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

Analytic integrability for some degenerate planar systems

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.
Citation: Antonio Algaba, Cristóbal García, Jaume Giné. Analytic integrability for some degenerate planar systems. Communications on Pure and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

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