Shape and period of limit cycles bifurcating from a class of Hamiltonian period annulus

Abstract

Abstract. In this work we are concerned with the problem of shape and period of isolated periodic solutions of perturbed analytic radial Hamiltonian vector fields in the plane. Françoise develop a method to obtain the first non vanishing Poincaré-Pontryagin-Melnikov function. We generalize this technique and we apply it to know, up to any order, the shape of the limit cycles bifurcating from the period annulus of the class of radial Hamiltonians. We write any solution, in polar coordinates, as a power series expansion in terms of the small parameter. This expansion is also used to give the period of the bifurcated periodic solutions. We present the concrete expression of the solutions up to third order of perturbation of Hamiltonians of the form H = H(r). Necessary and sufficient conditions that show if a solution is simple or double are also presented.

Periodic orbits from second order perturbation via rational trigonometric integrals

“…As it can be seen in [32], where polynomial perturbations of a linear center are considered, the Poincaré-Pontryagin-Melnikov functions of higher order can be obtained without additionally difficulties. Such a generating functions can be obtained because the involved integrals only depend polynomially on r, sin θ and cos θ.…”

“…In the latter one, the method is extended to radial Hamiltonians to study the shape of an orbit as well. A higher order study when ω is a polynomial one form is carried out in [32] because all the iterated primitives are obtained in terms of elementary functions.…”

“…This case is also studied in [32]. All the families that are considered in this second application can be transformed into (8) where the perturbations of the linear center are rational functions.…”

“…Note that, as the time of both equations is the same, the period of the periodic solutions also coincides. Only the case when the linearization and its inverse are polynomial is considered in [32]. Quadratic isochronous centers are classified in [29] in four families called S 1 , S 2 , S 3 and S 4 .…”

Corrigendum to “Shape and period of limit cycles bifurcating from a class of Hamiltonian period annulus” [Nonlinear Anal. 81 (2013) 130–148]

“…The corrected expression is done in the corollary below. We remark that only the general expression for the period is used in [1]. Hence, all the expressions for the period described in the applications are correct.…”