Higher order branching of periodic orbits from polynomial isochrones

Abstract

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.

Keywords

• Limit cycles
• Isochrones
• Perturbations
• Cohomology Decomposition.