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January  2020, 19(1): 407-423. doi: 10.3934/cpaa.2020021

Analytic integrability around a nilpotent singularity: The non-generic case

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

1. 

Dept. Matemáticas, Facultad de Ciencias, Univ. of Huelva, Spain
2. 

Departament de Matemàtica, Inspires Research Centre, Universitat de Lleida, Avda. Jaume Ⅱ, 69, 25001 Lleida, Catalonia, Spain

* Corresponding author

Received  December 2018 Revised  May 2019 Published  July 2019

Recently, in [9] is characterized the analytic integrability problem around a nilpotent singularity for differential systems in the plane under generic conditions. In this work we solve the remaining case completing the analytic integrability problem for such singularity.

Keywords: Analytic integrability, nilpotent singularity, center problem, orbitally equivalence.
Mathematics Subject Classification: Primary: 34C25; Secondary: 34C14, 37C10, 37C27.
Citation: Antonio Algaba, María Díaz, Cristóbal García, Jaume Giné. Analytic integrability around a nilpotent singularity: The non-generic case. Communications on Pure & Applied Analysis, 2020, 19 (1) : 407-423. doi: 10.3934/cpaa.2020021
References:
[1]

A. Algaba, I. Checa, C. García and J. Giné, Analytic integrability inside a family of degenerate centers, Nonlinear Anal. Real World Appl., 31 (2016), 288-307.  doi: 10.1016/j.nonrwa.2016.02.003.  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. Freire, E. Gamero and C. García, An Algorithm for computing quasi-homogeneous formal normal forms under equivalence, Acta Appl. Math., 80 (2004), 335-339.  doi: 10.1023/B:ACAP.0000018769.73927.a4.  Google Scholar

[4]

Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems, Nonlinear Analysis, 72 (2010), 1726–1736. doi: 10.1016/j.na.2009.09.012.  Google Scholar

[5]

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

[6]

A. Algaba, C. García and J. Giné, Analytic integrability for some degenerate planar systems, Commun. Pure Appl. Anal., 12 (2013), 2797-2809.  doi: 10.3934/cpaa.2013.12.2797.  Google Scholar

[7]

A. Algaba, C. García and J. Giné, Analytic integrability for some degenerate planar vector fields, J. Differential Equations, 257 (2014), 549-565.  doi: 10.1016/j.jde.2014.04.010.  Google Scholar

[8]

A. Algaba, C. García and J. Giné, Analytic integrability of some examples of degenerate planar vector fields, Acta Appl. Math., 141 (2016), 1-15.  doi: 10.1007/s10440-014-0001-2.  Google Scholar

[9]

A. Algaba, C. García and J. Giné, Analytic integrability around a nilpotent singularity, J. Differential Equations, 267 (2019), 443-467.  doi: 10.1016/j.jde.2019.01.015.  Google Scholar

[10]

A. Algaba, C. García and J. Giné, Integrability of planar nilpotent differential systems through the existence of an inverse integrating factor, Commun. Nonlinear Sci. Numer. Simul., 71 (2019), 130-140.  doi: 10.1016/j.cnsns.2018.09.018.  Google Scholar

[11]

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

[12]

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

[13]

A. Algaba, C. García and M. Reyes, Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems, Chaos Solitons Fractals, 45 (2012), 869-878.  doi: 10.1016/j.chaos.2012.02.016.  Google Scholar

[14]

A. Baider and J. A. Sanders, Further reduction of the Takens-Bogdanov normal form, J. Differential Equations, 99 (1992), 205-244.  doi: 10.1016/0022-0396(92)90022-F.  Google Scholar

[15]

C. B. Collins, Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree, J. Math Anal. Appl., 195 (1995), 719-735.  doi: 10.1006/jmaa.1995.1385.  Google Scholar

[16]

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

[17]

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

[18]

A. Ferragut, J. Llibre and A. Mahdi, Polynomial inverse integrating factors for polynomial vector fields, Discrete Contin. Dyn. Syst., 17 (2006), 387-395.  doi: 10.3934/dcds.2007.17.387.  Google Scholar

[19]

J. Giné, Analytic integrability and characterization of centers for nilpotent singular points, Z. Angew. Math. Phys., 55 (2004), 725-740.  doi: 10.1007/s00033-004-1093-8.  Google Scholar

[20]

J. Giné, Analytic integrability and characterization of center 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, Reduction of integrable planar polynomial differential systems, Appl. Math. Lett., 25 (2012), 1862-1865.  doi: 10.1016/j.aml.2012.02.047.  Google Scholar

[22]

J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547.  doi: 10.3934/dcds.2013.33.4531.  Google Scholar

[23]

A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893.  doi: 10.1063/1.531484.  Google Scholar

[24]

M. Han and K. Jiang, Normal forms of integrable systems at a resonant saddle, Ann. Differential Equations, 14 (1998), 150-155.   Google Scholar

[25]

A. M. Lyapunov, Stability of Motion, With a contribution by V. A. Pliss and an introduction by V. P. Basov. Translated from the Russian by Flavian Abramovici and Michael Shimshoni  Google Scholar

[26]

