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May  2015, 14(3): 1073-1095. doi: 10.3934/cpaa.2015.14.1073

Essential perturbations of polynomial vector fields with a period annulus

Adriana Buică 1, , Jaume Giné 2, and  Maite Grau 2,

1. 

Department of Applied Mathematics, Babeş-Bolyai University, 1 Kogălniceanu str., Cluj-Napoca, 400084
2. 

Departament de Matemàtica, Universitat de Lleida, Avda. Jaume II, 69, 25001 Lleida

Received  October 2014 Revised  January 2015 Published  March 2015

Chicone--Jacobs and Iliev found the essential perturbations of quadratic systems when considering the problem of finding the cyclicity of a period annulus. Given a perturbation of a particular family of centers of polynomial differential systems of arbitrary degree for which the expressions of its Poincaré--Liapunov constants are known, we give the structure of its $k$-th Melnikov function. This allows to find the essential perturbations in concrete cases. We study here in detail the essential perturbations for all the centers of the differential systems \begin{eqnarray} \dot{x} = -y + P_{\rm d}(x,y), \quad \dot{y} = x + Q_{d}(x,y), \end{eqnarray} where $P_d$ and $Q_d$ are homogeneous polynomials of degree $d$, for $ d=2$ and $ d=3$.
Keywords: bifurcation., non-degenerated center, Melnikov functions, essential perturbation, Cyclicity.
Mathematics Subject Classification: Primary: 34C23, 37G15; Secondary: 34C05, 34C25, 34C0.
Citation: Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073
References:
[1]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type,, \emph{Amer. Math. Soc. Transl.}, 100 (1954), 1.   Google Scholar

[2]

A. Buică, A. Gasull and J. Yang, The third order Melnikov function of a quadratic center under quadratic perturbations,, \emph{J. Math. Anal. Appl.}, 331 (2007), 443.  doi: 10.1016/j.jmaa.2006.09.008.  Google Scholar

[3]

M. Caubergh and F. Dumortier, Algebraic curves of maximal cyclicity,, \emph{Math. Proc. Camb. Phil. Soc.}, 140 (2006), 47.  doi: 10.1017/S0305004105008807.  Google Scholar

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M. Caubergh and A. Gasull, Absolute cyclicity, Lyapunov quantities and center conditions,, \emph{J. Math. Anal. Appl.}, 366 (2010), 297.  doi: 10.1016/j.jmaa.2010.01.010.  Google Scholar

[5]

C. Chicone and M. Jacobs, Bifurcations of limit cycles from quadratic isochrones,, \emph{J. Differential Equations}, 91 (1991), 268.  doi: 10.1016/0022-0396(91)90142-V.  Google Scholar

[6]

C. Christopher, Estimating limit cycle bifurcations from centers,, in \emph{Trends in Mathematics: Differential equations with symbolic computation}, (2005), 23.  doi: 10.1007/3-7643-7429-2_2.  Google Scholar

[7]

F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields,, \emph{J. Differential Equations}, 110 (1994), 86.  doi: 10.1006/jdeq.1994.1061.  Google Scholar

[8]

J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac,, in \emph{Actualit\'es Math\'ematiques}, (1992).   Google Scholar

[9]

A. Gasull and J. Giné, Cyclicity versus Center problem,, \emph{Qual. Theory Dyn. Syst.}, 9 (2010), 101.  doi: 10.1007/s12346-010-0022-9.  Google Scholar

[10]

A. Gasull and J. Torregrosa, A relation between small amplitude and big limit cycles,, \emph{Rocky Mountain J. Math.}, 31 (2001), 1277.  doi: 10.1216/rmjm/1021249441.  Google Scholar

[11]

A. Gasull and J. Torregrosa, A new approach to the computation of the Lyapunov constants,, \emph{Comput. Appl. Math.}, 20 (2001), 149.   Google Scholar

[12]

L. Gavrilov, Cyclicity of period annuli and principalization of Bautin ideals,, \emph{Ergodic Theory Dynam. Systems}, 28 (2008), 1497.  doi: 10.1017/S0143385707000971.  Google Scholar

[13]

L. Gavrilov and D. Novikov, On the finite cyclicity of open period annuli,, \emph{Duke Math. J.}, 152 (2010), 1.  doi: 10.1215/00127094-2010-005.  Google Scholar

