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Normal forms near a saddle-node and applications to finite cyclicity of graphics

Published online by Cambridge University Press:  19 June 2002

F. DUMORTIER ,
Y. ILYASHENKO  and
C. ROUSSEAU
F. DUMORTIER
Affiliation:
Limburgs Universitair Centrum, Universitaire Campus, B-3590, Diepenbeek, Belgium and Department of Mathematics, Cornell University, Ithaca, NY 14853, USA (e-mail: [email protected])
Y. ILYASHENKO
Affiliation:
State and Independent Moscow Universities, Steklov Mathematical Institute, Moscow, Russia (e-mail: [email protected])
C. ROUSSEAU
Affiliation:
Département de Mathématiques et de statistique and CRM, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, QC, H3C 3J7, Canada (e-mail: [email protected])
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Abstract

We refine the transformation to smooth normal form for an analytic family of vector fields in the neighbourhood of a saddle-node. This refinement is very powerful and allows us to prove the finite cyclicity of families of graphics (‘ensembles’) occurring inside analytic families of vector fields. In [ZR1] and [ZR2] it is used to prove the finite cyclicity of graphics through a nilpotent singular point of elliptic type. Several examples are presented: lips, graphics with two subsequent lips, graphics with a nilpotent point of elliptic type and a saddle-node. We also discuss the bifurcation diagram of limit cycles for a graphic in the lips.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 3 , June 2002 , pp. 783 - 818
Copyright
2002 Cambridge University Press

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