Essential perturbations of polynomial vector fields with a period annulus

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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

Abstract
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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 and 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, Amer. Math. Soc. Transl., 100 (1954), 1-19.
[2]	A. Buică, A. Gasull and J. Yang, The third order Melnikov function of a quadratic center under quadratic perturbations, J. Math. Anal. Appl., 331 (2007), 443-454. doi: 10.1016/j.jmaa.2006.09.008.
[3]	M. Caubergh and F. Dumortier, Algebraic curves of maximal cyclicity, Math. Proc. Camb. Phil. Soc., 140 (2006), 47-70. doi: 10.1017/S0305004105008807.
[4]	M. Caubergh and A. Gasull, Absolute cyclicity, Lyapunov quantities and center conditions, J. Math. Anal. Appl., 366 (2010), 297-309. doi: 10.1016/j.jmaa.2010.01.010.
[5]	C. Chicone and M. Jacobs, Bifurcations of limit cycles from quadratic isochrones, J. Differential Equations, 91 (1991), 268-326. doi: 10.1016/0022-0396(91)90142-V.
[6]	C. Christopher, Estimating limit cycle bifurcations from centers, in Trends in Mathematics: Differential equations with symbolic computation, Birkhäuser (2005), 23-35. doi: 10.1007/3-7643-7429-2_2.
[7]	F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133. doi: 10.1006/jdeq.1994.1061.
[8]	J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, in Actualités Mathématiques, Hermann, Paris, 1992.
[9]	A. Gasull and J. Giné, Cyclicity versus Center problem, Qual. Theory Dyn. Syst., 9 (2010), 101-113. doi: 10.1007/s12346-010-0022-9.
[10]	A. Gasull and J. Torregrosa, A relation between small amplitude and big limit cycles, Rocky Mountain J. Math., 31 (2001), 1277-1303. doi: 10.1216/rmjm/1021249441.
[11]	A. Gasull and J. Torregrosa, A new approach to the computation of the Lyapunov constants, Comput. Appl. Math., 20 (2001), 149-177.
[12]	L. Gavrilov, Cyclicity of period annuli and principalization of Bautin ideals, Ergodic Theory Dynam. Systems, 28 (2008), 1497-1507. doi: 10.1017/S0143385707000971.
[13]	L. Gavrilov and D. Novikov, On the finite cyclicity of open period annuli, Duke Math. J., 152 (2010), 1-26. doi: 10.1215/00127094-2010-005.
[14]	J. Giné, The nondegenerate center problem and the inverse integrating factor, Bull. Sci. Math., 130 (2006), 152-161. doi: 10.1016/j.bulsci.2005.09.001.
[15]	D. Hilbert, Mathematical problems, Bull. Am. Math. Soc., 8 (1902), 437-479. doi: 10.1090/S0002-9904-1902-00923-3.
[16]	I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161. doi: 10.1016/S0007-4497(98)80080-8.
[17]	Yu. S. Il'yashenko, Finiteness theorems for limit cycles, in Translations of Mathematical Monographs, 94. American Mathematical Society, Providence, RI, 1991.
[18]	Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, in Graduate Studies in Mathematics 86, American Mathematical Society, Providence, Rhode Island, 2008.
[19]	Jibin Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106. doi: 10.1142/S0218127403006352.
[20]	P. Mardešić, M. Saavedra, M. Uribe and M. Wallace, Unfolding of the Hamiltonian triangle vector field, J. Dyn. Control Syst., 17 (2011), 291-310. doi: 10.1007/s10883-011-9120-5.
[21]	H. Poincaré, Méemoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 37 (1881), 375-422; 8 (1882), 251-296; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, pp 3-84.
[22]	G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, pp. 91-158 in Sur les espaces fibrés et les variétés feuilletées by W.-T. Wu, G. Reeb, Actualités Sci. Industr., 1183, Tome XI, Paris, Hermann et Cie, Éditeurs, Paris, 1952.
[23]	V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.
[24]	R. Roussarie, Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem, Progress in Mathematics 164 Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8798-4.
[25]	R. Roussarie, Melnikov functions and Bautin ideal, Qual. Theory Dyn. Syst., 2 (2001), 67-78. doi: 10.1007/BF02969382.
[26]	D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields, in Bifurcations and Periodic Orbits of Vector Fields (Montreal, PQ, 1992), 429-467, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 408, Kluwer Acad. Publ., Dordrecht, 1993.
[27]	K. S. Sibirskiĭ, On the number of limit cycles in the neighborhood of a singular point, (Russian) Differencial'nye Uravnenija, 1 (1965) 53-66. English translation: Differential Equations 1 (1965), 36-47.
[28]	S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15. doi: 10.1007/BF03025291.

