Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

   
 
 

 

Finite cyclicity of some center graphics through a nilpotent point inside quadratic systems


Authors: Robert Roussarie and Christiane Rousseau
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 76 (2015), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2015, 181-218
MSC (2010): Primary 34C07, 37G15
DOI: https://doi.org/10.1090/mosc/248
Published electronically: November 17, 2015
MathSciNet review: 3467264
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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_{14}^1)$, 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_{6b}^1)$, $ (H_{13}^3)$, and $ (DI_{2b})$.


References [Enhancements On Off] (What's this?)

  • 1. Dumortier F., Roussarie R., Canard cycles and centre manifolds, Memoirs of AMS. 1996. Vol.121, no. 577. P.1-100. MR 1327208 (96k:34113)
  • 2. Dumortier F., Roussarie R., Rousseau C., Hilbert's 16th problem for quadratic vector fields, J. Diff. Equat. 1994. Vol. 110, no. 1. P.86-133. MR 1275749 (95g:58179)
  • 3. Dumortier F., El Morsalani M., Rousseau C., Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics, Nonlinearity. Vol. 9, no. 5 (1996), 1209-1261. MR 1416474 (97g:58136)
  • 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. Berlin: Springer-Verlag, 1991. (Lecture Notes in Math.; Vol.1480). MR 1166189 (93f:58165)
  • 5. Ilyashenko Y., Yakovenko S., Finitely-smooth normal forms of local families of diffeomorphisms and vector fields, Russian Math. Surv. Vol. 46 (1991), 1-43. MR 1109035 (92i:58165)
  • 6. Malgrange B., Ideals of differentiable functions. London: Oxford University Press, 1967. (Tata Institute of Fundamental Research Studies in Math.; no. 3). MR 0212575 (35:3446)
  • 7. Roussarie R., Desingularization of unfoldings of cuspidal loops, Geometry and analysis in nonlinear dynamics. Harlow: Longman Sci. and Tech., 1992. (Pitman Res. Notes Math. Series; Vol. 222). P.41-55. MR 1173667 (93g:58109)
  • 8. Roussarie R., Rousseau C., Finite cyclicity of nilpotent graphics of pp-type surrounding a center, Bull. Belg. Math. Soc. Simon Stevin. 2008. Vol. 15, no. 5. P.889-920. MR 2484139 (2009k:34070)
  • 9. Zhu H., Rousseau C., Finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type, J. Diff. Equat. Vol. 178 (2002), 325-436. MR 1879832 (2002m:34048)

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 34C07, 37G15

Retrieve articles in all journals with MSC (2010): 34C07, 37G15


Additional Information

Robert Roussarie
Affiliation: Institut de Mathématiques de Bourgogne, Université de Bourgogne, BP 47870, 21078 Dijon, France
Email: Robert.Roussarie@u-bourgogne.fr

Christiane Rousseau
Affiliation: Département de Mathématiques et Statistique, Université de Montréal, CP 6128 Succ Centre-Ville, Montreal QC H3C 3J7, Canada
Email: rousseac@dms.umontreal.cs

DOI: https://doi.org/10.1090/mosc/248
Keywords: Hilbert's 16th problem, finite cyclicity, graphic through a nilpotent point, center graphic, quadratic vector fields
Published electronically: November 17, 2015
Additional Notes: This research was supported by NSERC in Canada.
Article copyright: © Copyright 2015 R. Roussarie, C. Rousseau

American Mathematical Society