Cubic homogeneous polynomial centers

Abstract

Abstract. First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center.After using the averaging method of first order we show that we can obtain at most 1 limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with 1 limit cycles.

Limit cycles bifurcating from planar polynomial quasi-homogeneous centers

“…In fact for homogeneous cubic systems perturbed inside the class of all cubic polynomial systems these explicit formulas were given [6]. Some upper bounds for the number of limit cycles which bifurcate from the period annulus of a quasi-homogeneous polynomial differential system with a center have been given in [2]; see also the references therein.…”