Abstract
This survey paper is devoted to introducing some basic concepts and methods about the application of Abelian integral to study the number of limit cycles, especially to the weak Hilbert’s 16th problem. We will introduce some recent results in this field.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Arnold V.I.: Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields. Funct. Anal. Appl 11, 85–92 (1977)
Arnold V.I.: Ten problems. Adv. Soviet Math. 1, 1–8 (1990)
Binyamini G., Novikov D., Yakovenko S.: On the number of zeros of Abelian integrals : a constructive solution of the infinitesimal Hilbert sixteenth problem. Invent. Math. 181, 227–289 (2010)
Caubergh M., Dumortier F.: Hopf-Takens bifurcations and centres. J. Diff. Eqs. 202, 1–31 (2004)
Caubergh M., Dumortier F., Roussarie R.: Alien limit cycles near a Hamiltonian 2-saddle cycle. C. R. Math. Acad. Sci. Paris 340, 587–592 (2005)
Chen F., Li C., Llibre J., Zhang Z.: A uniform proof on the weak Hilbert’s 16th problem for n = 2. J. Diff. Eqs. 221, 309–342 (2006)
Chen, L., Ma, X., Zhang, G., Li, C.: Cyclicity of several quadratic reversible systems with center of genus one. J. Appl. Anal. Comp. (to appear)
Chen G., Li C., Liu C., Llibre J.: The cyclicity of period annuli of some classes of reversible quadratic systems. Disc. Contin. Dyn. Sys 16, 157–177 (2006)
Coll B., Li C., Prohens R.: Quadratic perturbations of a class of quadratic reversible systems with two centers. Disc. Contin. Dyn. Sys 24, 699–729 (2009)
Chow S.-N., Li C., Wang D.: Normal forms and bifurcations of planar vector fields. Cambridge University Press, Cambridge (1994)
Chow S.-N., Li C., Yi Y.: The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment loop. Erg. Th. Dyn. Syst 22, 1233–1261 (2002)
Christopher C., Li C.: Limit cycles of differential equations. Birkhäuser Verlag, Berlin (2007)
Christopher C.J., Lloyd N.G.: Polynomial systems: a lower bound for the Hilbert numbers. Proc. Royal Soc. Lond. Ser. A 450, 219–224 (1995)
Dulac H.: Sur les cycles limites. Bull. Soc. Math. France 51, 45–188 (1923)
Dumortier F., Li C.: Perturbations from an elliptic Hamiltonian of degree four: (I) saddle loop and two saddle cycle. J. Diff. Eqs. 176, 114–157 (2001)
Dumortier F., Li C.: Perturbations from an elliptic Hamiltonian of degree four: (II) cuspidal loop. J. Diff. Eqs. 175, 209–243 (2001)
Dumortier F., Li C.: Perturbations from an elliptic Hamiltonian of degree four: (III) Global center. J. Diff. Eqs. 188, 473–511 (2003)
Dumortier F., Li C.: Perturbations from an elliptic Hamiltonian of degree four: (IV) figure eight–loop. J. Diff. Eqs. 188, 512–554 (2003)
Dumortier F., Li C., Zhang Z.: Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J. Diff. Eqs. 139, 146–193 (1997)
Dumortier F., Roussarie R.: Abelian integrals and limit cycles. J. Diff. Eqs. 227, 116–165 (2006)
Écalle, J.: Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Actualitiées Math. Hermann, Paris (1992)
Françoise J.-P.: Successive derivatives of a first return map, application to the study of quadratic vector fields. Erg. Th. Dyn. Syst 16, 87–96 (1996)
Gasull, A., Li, C., Torregrosa, J.: A new Chebyshev family with applications to Abel equations. J. Diff. Eqs. (to appear)
Gasull A., Li W., Llibre J., Zhang Z.: Chebyshev property of complete elliptic integrals and its application to Abelian integrals. Pacif. J. Math. 202, 341–361 (2002)
Gautier S., Gavrilov L., Iliev I.D.: Perturbations of quadratic centers of genus one. Disc. Contin. Dyn. Sys 25, 511–535 (2009)
Gavrilov L.: Petrov modules and zeros of Abelian integrals. Bull. Sci. Math 122, 571–584 (1998)
Gavrilov L.: The infinitesimal 16th Hilbert problem in the quadratic case. Invent. Math. 143, 449–497 (2001)
Gavrilov L., Iliev I.D.: Second order analysis in polynomially perturbed reversible quadratic Hamiltonian systems. Erg. Th. Dyn. Syst 20, 1671–1686 (2000)
Gavrilov L., Iliev I.D.: Quadratic perturbations of quadratic codimension-four centers. J. Math. Anal. Appl. 357, 69–76 (2009)
Gavrilov L., Iliev I.D.: The displacement map associated to polynomial unfoldings of planar Hamiltonian vector field. Am. J. Math. 127, 1153–1190 (2005)
Grau M., Mañosas F., Villadelprat J.: A Chebyshev criterion for Abelian integrals. Trans. Am. Math. Soc 363, 109–129 (2011)
