Hostname: page-component-5c6d5d7d68-xq9c7 Total loading time: 0 Render date: 2024-08-20T06:51:06.335Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert’s 16th problem on a period annulus and Nash space of arcs

Part of: Qualitative theory Birational geometry

Published online by Cambridge University Press:  12 July 2019

JEAN–PIERRE FRANÇOISE ,
LUBOMIR GAVRILOV  and
DONGMEI XIAO
JEAN–PIERRE FRANÇOISE
Affiliation:
Université P.-M. Curie, Paris 6, Laboratoire Jacques–Louis Lions UMR 7598 CNRS, 4 Place Jussieu, 75252, Paris, France, and School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, PR China. e-mail: Jean-Pierre.Francoise@upmc.fr
LUBOMIR GAVRILOV
Affiliation:
Institut de Mathématiques de Toulouse; UMR 5219 Université de Toulouse; CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France. e-mail: lubomir.gavrilov@math.univ-toulouse.fr
DONGMEI XIAO
Affiliation:
School of Mathematical Sciences Shanghai Jiao Tong University, Shanghai, 200240, PR China. e-mail: xiaodm@sjtu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert’s 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor E of the canonical blow-up BI ℂn of the Bautin ideal I. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs Arc(BI ℂn, E). The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

MSC classification

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
Secondary: 14E18: Arcs and motivic integration
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 2 , September 2020 , pp. 377 - 409
Copyright
© Cambridge Philosophical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Andronova, E. A.. On the topology of quadratic systems with four (or more) limit cycles. Uspekhi Mat. Nauk 41 (1986), no. 2(248), 183–184.Google Scholar
Andronova, E. A.. Some topological structures of quadratic systems with at least four limit cycles. Methods of qualitative theory of differential equations and related topics, Supplement, 197–204, Amer. Math. Soc. Transl. Ser. 2, 200 (Amer. Math. Soc., Providence, RI, 2000).Google Scholar
Bautin, N. N.. Du nombre de cycles limites en cas de variation de coefficients d’un état d’équation du type foyer ou centre. Dokl. Acad. Nauk USSR 24 (1939), 669–672.Google Scholar
Bautin, N. N.. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, American Math. Soc. Translation (1954) 100, 19 pp(Russian original: Mat. Sb. 30 (1952), 181–196).Google Scholar
Binyamini, G., Novikov, D. and Yakovenko, S.. On the number of zeros of Abelian integrals. A constructive solution of the infinitesimal Hilbert sixteenth problem. Invent. Math. 181 (2010), no. 2, 227–289.CrossRefGoogle Scholar
Briskin, M., Françoise, J.–P. and Yomdin, Y.. The Bautin ideal of the Abel equation. Nonlinearity 11(1998), 431–443.CrossRefGoogle Scholar
Buica, A., Giné, J. and Grau, M.. Essential perturbations of polynomial vector fields with a period annulus. Commun. Pure Appl. Anal., vol. 14 (2015), 1073–1095.CrossRefGoogle Scholar
Cerveau, D. and Lins Neto, A.. Irreducible components of the space of holomorphic foliations of degree two in ℂP(n), n ≥ 3. Ann. of Math. (2), 143(3) (1996) 577–612.CrossRefGoogle Scholar
Chicone, C. and Jacobs, M.. Bifurcation of limit cycles from quadratic isochrones. J. Differential Equations 91 (1991), 268–326.CrossRefGoogle Scholar
Chow, S. N., Li, C. and Yi, Y.. The cyclicity of period annuli of degenerate quadratic Hamiltonian systems with elliptic segment loops. Ergodic Theory Dynam. Systems 22 (2002), no. 2, 349–374.CrossRefGoogle Scholar
Dulac, H.. Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre. Bull. Sci. Math. Sér. 2, 32 (1908), 230–252.Google Scholar
Françoise, J.–P.. Successive derivatives of a first return map, application to the study of quadratic vector fields Ergodic Theory Dynam. Systems 16 (1996), 87–96.CrossRefGoogle Scholar
