Analytic integrability for some degenerate planar systems

Antonio Algaba; Cristóbal García; Jaume Giné

doi:10.3934/cpaa.2013.12.2797

Article Contents

2013, Volume 12, Issue 6: 2797-2809. Doi: 10.3934/cpaa.2013.12.2797

This issue Previous Article Convergence rates for elliptic reiterated homogenization problems Next Article Well-posedness and long time behavior of an Allen-Cahn type equation

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: September 30, 2012

Revised: January 31, 2013

Published: October 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 34C05; Secondary: 34C23, 37G15.

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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.