Essential perturbations of polynomial vector fields with a period annulus

Previous Article
Asymptotic behavior for the unique positive solution to a singular elliptic problem
CPAA Home
This Issue
Next Article
Klein-Gordon-Maxwell equations in high dimensions

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
Related Papers

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
[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

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

[1]	Isaac A. García, Douglas S. Shafer. Cyclicity of a class of polynomial nilpotent center singularities. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2497-2520. doi: 10.3934/dcds.2016.36.2497
[2]	Jianfeng Huang, Yulin Zhao. Bifurcation of isolated closed orbits from degenerated singularity in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2861-2883. doi: 10.3934/dcds.2013.33.2861
[3]	Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457
[4]	Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437
[5]	Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735
[6]	Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203
[7]	Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks & Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897
[8]	Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks & Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009
[9]	Chunyan Ji, Daqing Jiang. Persistence and non-persistence of a mutualism system with stochastic perturbation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 867-889. doi: 10.3934/dcds.2012.32.867
[10]	Ana Paula S. Dias, Paul C. Matthews, Ana Rodrigues. Generating functions for Hopf bifurcation with $ S_n$-symmetry. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 823-842. doi: 10.3934/dcds.2009.25.823
[11]	Felipe Rivero. Time dependent perturbation in a non-autonomous non-classical parabolic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 209-221. doi: 10.3934/dcdsb.2013.18.209
[12]	Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259
[13]	Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523
[14]	Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701
[15]	Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031
[16]	José Caicedo, Alfonso Castro, Arturo Sanjuán. Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1857-1865. doi: 10.3934/dcds.2017078
[17]	Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
[18]	Sergio R. López-Permouth, Benigno R. Parra-Avila, Steve Szabo. Dual generalizations of the concept of cyclicity of codes. Advances in Mathematics of Communications, 2009, 3 (3) : 227-234. doi: 10.3934/amc.2009.3.227
[19]	Ayça Çeşmelioǧlu, Wilfried Meidl, Alexander Pott. On the dual of (non)-weakly regular bent functions and self-dual bent functions. Advances in Mathematics of Communications, 2013, 7 (4) : 425-440. doi: 10.3934/amc.2013.7.425
[20]	Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences