Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics

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.

[1]  M. E. Morsalani.  Perturbations of graphics with semi-hyperbolic singularities , 1996 .

[2]  H. Zoladek,et al.  The Cyclicity of Triangles and Segments in Quadratic Systems , 1995 .

[3]  M El Morsalani,et al.  Degenerate and non-trivial hyperbolic 2-polycycles: appearance of two independent Ecalle-Roussarie compensators and Khovanskii's theory , 1994 .

[4]  A. Mourtada,et al.  Degenerate and Non-trivial Hyperbolic Polycycles with Two Vertices , 1994 .

[5]  P. Joyal.  Un théorème de préparation pour fonctions à développement Tchébychévien , 1994, Ergodic Theory and Dynamical Systems.

[6]  Freddy Dumortier,et al.  Hilbert′s 16th Problem for Quadratic Vector Fields , 1994 .

[7]  Freddy Dumortier,et al.  Elementary graphics of cyclicity 1 and 2 , 1994 .

[8]  Mohamed El Morsalani.  Bifurcations de polycycles infinis pour les champs de vecteurs polynomiaux du plan , 1994 .

[9]  Y. Ilyashenko,et al.  Finitely-smooth normal forms of local families of diffeomorphisms and vector fields , 1991 .

[10]  A. Mourtada,et al.  Cyclicite finie des polycycles hyperboliques de champs de vecteurs du plan mise sous forme normale , 1990 .

[11]  Christiane Rousseau,et al.  Saddle quantities and applications , 1989 .

[12]  R. Roussarie.  Finite cyclicity of loops and cusps , 1989 .

[13]  R. Roussarie,et al.  A note on finite cyclicity property and Hilbert's 16th. Problem , 1988 .

[14]  Charles Pugh,et al.  Keeping track of limit cycles , 1986 .

[15]  Robert Roussarie,et al.  On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields , 1986 .

[16]  I. Taimanov,et al.  Closed geodesics on non-simply-connected manifolds , 1985 .

[17]  Askold Khovanskii,et al.  Real analytic varieties with the finiteness property and complex abelian integrals , 1984 .

[18]  W. A. Coppel,et al.  A survey of quadratic systems , 1966 .

[19]  Shlomo Sternberg,et al.  On the Structure of Local Homeomorphisms of Euclidean n-Space, II , 1958 .