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November  2013, 12(6): 2797-2809. doi: 10.3934/cpaa.2013.12.2797

Analytic integrability for some degenerate planar systems

Antonio Algaba 1, , Cristóbal García 1, and  Jaume Giné 2,

1. 

Department of Mathematics, University of Huelva, 21071-Huelva
2. 

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

Received  October 2012 Revised  February 2013 Published  May 2013

In the present paper we characterize the analytic integrability around the origin of a family of degenerate differential systems. Moreover, we study the analytic integrability of some degenerate systems through the orbital reversibility and from the existence of a Lie's symmetry for these systems. The results obtained for this family are similar to the results obtained when we characterize the analytic integrability of non-degenerate and nilpotent systems. The obtained results can be applied to compute the analytic integrable systems of any particular family of degenerate systems studied.
Keywords: integrability problem, degenerate center problem., Nonlinear differential systems.
Mathematics Subject Classification: Primary: 34C05; Secondary: 34C23, 37G1.
Citation: Antonio Algaba, Cristóbal García, Jaume Giné. Analytic integrability for some degenerate planar systems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2797-2809. doi: 10.3934/cpaa.2013.12.2797
References:
[1]

A. Algaba, E. Freire and E. Gamero, Isochronicity via normal form, Qual. Theory Dyn. Syst., 1 (2000), 133-156. doi: 10.1007/BF02969475.  Google Scholar

[2]

A. Algaba, E. Freire, E. Gamero and C. García, Quasi-homogeneous normal forms, J. Comput. Appl. Math., 150 (2003), 193-216. doi: 10.1016/S0377-0427(02)00660-X.  Google Scholar

[3]

A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420. doi: 10.1088/0951-7715/22/2/009.  Google Scholar

[4]

A. Algaba, C. García and M. Reyes, Like-linearizations of vector fields, Bull. Sci. Math., 133 (2009), 806-816. doi: 10.1016/j.bulsci.2009.09.006.  Google Scholar

[5]

A. Algaba, C. García and M. A. Teixeira, Reversibility and quasi-homogeneous normal forms of vector fields, Nonlinear Anal., 73 (2010), 510-525. doi: 10.1016/j.na.2010.03.046.  Google Scholar

[6]

A. Algaba, E. Freire, E. Gamero and C. García, Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems, Nonlinear Anal., 72 (2010), 1726-1736. doi: 10.1016/j.na.2009.09.012.  Google Scholar

[7]

A. Algaba, C. García and M. Reyes, Integrability of two dimensional quasi-homogeneous polynomial differential systems, Rocky Mountain J. Math., 41 (2011), 1-22. doi: 10.1216/RMJ-2011-41-1-1.  Google Scholar

[8]

A. Algaba, N. Fuentes and C. García, Centers of quasi-homogeneous polynomial planar systems, Nonlinear Anal. Real World Appl., 13 (2012), 419-431. doi: 10.1016/j.nonrwa.2011.07.056.  Google Scholar

[9]

A. Algaba, C. García and M. Reyes, A note on analitic integrability of planar vector fields, European J. Appl. Math., 23 (2012), 555-562. doi: 10.1017/S0956792512000113.  Google Scholar

[10]

A. Algaba, E. Gamero and C. García, The reversibility problems for quasi-homogeneous dynamical systems, Discrete Contin. Dyn. Syst., 33 (2013), 3225-3236. Google Scholar

[11]

M. Berthier and R. Moussu, Réversibilité et classification des centres nilpotents, Ann. Inst. Fourier (Grenoble), 44 (1994), 465-494. doi: 10.5802/aif.1406.  Google Scholar

[12]

A.D. Bruno, "Local Methods in Nonlinear Differential Equations," Springer Verlag, Berlin, 1989.  Google Scholar

[13]

J. Chavarriga, I. García, and J. Giné, Integrability of centers perturbed by quasi-homogeneous polynomials, J. Math. Anal. Appl., 210 (1997), 268-278. doi: 10.1006/jmaa.1997.5402.  Google Scholar

[14]

J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows, J. Differential Equations, 157 (1999), 163-182. doi: 10.1006/jdeq.1998.3621.  Google Scholar

