Branching of Periodic Orbits from Kukles Isochrones
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We study local bifurcations of limit cycles from isochronous (or linearizable) centers. The isochronicity has been determined using the method of Darboux linearization, which provides a birational linearization for the examples that we analyze. This transformation simplifies the analysis by avoiding the complexity of the Abelian integrals appearing in other approaches. As an application of this approach, we show that the Kukles isochrone (linear and nonlinear) has at most one branch point of limit cycles. Moreover, for each isochrone, there are small perturbations with exactly one continuous family of limit cycles.