Splitting of Separatrices in Hamiltonian Systems and Symplectic Maps Amadeu Delshams, Rafael Ram Rez-ros and Tere

Splitting of Separatrices in Hamiltonian Systems and Symplectic Maps Amadeu Delshams Rafael Ram Rez Ros and Tere M Seara

Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom

The splitting of separatrices for Hamiltonians with 1 2 degrees of freedom h(x; t=") = h0(x) + "ph1(x; t=") is measured. We assume that h(x) = h0(x1; x2) = x 2 2 =2 + V (x1) has a separatrix x(t), h(x; ) is 2 -periodic in , and " > 0 are independent small parameters, and p 0. Under suitable conditions of meromorphicity for x 2 (u) and the perturbation h(x(u); ), the order ` of the perturbation ...

SINGULAR SPLITTING OF SEPARATRICES FOR THE PERTURBED MCMILLAN MAP AMADEU DELSHAMS y RAFAEL RAM IREZ ROS

where V y P n Vny n is an even entire function Provided that V the origin O is a hyperbolic xed point with Spec dF O n e h o and its characteristic exponent h is given by cosh h V For F is an integrable map called McMillan map whose stable and unstable invariant curves to the origin coincide giving rise to a separatrix Thus the map F can be considered as a perturbation of the McMillan map being...

POINCAR E MELNIKOV ARNOLD METHOD FOR TWIST MAPS AMADEU DELSHAMS AND RAFAEL RAM REZ ROS Introduction A general theory for perturbations of an integrable planar map with a separatrix

A general theory for perturbations of an integrable planar map with a separatrix to a hyperbolic xed point has been developed in a previous lecture The splitting of the perturbed invariant curves was measured in rst order with respect to the parameter of perturbation by means of a periodic Melnikov function M de ned on the unperturbed separatrix In the case of planar twist maps M has zero mean ...

Lower and Upper Bounds for the Splitting of Separatrices of the Pendulum under a Fast Quasiperiodic Forcing Amadeu Delshams Vassili Gelfreich Angel Jorba and Tere M Seara

Quasiperiodic perturbations with two frequencies of a pendulum are considered where is the golden mean number We study the splitting of the three dimensional invariant manifolds associated to a two dimensional invariant torus in a neighbourhood of the saddle point of the pendulum Provided that some of the Fourier coe cients of the perturba tion the ones associated to Fibonacci numbers are separ...