Bifurcartions of limit cycles from hamiltonian isochrones: a

In this paper, we study the local bifurcations of limit cycles from isochrones. These isochrones have been de-termined using the method of Darboux linearization which provides a rational linearizing change of coordinates for the examples we analyze. By means of this linearizing transformation, the perturbation of a non-linear isochrone is actually reduced to that of the linear one, simplifying the analysis and avoiding the complexity of the Abelian integrals appearing in other approaches. As an application, we show that no more than two families of limit cycles can bifurcate from cubic Hamiltonian nonlinear isochrones. From the linear isochrone the maximum number is one. We also give the upper bound from an arbitrarily n−degree polynomial autonomous perturbation of the linear isochrone. Moreover there are small perturbations leading to the maximum number in each case