American Institute of Mathematical Sciences

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July  2009, 8(4): 1133-1157. doi: 10.3934/cpaa.2009.8.1133

Study of the cyclicity of some degenerate graphics inside quadratic systems

Freddy Dumortier 1, and  Christiane Rousseau 2,

1. 

Universiteit Hasselt, Campus Diepenbeek, Agoralaan–gebouw D, 3590 Diepenbeek
2. 

DMS and CRM, Université de Montréal, Canada

Received  January 2008 Revised  November 2008 Published  March 2009

In this paper we make essential steps in proving the finite cyclicity of degenerate graphics in quadratic systems, having a line of singular points in the finite plane. In particular we consider the graphics $(DF_{1 a})$, $(DF_{2 a})$ of the program of [8] to prove the finiteness part of Hilbert's 16th problem for quadratic vector fields. We make a complete treatment except for one very specific problem that we clearly identify.
Keywords: singular perturbations, Finite cyclicity, Hilbert’s 16-th problem for quadratic systems., blow-up of the family, degenerate graphics.
Mathematics Subject Classification: 34C07, 34C26, 37G9.
Citation: Freddy Dumortier, Christiane Rousseau. Study of the cyclicity of some degenerate graphics inside quadratic systems. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1133-1157. doi: 10.3934/cpaa.2009.8.1133
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