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 systems which are perturbations of one degree of freedom Hamiltonians with a homoclinic connection. We derive the inner equation for this class of Hamiltonian system which is expressed as the Hamiltonian–Jacobi equation of a 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 behaviour 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 with our initial Hamiltonian.