[14]

J. Giné, The nondegenerate center problem and the inverse integrating factor,, \emph{Bull. Sci. Math.}, 130 (2006), 152.  doi: 10.1016/j.bulsci.2005.09.001.  Google Scholar

[15]

D. Hilbert, Mathematical problems,, \emph{Bull. Am. Math. Soc.}, 8 (1902), 437.  doi: 10.1090/S0002-9904-1902-00923-3.  Google Scholar

[16]

I. D. Iliev, Perturbations of quadratic centers,, \emph{Bull. Sci. Math.}, 122 (1998), 107.  doi: 10.1016/S0007-4497(98)80080-8.  Google Scholar

[17]

Yu. S. Il'yashenko, Finiteness theorems for limit cycles,, in \emph{Translations of Mathematical Monographs}, 94 (1991).   Google Scholar

[18]

Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations,, in Graduate Studies in Mathematics {\bf 86}, 86 (2008).   Google Scholar

[19]

Jibin Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields,, \emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, 13 (2003), 47.  doi: 10.1142/S0218127403006352.  Google Scholar

[20]

P. Mardešić, M. Saavedra, M. Uribe and M. Wallace, Unfolding of the Hamiltonian triangle vector field,, \emph{J. Dyn. Control Syst.}, 17 (2011), 291.  doi: 10.1007/s10883-011-9120-5.  Google Scholar

[21]

H. Poincaré, Méemoire sur les courbes définies par les équations différentielles,, \emph{Journal de Math\'ematiques}, 37 (1881), 375.   Google Scholar

[22]

G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, pp. 91-158, in \emph{Sur les espaces fibr\'es et les vari\'et\'es feuillet\'ees} by W.-T. Wu, (1183).   Google Scholar

[23]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach,, Birkh\, (2009).  doi: 10.1007/978-0-8176-4727-8.  Google Scholar

[24]

R. Roussarie, Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem,, Progress in Mathematics \textbf{164} Birkh\, 164 (1998).  doi: 10.1007/978-3-0348-8798-4.  Google Scholar

[25]

R. Roussarie, Melnikov functions and Bautin ideal,, \emph{Qual. Theory Dyn. Syst.}, 2 (2001), 67.  doi: 10.1007/BF02969382.  Google Scholar

[26]

D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields,, in \emph{Bifurcations and Periodic Orbits of Vector Fields} (Montreal, (1992), 429.   Google Scholar

[27]

K. S. Sibirskiĭ, On the number of limit cycles in the neighborhood of a singular point,, (Russian) \emph{Differencial'nye Uravnenija}, 1 (1965), 53.   Google Scholar

[28]

S. Smale, Mathematical problems for the next century,, \emph{Math. Intelligencer}, 20 (1998), 7.  doi: 10.1007/BF03025291.  Google Scholar

show all references

References:
[1]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type,, \emph{Amer. Math. Soc. Transl.}, 100 (1954), 1.   Google Scholar

[2]

A. Buică, A. Gasull and J. Yang, The third order Melnikov function of a quadratic center under quadratic perturbations,, \emph{J. Math. Anal. Appl.}, 331 (2007), 443.  doi: 10.1016/j.jmaa.2006.09.008.  Google Scholar

[3]

M. Caubergh and F. Dumortier, Algebraic curves of maximal cyclicity,, \emph{Math. Proc. Camb. Phil. Soc.}, 140 (2006), 47.  doi: 10.1017/S0305004105008807.  Google Scholar

[4]

M. Caubergh and A. Gasull, Absolute cyclicity, Lyapunov quantities and center conditions,, \emph{J. Math. Anal. Appl.}, 366 (2010), 297.  doi: 10.1016/j.jmaa.2010.01.010.  Google Scholar

[5]

C. Chicone and M. Jacobs, Bifurcations of limit cycles from quadratic isochrones,, \emph{J. Differential Equations}, 91 (1991), 268.  doi: 10.1016/0022-0396(91)90142-V.  Google Scholar

[6]

C. Christopher, Estimating limit cycle bifurcations from centers,, in \emph{Trends in Mathematics: Differential equations with symbolic computation}, (2005), 23.  doi: 10.1007/3-7643-7429-2_2.  Google Scholar

