Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems

We classify new classes of centers and of isochronous centers for polynomial differential systems in ℝ2 of arbitrary odd degree d ≥ 7 that in complex notation z = x + iy can be written as where j is either 0 or 1, λ ε ℝ and A,B,C ε ℂ. Note that if j = 0 and d = 7 we obtain a special case of seventh polynomial differential systems which can have a center at the origin, and if j = 1 and d = 9 we obtain a special case of ninth polynomial differential systems which can have a center at the origin. © 2010 Springer Science+Business Media, LLC.