The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems

We consider families of one and a half degrees of freedom rapidly forced Hamiltonian system which are perturbations of one degree of freedom Hamiltonians having a homoclinic connection. We derive the inner equation for this class of Hamiltonian system which is expressed as the Hamiltonian-Jacobi equation of one a half degrees of freedom Hamiltonian. The inner equation depends on a parameter not necessarily small. We prove the existence of special solutions of the inner equation with a given behavior at infinity. We also compute the asymptotic expression for the difference between these solutions. In some perturbative cases, this asymptotic expression is strongly related with the Melnikov function associated to our initial Hamiltonian. AMS classification scheme numbers: 37J45, 37G20, 35B40, 34E10, 34M99 Submitted to: Nonlinearity Inner equation for 1 2 degrees of freedom Hamiltonian systems 2