Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics
- 1 September 1996
- journal article
- Published by IOP Publishing in Nonlinearity
- Vol. 9 (5) , 1209-1261
- https://doi.org/10.1088/0951-7715/9/5/008
Abstract
In this paper we study the finite cyclicity of several elementary graphics appearing in quadratic systems. This makes substantial progress in the study of the finite cyclicity of the elementary graphics with non-identical return map listed in Dumortier et al J. Diff. Eqns 110 86 - 133 . The main tool we use is the method of Khovanskii. We also use the fact that some graphics have unbroken connections and we calculate normal forms for elementary singular points in the graphics. Several arguments use the fact that two singular points `compensate' each other precisely when the graphic surrounds a centre. One originality of the paper is to prove that for certain graphics among quadratic systems some regular transition maps are not tangent to the identity.Keywords
This publication has 17 references indexed in Scilit:
- Un théorème de préparation pour fonctions à développement TchébychévienErgodic Theory and Dynamical Systems, 1994
- Hilbert′s 16th Problem for Quadratic Vector FieldsJournal of Differential Equations, 1994
- Elementary graphics of cyclicity 1 and 2Nonlinearity, 1994
- Bifurcations de polycycles infinis pour les champs de vecteurs polynomiaux du planAnnales de la Faculté des sciences de Toulouse : Mathématiques, 1994
- Finitely-smooth normal forms of local families of diffeomorphisms and vector fieldsRussian Mathematical Surveys, 1991
- Cyclicite finie des polycycles hyperboliques de champs de vecteurs du plan mise sous forme normaleLecture Notes in Mathematics, 1990
- Saddle quantities and applicationsJournal of Differential Equations, 1989
- Keeping track of limit cyclesJournal of Differential Equations, 1986
- Real analytic varieties with the finiteness property and complex abelian integralsFunctional Analysis and Its Applications, 1984
- A survey of quadratic systemsJournal of Differential Equations, 1966