Higher order branching of periodic orbits from polynomial isochrones

Toni, B.

Electronic Journal of Differential Equations, Tome 1999 (1999).

Résumé

Summary: We discuss the higher order local bifurcations of limit cycles from polynomial isochrones (linearizable centers) when the linearizing transformation is explicitly known and yields a polynomial perturbation one-form. Using a method based on the relative cohomology decomposition of polynomial one-forms complemented with a step reduction process, we give an explicit formula for the overall upper bound of branch points of limit cycles in an arbitrary $n$ degree polynomial perturbation of the linear isochrone, and provide an algorithmic procedure to compute the upper bound at successive orders. We derive a complete analysis of the nonlinear cubic Hamiltonian isochrone and show that at most nine branch points of limit cycles can bifurcate in a cubic polynomial perturbation. Moreover, perturbations with exactly two, three, four, six, and nine local families of limit cycles may be constructed.

Détail

Classification : 34C15, 34C25, 58F14, 58F21, 58F30
Keywords: limit cycles, isochrones, perturbations, cohomology decomposition