Limit cycles near hyperbolas in quadratic systems |
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Authors: | Joan C. Arté s,Freddy Dumortier,Jaume Llibre |
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Affiliation: | a Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain b Mathematics Department, Universiteit Hasselt, Campus Diepenbeek, Agoralaan-Gebouw D, 3590 Diepenbeek, Belgium |
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Abstract: | In this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane. We study a strip of hyperbolas occurring in some quadratic systems. We deal with the cyclicity of the degenerate graphics DI2a from the programme, set up in [F. Dumortier, R. Roussarie, C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations 110 (1994) 86-133], to solve the finiteness part of Hilbert's 16th problem for quadratic systems. Techniques from geometric singular perturbation theory are combined with the use of the Bautin ideal. We also rely on the theory of Darboux integrability. |
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Keywords: | 58F14 58F21 58F30 34C05 |
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