Bounding the lenght of the first non-zero Melnikov function
In this talk I will present some results obtained with D. Novikov, L. Ortiz-Bobadilla and J. Pontigo-Herrera.
We study small polynomial deformations of a Hamiltonian system in the plane of the form dF+ϵω=0. We consider a family of regular cycles γ(t)∈F−1(t) and the displacement function Δ(t) along this family on a transversal parametrized by the values t of F.
Then Δ(t)=∑j=μMj(t)ϵj. The functions Mj are called Melnikov functions and we assume that Mμ≢0. It is the first nonzero Melnikov function. It is known that Mμ is given by an iterated integral of length at most μ. In general, the minimal length depends on the perturbation ω.
We want to give a bound for this minimal length, independent of the perturbation ω and depending only on the Hamiltonian F and the family of cycles γ.