Han. M., Li, J. Lower bounds for the Hilbert number of polynomial systems (Submitted)
Horozov E., Iliev I.D.: On the number of limit cycles in perturbations of quadratic Hamiltonian systems. Proc. Lond. Math. Soc. 69, 198–224 (1994)
Iliev I.D.: High-order Melnikov functions for degenerate cubic Hamiltonians. Adv. Diff. Eqs. 1, 689–708 (1996)
Iliev I.D.: The cyclicity of the period annulus of the quadratic Hamiltonian triangle. J. Diff. Eqs. 128, 309–326 (1996)
Iliev I.D.: Inhomogeneous Fuchs equations and the limit cycles in a class of near-integrable quadratic systems. Proc. Roy. Soc. Edinb. A 127, 1207–1217 (1997)
Iliev I.D.: Perturbations of quadratic centers. Bull. Sci. Math 122, 107–161 (1998)
Iliev I.D.: The number of limit cycles due to polynomial perturbations of the harmonic oscillator. Math. Proc. Cambridge Philos. Soc. 127(2), 317–322 (1999)
Iliev I.D., Li C., Yu J.: Bifurcation of limit cycles from quadratic non-Hamiltonian systems with two centers and two unbounded heteroclinic loops. Nonlinearity 18, 305–330 (2005)
Iliev I.D., Li C., Yu J.: Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Com. Pure Appl. Anal. 9, 583–610 (2010)
Ilyashenko, Y.S.: Finiteness theorems for limit cycles, Uspekhi Mat. Nauk 45, no.2 (272), 143–200 (1990, Russian). English transl. Russian Math. Surveys 45, 129–203 (1990)
Ilyashenko, Y., Yakovenko, S.: Lectures on analytic differential equations, graduate studies in mathematics, 86, Am. Math. Soc., Providence (2008)
Khovansky A.G.: Real analytic manifolds with finiteness properties and complex Abelian integrals. Funct. Anal. Appl. 18, 119–128 (1984)
Li B., Zhang Z.: A note of a G.S. Petrov’s result about the weakened 16th Hilbert problem. J. Math. Anal. Appl. 190, 489–516 (1995)
Li C.: Two problems of planar quadratic systems. Scientia Sinica (Series A) 12, 1087–1096 (1982) (in Chinese)
Li C., Li W.: Weak Hilbert’s 16th problem and relative reserch. Adv. Math. (China) 39(5), 513–526 (2010)
Li C., Liu C., Yang J.: A cubic system with thirteen limit cycles. J. Diff. Eqs. 246, 3609–3619 (2009)
Li C., Llibre J.: A unified study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems. J. Dyn. Diff. Eqs. 16, 271–295 (2004)
Li C., Llibre J.: The cyclicity of period annulus of a quadratic reversible Lotka–Volterra system. Nonlinearity 22, 2971–2979 (2009)
Li C., Llibre J.: Quadratic perturbations of a quadratic reversible Lotka–Volterra system. Qual. Theory Dyn. Syst. 9, 235–249 (2010)
Li C., Zhang Z.-F.: A criterion for determining the monotonicity of ratio of two Abelian integrals. J. Diff. Eqs. 124, 407–424 (1996)
Li C., Zhang Z.-H.: Remarks on 16th weak Hilbert problem for n = 2. Nonlinearity 15, 1975–1992 (2002)
Li J.: Hilbert’s 16th problem and bifurcations of planar vector fields. Inter. J. Bifur. Chaos 13, 47–106 (2003)
Li J., Liu Y.: New results on the study of Z q −equivariant planar polynomial vector fields. Qual. Theory Dyn. Syst. 9, 167–219 (2010)
Li J.-M.: Limit cycles bifurcated from a reversible quadratic center. Qual. Theory Dyn. Syst. 6, 205–215 (2005)
Li, W.: Normal Form Theory and Its Applications, Science Press, Beijing (2000, in Chinese)
Liu, C.: A class of quadratic reversible centers can perturb four limit cycles under quadratic perturbations (Submitted)
Liu, C.: Limit cycles bifurcated from some reversible quadratic centers with non-algebraic first integral (2011, preprint)
Liang H., Zhao Y.: Quadratic perturbations of a class of quadratic reversible systems with one center. Disc. Contin. Dyn. Syst. 27, 325–335 (2010)
Llibre J.: Averaging theory and limit cycles for quadratic systems. Radovi Math. 11, 1–14 (2002)
Llibre J., Rodríguez G.: Configuration of limit cycles and planar polynomial vector fields. J. Diff. Eqs. 198, 374–380 (2004)
Mardesic P.: An explicit bound for the multiplicity of zeros of generic Abelian integrals. Nonlinearity 4, 845–852 (1991)
Markov Y.: Limit cycles of perturbations of a class of quadratic Hamiltonian vector fields. Serdica Math. J. 22, 91–108 (1996)
Mañosas F., Villadelprat J.: Bounding the number of zeros of certain Abelian integrals. J. Diff. Eqs. 251, 1656–1669 (2011)
