Global centers of a family of cubic systems

Consider the family of polynomial differential systems of degree 3, or simply cubic systems x' = y, y' = -x + a₁ x² + a₂ xy + a₃ y² + a₄ x³ + a₅ x² y + a₆ xy² + a₇ y³, x ′ = y , y ′ = - x + a 1 x 2 + a 2 x y + a 3 y 2 + a 4 x 3 + a 5 x 2 y + a 6 x y 2 + a 7 y 3 , in the plane R² R 2 . An equilibrium point (x₀,y₀) ( x 0 , y 0 ) of a planar differential system is a center if there is a neighborhood U U of (x₀,y₀) ( x 0 , y 0 ) such that U \(x₀,y₀)\ U \ { ( x 0 , y 0 ) } is filled with periodic orbits. When R² \(x₀,y₀)\ R 2 \ { ( x 0 , y 0 ) } is filled with periodic orbits, then the center is a global center . For this family of cubic systems Lloyd and Pearson characterized in Lloyd and Pearson (Comput Math Appl 60:2797–2805, 2010) when the origin of coordinates is a center. We classify which of these centers are global centers.