Global centers of a family of cubic systems

Abstract

AbstractConsider the family of polynomial differential systems of degree 3, or simply cubic systems $$ x' = y, \quad y' = -x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 + a_7 y^3, $$ 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 $$\mathbb {R}^2$$ R 2 . An equilibrium point $$(x_0,y_0)$$ ( x 0 , y 0 ) of a planar differential system is a center if there is a neighborhood $$\mathcal {U}$$ U of $$(x_0,y_0)$$ ( x 0 , y 0 ) such that $$\mathcal {U} \backslash \{(x_0,y_0)\}$$ U \ { ( x 0 , y 0 ) } is filled with periodic orbits. When $$\mathbb {R}^2\setminus \{(x_0,y_0)\}$$ 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.