Cubic homogeneous polynomial centers (Q482178)

The weakened 16th Hilbert's problem for perturbed mainly non-Hamiltonian centers is under consideration using the technique of the averaging method of first order. The first result states that any planar cubic homogeneous polynomial differential system having a center can be rewritten into the form \[ \dot{x}=ax^3+(b-3 \alpha \mu)x^2y-axy^2-\alpha y^3, \quad \dot{y}=\alpha x^3+ax^2y+(b+3 \alpha \mu)xy^2-a y^3, \] where \(\alpha = \pm 1\), \(a, b, \mu \in\mathbb{R}\) and \(\mu > -1/3\). Then the authors prove the possibility to obtain at most one 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 one limit cycle.