Geodesic
Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 76 (2015) no. 2, pp. 205-248 
R. Roussarie; C. Rousseau. Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 76 (2015) no. 2, pp. 205-248. http://geodesic.mathdoc.fr/item/MMO_2015_76_2_a3/
@article{MMO_2015_76_2_a3,
     author = {R. Roussarie and C. Rousseau},
     title = {Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {205--248},
     year = {2015},
     volume = {76},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/MMO_2015_76_2_a3/}
}
TY  - JOUR
AU  - R. Roussarie
AU  - C. Rousseau
TI  - Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems
JO  - Trudy Moskovskogo matematičeskogo obŝestva
PY  - 2015
SP  - 205
EP  - 248
VL  - 76
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/MMO_2015_76_2_a3/
LA  - en
ID  - MMO_2015_76_2_a3
ER  - 
%0 Journal Article
%A R. Roussarie
%A C. Rousseau
%T Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems
%J Trudy Moskovskogo matematičeskogo obŝestva
%D 2015
%P 205-248
%V 76
%N 2
%U http://geodesic.mathdoc.fr/item/MMO_2015_76_2_a3/
%G en
%F MMO_2015_76_2_a3

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In this paper we introduce new methods to prove the finite cyclicity of some graphics through a triple nilpotent point of saddle or elliptic type surrounding a center. After applying a blow-up of the family, yielding a singular $3$-dimensional foliation, this amounts to proving the finite cyclicity of a family of limit periodic sets of the foliation. The boundary limit periodic sets of these families were the most challenging, but the new methods are quite general for treating such graphics. We apply these techniques to prove the finite cyclicity of the graphic $(I^1_{14})$, which is part of the program started in $1994$ by Dumortier, Roussarie and Rousseau (and called DRR program) to show that there exists a uniform upper bound for the number of limit cycles of a planar quadratic vector field. We also prove the finite cyclicity of the boundary limit periodic sets in all graphics but one through a triple nilpotent point at infinity of saddle, elliptic or degenerate type (with a line of zeros) and surrounding a center, namely the graphics $(I^1_{6b})$, $(H^3_{13})$, and $(DI_{2b})$. References: 9 entries.

[1] Dumortier F., Roussarie R., “Canard cycles and centre manifolds”, Memoirs of AMS, 121, no. 577, 1996, 1–100 | DOI | MR

[2] Dumortier F., Roussarie R., Rousseau C., “Hilbert's 16th problem for quadratic vector fields”, J. Diff. Equat., 110:1 (1994), 86–133 | DOI | MR | Zbl

[3] Dumortier F., El Morsalani M., Rousseau C., “Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics”, Nonlinearity, 9:5 (1996), 1209–1261 | DOI | MR | Zbl

[4] Dumortier F., Roussarie R., Sotomayor S., “Generic 3-parameter families of planar vector fields, unfolding of saddle, focus and elliptic singularities with nilpotent linear parts”, Bifurcations of planar vector fields, Lecture Notes in Math., 1480, Springer-Verlag, Berlin, 1991 | MR | Zbl

[5] Ilyashenko Y., Yakovenko S., “Finitely-smooth normal forms of local families of diffeomorphisms and vector fields”, Russian Math. Surv., 46 (1991), 1–43 | DOI | MR | Zbl

[6] Malgrange B., Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Math., 3, Oxford University Press, London, 1967 | MR

[7] Roussarie R., “Desingularization of unfoldings of cuspidal loops”, Geometry and analysis in nonlinear dynamics, Pitman Res. Notes Math. Series, 222, Longman Sci. and Tech., Harlow, 1992, 41–55 | MR

[8] Roussarie R., Rousseau C., “Finite cyclicity of nilpotent graphics of pp-type surrounding a center”, Bull. Belg. Math. Soc. Simon Stevin, 15:5 (2008), 889–920 | MR | Zbl

[9] Zhu H., Rousseau C., “Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type”, J. Diff. Equat., 178 (2002), 325–436 | DOI | MR | Zbl