Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Examples of center cyclicity bounds using the reduced Bautin depth


Author: Isaac A. García
Journal: Proc. Amer. Math. Soc. 145 (2017), 4363-4370
MSC (2010): Primary 37G15, 37G10, 34C07
DOI: https://doi.org/10.1090/proc/13570
Published electronically: March 23, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: There is a method for bounding the cyclicity of non-degenerate monodromic singularities of polynomial planar families of vector fields $ \mathcal {X}_\lambda $ which can work even in the case that the Poincaré first return map has associated a non-radical Bautin ideal $ \mathcal {B}$. The method is based on the stabilization of the integral closures of an ascending chain of polynomial ideals in the ring of polynomials in the parameters $ \lambda $ of the family that stabilizes at $ \mathcal {B}$. In this work we use computational algebra methods to provide an explicit example in which the classical procedure to find the Bautin depth of $ \mathcal {B}$ fails but the new approach is successful.


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

  • [1] N. N. Bautin, On the number of limit cycles appearing with variation of the coefficients from an equilibrium state of the type of a focus or a center, Mat. Sbornik N.S. 30(72) (1952), 181-196 (Russian). MR 0045893
  • [2] G. R. Belitskiĭ, Smooth equivalence of germs of vector fields with one zero or a pair of purely imaginary eigenvalues, Funktsional. Anal. i Prilozhen. 20 (1986), no. 4, 1-8, 96 (Russian). MR 878039
  • [3] Colin Christopher, Estimating limit cycle bifurcations from centers, Differential equations with symbolic computation, Trends Math., Birkhäuser, Basel, 2005, pp. 23-35. MR 2187371, https://doi.org/10.1007/3-7643-7429-2_2
  • [4] David Cox, John Little, and Donal O'Shea, Ideals, varieties, and algorithms, An introduction to computational algebraic geometry and commutative algebra, 3rd ed., Undergraduate Texts in Mathematics, Springer, New York, 2007. MR 2290010
  • [5] Isaac A. García, The cyclicity of polynomial centers via the reduced Bautin depth, Proc. Amer. Math. Soc. 144 (2016), no. 6, 2473-2478. MR 3477063, https://doi.org/10.1090/proc/12896
  • [6] Isaac A. García, Jaume Llibre, and Susanna Maza, The Hopf cyclicity of the centers of a class of quintic polynomial vector fields, J. Differential Equations 258 (2015), no. 6, 1990-2009. MR 3302528, https://doi.org/10.1016/j.jde.2014.11.018
  • [7] Isaac A. García and Douglas S. Shafer, Cyclicity of a class of polynomial nilpotent center singularities, Discrete Contin. Dyn. Syst. 36 (2016), no. 5, 2497-2520. MR 3485407, https://doi.org/10.3934/dcds.2016.36.2497
  • [8] H. Hauser, J.-J. Risler, and B. Teissier, The reduced Bautin index of planar vector fields, Duke Math. J. 100 (1999), no. 3, 425-445. MR 1719738, https://doi.org/10.1215/S0012-7094-99-10015-9
  • [9] Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. MR 2266432
  • [10] Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, vol. 86, American Mathematical Society, Providence, RI, 2008. MR 2363178
  • [11] D. Katz, Generating ideals up to projective equivalence, Proc. Amer. Math. Soc. 120 (1994), no. 1, 79-83. MR 1176070, https://doi.org/10.2307/2160169
  • [12] Valery G. Romanovski and Douglas S. Shafer, The center and cyclicity problems: a computational algebra approach, Birkhäuser Boston, Inc., Boston, MA, 2009. MR 2500203
  • [13] Robert Roussarie, Bifurcation of planar vector fields and Hilbert's sixteenth problem, Progress in Mathematics, vol. 164, Birkhäuser Verlag, Basel, 1998. MR 1628014
  • [14] Wolmer Vasconcelos, Integral closure, Rees algebras, multiplicities, algorithms, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. Rees algebras, multiplicities, algorithms. MR 2153889

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37G15, 37G10, 34C07

Retrieve articles in all journals with MSC (2010): 37G15, 37G10, 34C07


Additional Information

Isaac A. García
Affiliation: Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida, Spain
Email: garcia@matematica.udl.cat

DOI: https://doi.org/10.1090/proc/13570
Keywords: Center, polynomial vector fields, Bautin ideal, cyclicity, limit cycle
Received by editor(s): March 3, 2016
Received by editor(s) in revised form: October 27, 2016
Published electronically: March 23, 2017
Additional Notes: The author was partially supported by MINECO grant number MTM2014-53703-P and by CIRIT grant number 2014 SGR 1204.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society