion near a resonance Poincar e studied the pendulum with a high frequency perturbation which can be described by the Hamiltonian y cosx sin x cos t His calculations of the splitting originally validated only for j j exponentially small with respect to predicted correctly the splitting up to j j p for any positive parameter p Gel Tre The main problem in studying such kind of systems is that the splitting is exponentially small with respect to Namely Neishtadt s theorem Nei implies that in a Hamiltonian of the form H x y t H x y H x y t where the Hamiltonian system of H has a saddle and an associated homoclinic orbit and the perturbation H is a periodic function of time with zero mean value the splitting can be bounded from above byO e const For this estimate to be valid all the functions have to be real analytic in x and y but C dependence on time is su cient Lately the constant in the exponent was related to the position of complex time singularities of the unperturbed homoclinic orbit HMS Fon Fon The above mentioned systems provide a realistic model for the motion near a reso nance only in the case of two degrees of freedom If one considers simple resonances of systems with more than two degrees of freedom one can choose all the angles except one to be fast variables The simplest case is a quasiperiodic perturbation of a planar Hamiltonian system Neishtadt s averaging theorem was generalized to this case by C Sim o Sim but the upper bounds provided for the splitting depend in an essential way on the frequency vector of the perturbation For a perturbation of the pendulum depending on two frequencies C Sim o Sim checked numerically that a proper modi cation of the Melnikov method gives the correct prediction for the splitting Autonomous models with perturbations that depend on time in a quasiperiodic way appear in several problems of Celestial Mechanics For instance the motion of a spacecraft in the Earth Moon system can be modeled assuming that Earth and Moon Splitting of separatrices under fast quasiperiodic forcing revolve in circles around their common centre of masses this gives an autonomous model and the main perturbations di erence between the circular and the real motion of the Moon e ect of the Sun etc are modeled as a time dependent quasiperiodic function For more details see DJS or GJMS In the present paper we consider a quasiperiodic high frequency perturbation of the pendulum described by the Hamiltonian function I h x y