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

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

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.

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. AlgabaI. ChecaC. 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. AlgabaE. FreireE. 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. AlgabaE. FreireE. 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. AlgabaE. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. ChavarrigaH. GiacominiJ. 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. ChavarrigaH. GiacominiJ. 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. FerragutJ. 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. GinéM. 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. AlgabaI. ChecaC. 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. AlgabaE. FreireE. 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. AlgabaE. FreireE. 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. AlgabaE. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. AlgabaC. 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. ChavarrigaH. GiacominiJ. 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. ChavarrigaH. GiacominiJ. 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. FerragutJ. 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. GinéM. 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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