# Global centers of a family of cubic systems

## TL;DR

This paper identifies which equilibrium points in a specific type of cubic differential system are global centers, meaning their neighborhoods are filled with periodic orbits.

## Contribution

The paper provides a classification of global centers within a family of cubic systems previously studied by Lloyd and Pearson.

## Key findings

- The paper classifies which centers in the cubic system are global centers.
- It builds on prior work by Lloyd and Pearson to determine when the origin is a global center.

## Abstract

Consider the family of polynomial differential systems of degree 3, or simply cubic systems \documentclass[12pt]{minimal}
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\begin{document}$$ x' = y, \quad y' = -x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 + a_7 y^3, $$\end{document}x′=y,y′=-x+a1x2+a2xy+a3y2+a4x3+a5x2y+a6xy2+a7y3,in the plane \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^2$$\end{document}R2. An equilibrium point \documentclass[12pt]{minimal}
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\begin{document}$$(x_0,y_0)$$\end{document}(x0,y0) of a planar differential system is a center if there is a neighborhood \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {U}$$\end{document}U of \documentclass[12pt]{minimal}
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\begin{document}$$(x_0,y_0)$$\end{document}(x0,y0) such that \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {U} \backslash \{(x_0,y_0)\}$$\end{document}U\{(x0,y0)} is filled with periodic orbits. When \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^2\setminus \{(x_0,y_0)\}$$\end{document}R2\{(x0,y0)} 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.