[7]

F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields,, \emph{J. Differential Equations}, 110 (1994), 86.  doi: 10.1006/jdeq.1994.1061.  Google Scholar

[8]

J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac,, in \emph{Actualit\'es Math\'ematiques}, (1992).   Google Scholar

[9]

A. Gasull and J. Giné, Cyclicity versus Center problem,, \emph{Qual. Theory Dyn. Syst.}, 9 (2010), 101.  doi: 10.1007/s12346-010-0022-9.  Google Scholar

[10]

A. Gasull and J. Torregrosa, A relation between small amplitude and big limit cycles,, \emph{Rocky Mountain J. Math.}, 31 (2001), 1277.  doi: 10.1216/rmjm/1021249441.  Google Scholar

[11]

A. Gasull and J. Torregrosa, A new approach to the computation of the Lyapunov constants,, \emph{Comput. Appl. Math.}, 20 (2001), 149.   Google Scholar

[12]

L. Gavrilov, Cyclicity of period annuli and principalization of Bautin ideals,, \emph{Ergodic Theory Dynam. Systems}, 28 (2008), 1497.  doi: 10.1017/S0143385707000971.  Google Scholar

[13]

L. Gavrilov and D. Novikov, On the finite cyclicity of open period annuli,, \emph{Duke Math. J.}, 152 (2010), 1.  doi: 10.1215/00127094-2010-005.  Google Scholar

[14]

J. Giné, The nondegenerate center problem and the inverse integrating factor,, \emph{Bull. Sci. Math.}, 130 (2006), 152.  doi: 10.1016/j.bulsci.2005.09.001.  Google Scholar

[15]

D. Hilbert, Mathematical problems,, \emph{Bull. Am. Math. Soc.}, 8 (1902), 437.  doi: 10.1090/S0002-9904-1902-00923-3.  Google Scholar

[16]

I. D. Iliev, Perturbations of quadratic centers,, \emph{Bull. Sci. Math.}, 122 (1998), 107.  doi: 10.1016/S0007-4497(98)80080-8.  Google Scholar

[17]

Yu. S. Il'yashenko, Finiteness theorems for limit cycles,, in \emph{Translations of Mathematical Monographs}, 94 (1991).   Google Scholar

[18]

Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations,, in Graduate Studies in Mathematics {\bf 86}, 86 (2008).   Google Scholar

[19]

Jibin Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields,, \emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, 13 (2003), 47.  doi: 10.1142/S0218127403006352.  Google Scholar

[20]

P. Mardešić, M. Saavedra, M. Uribe and M. Wallace, Unfolding of the Hamiltonian triangle vector field,, \emph{J. Dyn. Control Syst.}, 17 (2011), 291.  doi: 10.1007/s10883-011-9120-5.  Google Scholar

[21]

H. Poincaré, Méemoire sur les courbes définies par les équations différentielles,, \emph{Journal de Math\'ematiques}, 37 (1881), 375.   Google Scholar

[22]

G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, pp. 91-158, in \emph{Sur les espaces fibr\'es et les vari\'et\'es feuillet\'ees} by W.-T. Wu, (1183).   Google Scholar

[23]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach,, Birkh\, (2009).  doi: 10.1007/978-0-8176-4727-8.  Google Scholar

[24]

R. Roussarie, Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem,, Progress in Mathematics \textbf{164} Birkh\, 164 (1998).  doi: 10.1007/978-3-0348-8798-4.  Google Scholar

[25]

R. Roussarie, Melnikov functions and Bautin ideal,, \emph{Qual. Theory Dyn. Syst.}, 2 (2001), 67.  doi: 10.1007/BF02969382.  Google Scholar

[26]

D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields,, in \emph{Bifurcations and Periodic Orbits of Vector Fields} (Montreal, (1992), 429.   Google Scholar

[27]

K. S. Sibirskiĭ, On the number of limit cycles in the neighborhood of a singular point,, (Russian) \emph{Differencial'nye Uravnenija}, 1 (1965), 53.   Google Scholar

[28]

S. Smale, Mathematical problems for the next century,, \emph{Math. Intelligencer}, 20 (1998), 7.  doi: 10.1007/BF03025291.  Google Scholar

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