[15]

J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory Dyn. Syst., 23 (2003), 417-428. doi: 10.1017/S014338570200127X.  Google Scholar

[16]

A. Gasull, J. Llibre, V. Mañosa and F. Mañosas, The focus-centre problem for a type of degenerate system, Nonlinearity, 13 (2000), 699-730. doi: 10.1088/0951-7715/13/3/311.  Google Scholar

[17]

H. Giacomini, J. Giné and J. Llibre, The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems, J. Differential Equations, 227 (2006), 406-426. doi: 10.1016/j.jde.2006.03.012.  Google Scholar

[18]

H. Giacomini, J. Giné and J. Llibre, Corrigendum to:"The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems", J. Differential Equations, 232 (2007), 702-702. doi: 10.1016/j.jde.2006.10.004.  Google Scholar

[19]

J. Giné, Sufficient conditions for a center at a completely degenerate critical point, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 1659-1666. doi: 10.1142/S0218127402005315.  Google Scholar

[20]

J. Giné, Analytic integrability and characterization of centers for generalized nilpotent singular points, Appl. Math. Comput., 148 (2004), 849-868. doi: 10.1016/S0096-3003(02)00941-4.  Google Scholar

[21]

J. Giné, On the centers of planar analytic differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 3061-3070. doi: 10.1142/S0218127407018865.  Google Scholar

[22]

J. Giné, On the degenerate center problem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 1383-1392. doi: 10.1142/S0218127411029082.  Google Scholar

[23]

J. Giné and M. Grau, Linearizability and integrability of vector fields via commutation, J. Math. Anal. Appl., 319 (2006), 326-332. doi: 10.1016/j.jmaa.2005.10.017.  Google Scholar

[24]

J. B. 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.  Google Scholar

[25]

J.-F. Mattei and R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. (4), 13 (1980), 469-523.  Google Scholar

[26]

J. M. Pearson, N. G. Lloyd and C. J. Christopher, Algorithmic derivation of centre conditions, SIAM Rev., 38 (1996), 619-636. doi: 10.1137/S0036144595283575.  Google Scholar

[27]

H. Poincaré, Mémoire 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. Google Scholar

[28]

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

[29]

E. Strózyna and H. Żoładek, The analytic and normal form for the nilpotent singularity, J. Differential Equations, 179 (2002), 479-537. doi: 10.1006/jdeq.2001.4043.  Google Scholar

[30]

M. A. Teixeira and J. Yang, The center-focus problem and reversibility, J. Differential Equations, 174 (2001), 237-251. doi: /10.1006/jdeq.2000.3931.  Google Scholar

show all references

References:
[1]

A. Algaba, E. Freire and E. Gamero, Isochronicity via normal form, Qual. Theory Dyn. Syst., 1 (2000), 133-156. doi: 10.1007/BF02969475.  Google Scholar

[2]

A. Algaba, E. Freire, E. Gamero and C. García, Quasi-homogeneous normal forms, J. Comput. Appl. Math., 150 (2003), 193-216. doi: 10.1016/S0377-0427(02)00660-X.  Google Scholar

[3]

A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420. doi: 10.1088/0951-7715/22/2/009.  Google Scholar

[4]

A. Algaba, C. García and M. Reyes, Like-linearizations of vector fields, Bull. Sci. Math., 133 (2009), 806-816. doi: 10.1016/j.bulsci.2009.09.006.  Google Scholar

[5]

A. Algaba, C. García and M. A. Teixeira, Reversibility and quasi-homogeneous normal forms of vector fields, Nonlinear Anal., 73 (2010), 510-525. doi: 10.1016/j.na.2010.03.046.  Google Scholar

[6]

A. Algaba, E. Freire, E. Gamero and C. García, Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems, Nonlinear Anal., 72 (2010), 1726-1736. doi: 10.1016/j.na.2009.09.012.  Google Scholar

[7]

A. Algaba, C. García and M. Reyes, Integrability of two dimensional quasi-homogeneous polynomial differential systems, Rocky Mountain J. Math., 41 (2011), 1-22. doi: 10.1216/RMJ-2011-41-1-1.  Google Scholar

[8]

A. Algaba, N. Fuentes and C. García, Centers of quasi-homogeneous polynomial planar systems, Nonlinear Anal. Real World Appl., 13 (2012), 419-431. doi: 10.1016/j.nonrwa.2011.07.056.  Google Scholar

[9]

A. Algaba, C. García and M. Reyes, A note on analitic integrability of planar vector fields, European J. Appl. Math., 23 (2012), 555-562. doi: 10.1017/S0956792512000113.  Google Scholar

[10]

A. Algaba, E. Gamero and C. García, The reversibility problems for quasi-homogeneous dynamical systems, Discrete Contin. Dyn. Syst., 33 (2013), 3225-3236. Google Scholar

[11]

M. Berthier and R. Moussu, Réversibilité et classification des centres nilpotents, Ann. Inst. Fourier (Grenoble), 44 (1994), 465-494. doi: 10.5802/aif.1406.  Google Scholar

[12]

A.D. Bruno, "Local Methods in Nonlinear Differential Equations," Springer Verlag, Berlin, 1989.  Google Scholar

[13]

J. Chavarriga, I. García, and J. Giné, Integrability of centers perturbed by quasi-homogeneous polynomials, J. Math. Anal. Appl., 210 (1997), 268-278. doi: 10.1006/jmaa.1997.5402.  Google Scholar

[14]

J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows, J. Differential Equations, 157 (1999), 163-182. doi: 10.1006/jdeq.1998.3621.  Google Scholar

[15]

J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory Dyn. Syst., 23 (2003), 417-428. doi: 10.1017/S014338570200127X.  Google Scholar

[16]

A. Gasull, J. Llibre, V. Mañosa and F. Mañosas, The focus-centre problem for a type of degenerate system, Nonlinearity, 13 (2000), 699-730. doi: 10.1088/0951-7715/13/3/311.  Google Scholar

[17]

H. Giacomini, J. Giné and J. Llibre, The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems, J. Differential Equations, 227 (2006), 406-426. doi: 10.1016/j.jde.2006.03.012.  Google Scholar

[18]

H. Giacomini, J. Giné and J. Llibre, Corrigendum to:"The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems", J. Differential Equations, 232 (2007), 702-702. doi: 10.1016/j.jde.2006.10.004.  Google Scholar

[19]

J. Giné, Sufficient conditions for a center at a completely degenerate critical point, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 1659-1666. doi: 10.1142/S0218127402005315.  Google Scholar

[20]

J. Giné, Analytic integrability and characterization of centers for generalized nilpotent singular points, Appl. Math. Comput., 148 (2004), 849-868. doi: 10.1016/S0096-3003(02)00941-4.  Google Scholar

[21]

J. Giné, On the centers of planar analytic differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 3061-3070. doi: 10.1142/S0218127407018865.  Google Scholar

[22]

J. Giné, On the degenerate center problem, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 1383-1392. doi: 10.1142/S0218127411029082.  Google Scholar

[23]

J. Giné and M. Grau, Linearizability and integrability of vector fields via commutation, J. Math. Anal. Appl., 319 (2006), 326-332. doi: 10.1016/j.jmaa.2005.10.017.  Google Scholar

[24]

J. B. 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.  Google Scholar

[25]

J.-F. Mattei and R. Moussu, Holonomie et intégrales premières, Ann. Sci. École Norm. Sup. (4), 13 (1980), 469-523.  Google Scholar

[26]

J. M. Pearson, N. G. Lloyd and C. J. Christopher, Algorithmic derivation of centre conditions, SIAM Rev., 38 (1996), 619-636. doi: 10.1137/S0036144595283575.  Google Scholar

[27]

H. Poincaré, Mémoire 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. Google Scholar

[28]

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

[29]

E. Strózyna and H. Żoładek, The analytic and normal form for the nilpotent singularity, J. Differential Equations, 179 (2002), 479-537. doi: 10.1006/jdeq.2001.4043.  Google Scholar

[30]

M. A. Teixeira and J. Yang, The center-focus problem and reversibility, J. Differential Equations, 174 (2001), 237-251. doi: /10.1006/jdeq.2000.3931.  Google Scholar

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