The successive derivatives of the period function of a plane vector field
Previously, the author provided an expression which generalized the classical Melnikov function to any order, for the first nonzero derivative of a return mapping [the author, Publ. Mat., Barc. 41, No. 1, 127-134 (1997; Zbl 0892.34023)]. This method relied on the decomposition of a 1-form associated to the relative cohomology of the perturbed Hamiltonian. With the same techniques, he gives a formula for the first nonzero derivative of a period function. He focuses on the particular example of \(H= (1/2)(x^2+ y^2)\) and then he defines a class of Hamiltonians for which the same computation remains valid. Finally, he investigates relations with the Birkhoff normal form.
scientific article; zbMATH DE number 1212364