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.

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.

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