Françoise, J.–P. and Xiao, D.. Perturbation of a symmetric center within Liénard equations. J. Differential Equations 259 (2015), 2408–2429.CrossRefGoogle Scholar
Françoise, J.–P. and Pugh, C. C.. Keeping track of limit cycles. J. Differential Equations, 65 (1986), 139–157.CrossRefGoogle Scholar
Françoise, J.–P. and Yomdin, Y.. Bernstein inequalities and applications to analytic geometry and differential equations. J. Funct. Anal 146 (1997), no. 1, 185–205.CrossRefGoogle Scholar
Gavrilov, L.. Cyclicity of period annuli and principalization of Bautin ideals. Ergodic Theory Dynam. Systems 28 (2008), 1497–1507.CrossRefGoogle Scholar
Gavrilov, L.. Higher order Poincaré–Pontryagin functions and iterated path integral. Annales Fac des Sciences de Toulouse, Maths. 14 (2005), 663–682.CrossRefGoogle Scholar
Gavrilov, L.. The infinitesimal 16th Hilbert problem in the quadratic case. Invent. Math. 143 (2001), no. 3, 449–497.CrossRefGoogle Scholar
Gavrilov, L. and Iliev, I. D.. The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields. Amer. J. Math. 12 (2005), 1153–1190.CrossRefGoogle Scholar
Gavrilov, L. and Iliev, I. D.Quadratic perturbations of quadratic codimension-four centers. J. Math. Anal. Appl. 357 (2009), no. 1, 69–76.CrossRefGoogle Scholar
Gavrilov, L. and Novikov, D.. On the finite cyclicity of open period annuli. Duke Math. J. 152(1) (2010), 1–26.CrossRefGoogle Scholar
Gentes, M.. Bifurcations d’ordre supérieur, cycles limites et intégrabilité. Thèse de doctorat de l’Université P.–M. Curie 2009.Google Scholar
Gentes, M.. Center conditions and limit cycles for the perturbation of an elliptic sector. Bull. Sci. Math. 133 (2009), 597–643.CrossRefGoogle Scholar
Hartshorne, R.. Algebraic Geometry. Graduate Texts in Mathematics, no. 52 (Springer–Verlag, New York-Heidelberg, 1977).CrossRefGoogle Scholar
Hickel, M.. Sur quelques aspects de la géométrie de l’espace des arcs tracés sur un espace analytique. Annales Fac des Sciences de Toulouse, Maths. 14 (2005), 1–50.CrossRefGoogle Scholar
Horozov, E. and Iliev, I. D.. On the number of limit cycles in perturbations of quadratic Hamiltonian systems. Proc. London Math. Soc. (3) 69 (1994), no. 1, 198–224.CrossRefGoogle Scholar
Iliev, I. D.. Perturbations of quadratic centers. Bull. Sci. Math. 122 (1998) 2, 107–161.CrossRefGoogle Scholar
Iliev, I. D.. The number of limit cycles due to polynomial perturbations of the harmonic oscillator. Math. Proc. Camb. Phils. Soc. 127 (1999), no. 2, 317–322.CrossRefGoogle Scholar
Iliev, I. D.. The cyclicity of the period annulus of the quadratic Hamiltonian triangle. J. Differential Equations 128 (1996), 309–326.CrossRefGoogle Scholar
Iliev, I. D.. Higher-order Melnikov functions for degenerate cubic hamiltonians. Adv. Differential Equations 1 (1996), 689–708.Google Scholar
Iliev, I. D., Chengzhi, Li and Jiang, Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Commun. Pure Appl. Anal. 9 (2010), no. 3, 583–610.CrossRefGoogle Scholar
Iliev, I. D., Li, C. and Yu, J.. Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops. Nonlinearity 18 (2005), no. 1, 305–330.CrossRefGoogle Scholar
Ilyashenko, Y. and Yakovenko, S.. Lectures on analytic differential equations. Graduate Studies in Mathematics, vol 86 (American Mathematical Society, Providence, RI, 2008).Google Scholar
Ishii, S. and Kollar, J.. The Nash problem on arc families of singularities. Duke Math. J. 120 (2003) 601–620.Google Scholar
Jebrane, A., Mardesic, P. and Pelletier, M.. A generalization of Francoise’s algorithm for calculating higher order Melnikov functions. Bull. Sci. Math. 126 (2002), 705–732.CrossRefGoogle Scholar
Jebrane, A., Mardesic, P. and Pelletier, M.. A note on a generalization of Francoise’s algorithm for calculating higher order Melnikov functions. Bull. Sci. Math. 128 (2004), 749–760.CrossRefGoogle Scholar
Jebrane, A. and Zoladek, H.. A note on higher-order Melnikov functions. Qual. Theory Dyn. Syst. 6, (2007) 273–287.CrossRefGoogle Scholar
