We consider families of one and a half degrees of freedom Hamiltonians with high frequency periodic dependence on time, which are perturbations of an autonomous system.

We suppose that the origin is a parabolic fixed point with non-diagonalizable linear part and that the unperturbed system has a homoclinic connection associated to it. We provide a set of hypotheses under which the splitting is exponentially small and is given by the Poincaré-Melnikov function.

Readership

Graduate students and research mathematicians interested in dynamical systems and ergodic theory.

Table of Contents

• Notation and main results
• Analytic properties of the homoclinic orbit of the unperturbed system
• Parameterization of local invariant manifolds
• Flow box coordinates
• The extension theorem
• Splitting of separatrices
• References