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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, Amer. Math. Soc. Transl., 100 (1954), 1-19.
[2]	A. Buică, A. Gasull and J. Yang, The third order Melnikov function of a quadratic center under quadratic perturbations, J. Math. Anal. Appl., 331 (2007), 443-454. doi: 10.1016/j.jmaa.2006.09.008.
[3]	M. Caubergh and F. Dumortier, Algebraic curves of maximal cyclicity, Math. Proc. Camb. Phil. Soc., 140 (2006), 47-70. doi: 10.1017/S0305004105008807.
[4]	M. Caubergh and A. Gasull, Absolute cyclicity, Lyapunov quantities and center conditions, J. Math. Anal. Appl., 366 (2010), 297-309. doi: 10.1016/j.jmaa.2010.01.010.
[5]	C. Chicone and M. Jacobs, Bifurcations of limit cycles from quadratic isochrones, J. Differential Equations, 91 (1991), 268-326. doi: 10.1016/0022-0396(91)90142-V.
[6]	C. Christopher, Estimating limit cycle bifurcations from centers, in Trends in Mathematics: Differential equations with symbolic computation, Birkhäuser (2005), 23-35. doi: 10.1007/3-7643-7429-2_2.
[7]	F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133. doi: 10.1006/jdeq.1994.1061.
[8]	J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, in Actualités Mathématiques, Hermann, Paris, 1992.
[9]	A. Gasull and J. Giné, Cyclicity versus Center problem, Qual. Theory Dyn. Syst., 9 (2010), 101-113. doi: 10.1007/s12346-010-0022-9.
[10]	A. Gasull and J. Torregrosa, A relation between small amplitude and big limit cycles, Rocky Mountain J. Math., 31 (2001), 1277-1303. doi: 10.1216/rmjm/1021249441.
[11]	A. Gasull and J. Torregrosa, A new approach to the computation of the Lyapunov constants, Comput. Appl. Math., 20 (2001), 149-177.
[12]	L. Gavrilov, Cyclicity of period annuli and principalization of Bautin ideals, Ergodic Theory Dynam. Systems, 28 (2008), 1497-1507. doi: 10.1017/S0143385707000971.
[13]	L. Gavrilov and D. Novikov, On the finite cyclicity of open period annuli, Duke Math. J., 152 (2010), 1-26. doi: 10.1215/00127094-2010-005.
[14]	J. Giné, The nondegenerate center problem and the inverse integrating factor, Bull. Sci. Math., 130 (2006), 152-161. doi: 10.1016/j.bulsci.2005.09.001.
[15]	D. Hilbert, Mathematical problems, Bull. Am. Math. Soc., 8 (1902), 437-479. doi: 10.1090/S0002-9904-1902-00923-3.
[16]	I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161. doi: 10.1016/S0007-4497(98)80080-8.
[17]	Yu. S. Il'yashenko, Finiteness theorems for limit cycles, in Translations of Mathematical Monographs, 94. American Mathematical Society, Providence, RI, 1991.
[18]	Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, in Graduate Studies in Mathematics 86, American Mathematical Society, Providence, Rhode Island, 2008.
[19]	Jibin Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 47-106. doi: 10.1142/S0218127403006352.
[20]	P. Mardešić, M. Saavedra, M. Uribe and M. Wallace, Unfolding of the Hamiltonian triangle vector field, J. Dyn. Control Syst., 17 (2011), 291-310. doi: 10.1007/s10883-011-9120-5.
[21]	H. Poincaré, Méemoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 37 (1881), 375-422; 8 (1882), 251-296; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, pp 3-84.
[22]	G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, pp. 91-158 in Sur les espaces fibrés et les variétés feuilletées by W.-T. Wu, G. Reeb, Actualités Sci. Industr., 1183, Tome XI, Paris, Hermann et Cie, Éditeurs, Paris, 1952.
[23]	V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4727-8.
[24]	R. Roussarie, Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem, Progress in Mathematics 164 Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8798-4.
[25]	R. Roussarie, Melnikov functions and Bautin ideal, Qual. Theory Dyn. Syst., 2 (2001), 67-78. doi: 10.1007/BF02969382.
[26]	D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields, in Bifurcations and Periodic Orbits of Vector Fields (Montreal, PQ, 1992), 429-467, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 408, Kluwer Acad. Publ., Dordrecht, 1993.
[27]	K. S. Sibirskiĭ, On the number of limit cycles in the neighborhood of a singular point, (Russian) Differencial'nye Uravnenija, 1 (1965) 53-66. English translation: Differential Equations 1 (1965), 36-47.
[28]	S. Smale, Mathematical problems for the next century, Math. Intelligencer, 20 (1998), 7-15. doi: 10.1007/BF03025291.

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