|
Trudy Moskovskogo Matematicheskogo Obshchestva, 2015, Volume 76, Issue 2, Pages 205–248
(Mi mmo576)
|
|
|
|
This article is cited in 2 scientific papers (total in 2 papers)
Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems
R. Roussariea, C. Rousseaub a Université de Bourgogne
b Université de Montréal
Abstract:
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.
Key words and phrases:
Hilbert’s 16th problem, finite cyclicity, graphic through a nilpotent point, center graphic, quadratic vector fields.
Received: 15.12.2014
Citation:
R. Roussarie, C. Rousseau, “Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems”, Tr. Mosk. Mat. Obs., 76, no. 2, MCCME, M., 2015, 205–248; Trans. Moscow Math. Soc., 76:2 (2015), 181–218
Linking options:
https://www.mathnet.ru/eng/mmo576 https://www.mathnet.ru/eng/mmo/v76/i2/p205
|
Statistics & downloads: |
Abstract page: | 143 | Full-text PDF : | 39 | References: | 46 |
|