The solution of some persistent p : −q resonant center problems

The notion of p : −q resonant center was introduced recently and studied by several authors. In this paper we generalize the notion of a persistent center to a persistent p : −q resonant center and find conditions for existence of a persistent p : −q resonant center for several p : −q resonant systems with quadratic nonlinearities. To prove the sufficiency of the obtained conditions we use either the Darboux theory of integrability or look for a formal first integral of the required form or we use the method based on the blow-up transformation

Linearizability problem of persistent centers

The concepts of persistent and weakly persistent centers were introduced in 2009 and the same concept was applied in the study of some families of differential equations in 2013. Such concept was generalized for complex planar differential systems in 2014. In this paper we extend the notion of persistent center to a linearizable persistent center and a linearizable weakly persistent center. Using the methods and algorithms of computational algebra we characterize the planar cubic differential system having linearizable persistent and linearizable weakly persistent centers at the origin