Novikov D., Yakovenko S.: Tangential Hilbert broblem for perturbations of hyperelliptic Hamiltonian systems. Electron. Res. Announc. Am. Math. Soc. 5, 55–65 (1999)
Peng L.: Unfolding of a quadratic integrable system with a homoclinic loop. Acta. Math. Sinica (English Series) 18, 737–754 (2002)
Peng, L.: Quadratic perturbations of a quadratic reversible center of genus one (Submitted)
Peng, L., Lei, Y.: Bifurcation of limit cycles from a quadratic reversible center with the unbounded elliptic separatrix (Submitted)
Peng L., Sun Y.: The cyclicity of the period annulus of a quadratic reversible system with one center of genus one. Turk. J. Math. 35, 1–19 (2011)
Peng L., Lei Y.: The cyclicity of the period annulus of a quadratic reversible system with a hemicycle. Disc. Contin. Dyn. Syst. 30, 873–890 (2011)
Petrov G.S.: Number of zeros of complete elliptic integrals. Funct. Anal. Appl. 18(2), 73–74 (1984). English transl., Funct. Anal. Appl. 18(3), 148–149 (1984)
Pontryagin L.: On dynamical systems close to hamiltonian ones. Zh. Exp. Theor. Phys. 4, 234–238 (1934)
Roussarie R.: On the number of limit cycles which appear by perturbation of separate loop of planar vector fields. Bol. Soc. Bras. Mat. 17, 67–101 (1986)
Schlomiuk D.: Algebraic particular integrals, integrability and the problem of the center. Trans. Am. Math. Soc. 338, 799–841 (1993)
Shao Y., Zhao Y.: The cyclicity and period function of a class of quadratic reversible LotkaCVolterra system of genus one. J. Math. Anal. Appl. 377, 817–827 (2011)
Shao, Y., Zhao, Y.: The cyclicity of a class of quadratic reversible system of genus one. (2011, preprint)
Takens, F.: Forced oscillations and bifurcations: applications of global analysis. In: Commun. Math., Vol. 3, Inst. Rijksuniv. Utrecht. (1974). Also In: Broer, H.W., Krauskopf, B., Vegter, G. (ed.) Global Analysis of Dynamical Systems. IOP Publishing Ltd, London (2001)
Varchenko A.N.: An estimate of the number of zeros of an Abelian integral depending on a parameter and limiting cycles. Funct. Anal. Appl. 18, 98–108 (1984)
Wang J., Xiao D.: On the number of limit cycles in small perturbations of a class of hyperelliptic Hamitonian systems with one nilpotent saddle. J. Diff. Eqs. 250, 2227–2243 (2011)
Wu, J., Peng, L., Li, C-P.: On the number of limit cycles in perturbations of a quadratic reversible center (Submitted)
Yakovenko, S.: A Geometric Proof of the Bautin Theorem, Concerning the Hilbert 16th Problem, Amer. Math. Soc., Providence, pp. 203–219 (1995)
Yakovenko, S.: Qualitative theory of ordinary differential equations and tangential Hilbert 16th problem. In: Schlomiuk, D. (eds.) CRM Monograph Series, vol. 24, Amer. Math. Soc., Providence (2005)
Yu J., Li C.: Bifurcation of a class of planar non-Hamiltonian integrable systems with one center and one homoclinic loop. J. Math. Anal. Appl. 269, 227–243 (2002)
Yu P., Han M.: Twelve limit cycles in a cubic order planar system with Z2-symmetry. Comm. Pure Appl. Anal. 3, 515–526 (2004)
Yu, P., Han, M., Four limit cycles from perturbing quadratic integrable systems by quadratic polynomials (2010, preprint). Posted on arXiv.org since February 4
Zhang Z., Li C.: On the number of limit cycles of a class of quadratic Hamiltonian systems under quadratic perturbations. Adv. Math. 26(5), 445–460 (1997)
Zhao Y.: On the number of limit cycles in quadratic perturbations of quadratic codimension four center. Nonlinearity 24, 2505–2522 (2011)
Zhao Y., Liang Z., Lu G.: The cyclicity of period annulus of the quadratic Hamiltonian systems with non-Morsean point. J. Diff. Eqs. 162, 199–223 (2000)
Zhao, Y., Zhu, H.: Bifurcation of limit cycles from a non-Hamiltonian quadratic integrable system with homoclinic loop (2011, preprint)
Zhao Y., Zhu S.: Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment. Bull. Sci. Math. 125, 109–138 (2001)
Zoła¸dek H.: Quadratic systems with centers and their perturbations. J. Diff. Eqs. 109, 223–273 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is partially supported by NSFC-10831003 and by AGAUR Grant number 2009PIV00064.
Rights and permissions
About this article
Cite this article
Li, C. Abelian Integrals and Limit Cycles. Qual. Theory Dyn. Syst. 11, 111–128 (2012). https://doi.org/10.1007/s12346-011-0051-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12346-011-0051-z