Johnson, J. M. and Kollár., J.Arcology. Amer. Math. Monthly 123(6) (2016), 519–541.CrossRefGoogle Scholar
Kollár, J.. Lectures on Resolution of Singularities, vol. 166 (Princeton University Press, Princeton, NJ, 2007).Google Scholar
Lejeune–Jalabert, M.. Courbes tracées sur un germe d’hypersurface. Amer. J. Math., 112 (1990), 525–568.CrossRefGoogle Scholar
Lejeune–Jalabert, M.. On arcs and jets. Proceedings of Congreso del centenario de la RSME en Avilà (2011).Google Scholar
Lejeune–Jalabert, M. and Teissier, B.. Clôture intégrale des idéaux et équisingularité. Ann. Fac. Sci. Toulouse, Math. (6), 17(4) (2008), 781–859.CrossRefGoogle Scholar
Lejeune–Jalabert, M. and Reguera–Lopez, A. J.. Arcs and wedges on sandwiched surface singularities. Amer. J. Math. 121 (1999), no. 6, 1191–1213.CrossRefGoogle Scholar
Li, Chengzhi and Llibre, J.. Quadratic perturbations of a quadratic reversible Lotka–Volterra system. Qual. Theory Dyn. Syst. 9 (2010), no. 1-2, 235–249CrossRefGoogle Scholar
Li, Chengzhi and Llibre, J.. The cyclicity of period annulus of a quadratic reversible Lotka–Volterra system. Nonlinearity 22 (2009), no. 12, 2971–2979.CrossRefGoogle Scholar
Li, Chengzhi and Llibre, J.. A unified study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems. J. Dynam. Differential Equations 16 (2004), no. 2, 271–295.CrossRefGoogle Scholar
Lins Neto, A.. Foliations with a Morse center. J. Singul. 9 (2014), 82–100,Google Scholar
Loeser, F.. Seattle lectures on motivic integration. Proc. Sympo. Pure Math., vol. 80.2 (2009), 745–784.CrossRefGoogle Scholar
Lojasiewicz, S., Tougeron, J. and Zurro, M.–A.. Eclatements des coefficients des séries entières et deux théorèmes de Gabrielov. Manuscripta Math. 92 (1997), 325–337.CrossRefGoogle Scholar
Mardesic, P., Saavedra, M.Uribe, M. and Wallace, M.. Unfolding of the Hamiltonian triangle vector field, J. Dyn. Control Syst. 17 (2011), no. 2, 291–310.CrossRefGoogle Scholar
Movasati, H.. Center conditions: rigidity of logarithmic differential equations. J. Differential Equations 197 (2004), no. 1, 197–217.CrossRefGoogle Scholar
John, F.Nash, Jr.Arc structure of singularities. Preprint Fine Library, Princeton, (1968) A celebration of John F. Nash Jr. Duke Math. J. 81(1995), 31–38.Google Scholar
Nobile, A.. On Nash theory of arc structure of singularities. Ann. Mat. pura appl., (IV), vol. CLX (1991), 129–146.CrossRefGoogle Scholar
Perko, L.. Differential equations and dynamical systems. Third edition. Texts in Applied Mathematics, 7 (Springer-Verlag, New York, 2001), xiv+553 pp.CrossRefGoogle Scholar
Roussarie, R.. Bifurcation of planar vector fields and Hilbert’s sixteenth problem. Progr. Math. vol 164 (Birkhäuser Verlag, Basel, 1998).Google Scholar
Roussarie, R.. Melnikov Functions and Bautin Ideal. Qual. Theory Dynam. Syst. 2. (2001), 67–78.CrossRefGoogle Scholar
Schlomiuk, D.. Algebraic particular integrals, integrability and the problem of the center. Trans. Amer. Math. Soc., vol. 338 (1993), 799–841.CrossRefGoogle Scholar
Zhao, Y.. On the number of limit cycles in quadratic perturbations of quadratic codimension-four centres. Nonlinearity 24 (2011), no. 9, 2505–2522.CrossRefGoogle Scholar
Zhao, Y., Liang, Z. and Lu, G.. The cyclicity of the period annulus of the quadratic Hamiltonian systems with non-Morsean point. J. Differential Equations 162 (2000), no. 1, 199–223.CrossRefGoogle Scholar
Zhao, Y. and Zhu, S.. Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment. Bull. Sci. Math. 125 (2001), no. 2, 109–138.CrossRefGoogle Scholar
Zoladek, H.. Quadratic systems with center and their perturbations. J. Differential Equations 109, (1994), 223–273.CrossRefGoogle Scholar
Zoladek, H.. Melnikov functions in quadratic perturbations of generalized Lotka–Volterra systems. J. Dyn. Control Syst. 21 (2015), no. 4, 573–603.CrossRefGoogle Scholar