J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.  doi: 10.1088/0951-7715/15/4/313.  Google Scholar

[27]

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

[28]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 7 (1881), 375–422; 8 (1882), 251–296; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, pp. 3–84.  Google Scholar

[29]

M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229.  doi: 10.2307/1999380.  Google Scholar

[30]

M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333 (1992), 673-688.  doi: 10.2307/2154053.  Google Scholar

show all references

References:
[1]

A. Algaba, I. Checa, C. García and J. Giné, Analytic integrability inside a family of degenerate centers, Nonlinear Anal. Real World Appl., 31 (2016), 288-307.  doi: 10.1016/j.nonrwa.2016.02.003.  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. Freire, E. Gamero and C. García, An Algorithm for computing quasi-homogeneous formal normal forms under equivalence, Acta Appl. Math., 80 (2004), 335-339.  doi: 10.1023/B:ACAP.0000018769.73927.a4.  Google Scholar

[4]

Monodromy, center-focus and integrability problems for quasi-homogeneous polynomial systems, Nonlinear Analysis, 72 (2010), 1726–1736. doi: 10.1016/j.na.2009.09.012.  Google Scholar

[5]

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

[6]

A. Algaba, C. García and J. Giné, Analytic integrability for some degenerate planar systems, Commun. Pure Appl. Anal., 12 (2013), 2797-2809.  doi: 10.3934/cpaa.2013.12.2797.  Google Scholar

[7]

A. Algaba, C. García and J. Giné, Analytic integrability for some degenerate planar vector fields, J. Differential Equations, 257 (2014), 549-565.  doi: 10.1016/j.jde.2014.04.010.  Google Scholar

[8]

A. Algaba, C. García and J. Giné, Analytic integrability of some examples of degenerate planar vector fields, Acta Appl. Math., 141 (2016), 1-15.  doi: 10.1007/s10440-014-0001-2.  Google Scholar

[9]

A. Algaba, C. García and J. Giné, Analytic integrability around a nilpotent singularity, J. Differential Equations, 267 (2019), 443-467.  doi: 10.1016/j.jde.2019.01.015.  Google Scholar

[10]

A. Algaba, C. García and J. Giné, Integrability of planar nilpotent differential systems through the existence of an inverse integrating factor, Commun. Nonlinear Sci. Numer. Simul., 71 (2019), 130-140.  doi: 10.1016/j.cnsns.2018.09.018.  Google Scholar

[11]

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

[12]

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

[13]

A. Algaba, C. García and M. Reyes, Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems, Chaos Solitons Fractals, 45 (2012), 869-878.  doi: 10.1016/j.chaos.2012.02.016.  Google Scholar

[14]

A. Baider and J. A. Sanders, Further reduction of the Takens-Bogdanov normal form, J. Differential Equations, 99 (1992), 205-244.  doi: 10.1016/0022-0396(92)90022-F.  Google Scholar

[15]

C. B. Collins, Algebraic conditions for a centre or a focus in some simple systems of arbitrary degree, J. Math Anal. Appl., 195 (1995), 719-735.  doi: 10.1006/jmaa.1995.1385.  Google Scholar

[16]

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

[17]

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

[18]

A. Ferragut, J. Llibre and A. Mahdi, Polynomial inverse integrating factors for polynomial vector fields, Discrete Contin. Dyn. Syst., 17 (2006), 387-395.  doi: 10.3934/dcds.2007.17.387.  Google Scholar

[19]

J. Giné, Analytic integrability and characterization of centers for nilpotent singular points, Z. Angew. Math. Phys., 55 (2004), 725-740.  doi: 10.1007/s00033-004-1093-8.  Google Scholar

[20]

J. Giné, Analytic integrability and characterization of center 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, Reduction of integrable planar polynomial differential systems, Appl. Math. Lett., 25 (2012), 1862-1865.  doi: 10.1016/j.aml.2012.02.047.  Google Scholar

[22]

J. Giné, M. Grau and J. Llibre, Polynomial and rational first integrals for planar quasi-homogeneous polynomial differential systems, Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547.  doi: 10.3934/dcds.2013.33.4531.  Google Scholar

[23]

A. Goriely, Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations, J. Math. Phys., 37 (1996), 1871-1893.  doi: 10.1063/1.531484.  Google Scholar

[24]

M. Han and K. Jiang, Normal forms of integrable systems at a resonant saddle, Ann. Differential Equations, 14 (1998), 150-155.   Google Scholar

[25]

A. M. Lyapunov, Stability of Motion, With a contribution by V. A. Pliss and an introduction by V. P. Basov. Translated from the Russian by Flavian Abramovici and Michael Shimshoni  Google Scholar

[26]

J. Llibre and X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity, 15 (2002), 1269-1280.  doi: 10.1088/0951-7715/15/4/313.  Google Scholar

[27]

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

[28]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 7 (1881), 375–422; 8 (1882), 251–296; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, pp. 3–84.  Google Scholar

[29]

M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229.  doi: 10.2307/1999380.  Google Scholar

[30]

M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333 (1992), 673-688.  doi: 10.2307/2154053.  Google Scholar

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