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

Abstract

Abstract. 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.

Exponentially small splitting of separatrices beyond Melnikov analysis: Rigorous results

“…In [3], it is seen that the constant f 0 coincides with the constant that Melnikov theory gives in front of the exponential term (see Lemma 2.3).…”

“…The solutions of the Hamilton-Jacobi equation (72) were studied in [3] in the complex domains for κ > 0 and θ > 0 (see Fig. 9).…”

The inner equation for generic analytic unfoldings of the Hopf-zero singularity

“…In fact, in [3] the asymptotic expression for I n is proved only when I n has the form I n = +∞ −∞ e i αt (s + t) − −1 dt with ∈ Q, but it is immediate that the result also holds in this case. The estimation for J n also needs an extra argument to be done from the results in [3].…”

“…In this section we will recover Theorem 1.2 from Theorem 1.5. We will need a technical lemma, analogous to Lemma 2.6, which was proved in [3].…”

“…Introduction. We consider the family of autonomous differential equations in C 3 given by dφ dτ = −ηφ − (α + cη) i φ + εF 1 (φ, ϕ, η) dϕ dτ = −ηϕ + (α + cη) i ϕ + εF 2 (φ, ϕ, η) (1.1)…”