Generalized Hopf Bifurcation for Planar Vector Fields via the Inverse Integrating Factor |
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Authors: | Isaac A García Héctor Giacomini Maite Grau |
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Institution: | 1.Departament de Matemàtica,Universitat de Lleida,Lleida,Spain;2.Laboratoire de Mathématiques et Physique Théorique, C.N.R.S. UMR 6083, Faculté des Sciences et Techniques,Université de Tours,Tours,France |
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Abstract: | In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point p
0 of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p
0 being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and
nilpotent. In a neighborhood of p
0 the differential system can always be brought, by means of a change to (generalized) polar coordinates (r, θ), to an equation over a cylinder in which the singular point p
0 corresponds to a limit cycle γ
0. This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ
0. We define the notion of vanishing multiplicity of the inverse integrating factor over γ
0. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p
0 in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse
integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the
previous paragraph and for any isolated singular point with at least one non-zero eigenvalue. |
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Keywords: | |
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