Toni, Bourama2019-11-222019-11-221999-09-20Toni, B. (1999). Higher order branching of periodic orbits from polynomial isochrones. <i>Electronic Journal of Differential Equations, 1999</i>(35), pp. 1-15.1072-6691https://hdl.handle.net/10877/8880We 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 <i>n</i> 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.Text15 pages1 file (.pdf)enAttribution 4.0 InternationalLimit cyclesIsochronesPerturbationsCohomology decompositionArticle