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Nondegenerate and Nilpotent Centers for a Cubic System of Differential Equations

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Abstract

We consider the autonomous system of differential equations of the form

$$\begin{aligned} {\dot{x}}=P_1(x,y)+P_2(x,y),\quad {\dot{y}}=Q_1(x,y)+Q_3(x,y), \end{aligned}$$

where \(P_i\) and \(Q_i\) are homogeneous polynomials of degree i. For such systems we provide the necessary and sufficient conditions to have a center at the origin. In fact this family only has nondegenerate and nilpotent centers.

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Acknowledgements

The authors are grateful to the referee for his/her valuable comments and suggestions to improve this paper. The first and second authors are partially supported by a MINECO/ FEDER grant number MTM2014-56272-C2-2 and by the Consejería de Educación y Ciencia de la Junta de Andalucía (projects P12-FQM-1658, and FQM-276). The third author is partially supported by a MINECO grant number 2017-84383-P and an AGAUR (Generalitat de Catalunya) grant number 2017SGR 1276.

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Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Huelva, Huelva, Spain

    Antonio Algaba & Cristóbal García

  2. Departament de Matemàtica, Inspires Research Centre, Universitat de Lleida, Avda. Jaume II, 69, 25001, Lleida, Catalonia, Spain

    Jaume Giné

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  1. Antonio Algaba
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  2. Cristóbal García
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  3. Jaume Giné
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Correspondence to Jaume Giné.

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Algaba, A., García, C. & Giné, J. Nondegenerate and Nilpotent Centers for a Cubic System of Differential Equations. Qual. Theory Dyn. Syst. 18, 333–345 (2019). https://doi.org/10.1007/s12346-018-0301-4

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