American Institute of Mathematical Sciences

Advanced
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 and 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.

[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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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

[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.

[27]

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

[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.

[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.

[30]

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

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.

[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. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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

[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.

[27]

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

[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.

[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.

[30]

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

[1]

Isaac A. García, Claudia Valls. The three-dimensional center problem for the zero-Hopf singularity. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2027-2046. doi: 10.3934/dcds.2016.36.2027
[2]

Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557
[3]

Isaac A. García, Douglas S. Shafer. Cyclicity of a class of polynomial nilpotent center singularities. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2497-2520. doi: 10.3934/dcds.2016.36.2497
[4]

Jaume Llibre, Claudia Valls. Analytic integrability of a class of planar polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2657-2661. doi: 10.3934/dcdsb.2015.20.2657
[5]

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
[6]

I. Baldomá, Tere M. Seara. The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 323-347. doi: 10.3934/dcdsb.2008.10.323
[7]

Vladimir S. Gerdjikov, Rossen I. Ivanov, Aleksander A. Stefanov. Riemann-Hilbert problem, integrability and reductions. Journal of Geometric Mechanics, 2019, 11 (2) : 167-185. doi: 10.3934/jgm.2019009
[8]

Rafael Ortega, Andrés Rivera. Global bifurcations from the center of mass in the Sitnikov problem. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 719-732. doi: 10.3934/dcdsb.2010.14.719
[9]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
[10]

Armengol Gasull, Jaume Giné, Joan Torregrosa. Center problem for systems with two monomial nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (2) : 577-598. doi: 10.3934/cpaa.2016.15.577
[11]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834
[12]

Mitsuru Shibayama. Non-integrability of the collinear three-body problem. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 299-312. doi: 10.3934/dcds.2011.30.299
[13]

Zhong Tan, Jianfeng Zhou. Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1335-1350. doi: 10.3934/cpaa.2016.15.1335
[14]

P.K. Newton, M. Ruith, E. Upchurch. The constrained planar N-vortex problem: I. Integrability. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : 137-152. doi: 10.3934/dcdsb.2005.5.137
[15]

Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627
[16]

Balázs Boros, Josef Hofbauer, Stefan Müller, Georg Regensburger. Planar S-systems: Global stability and the center problem. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 707-727. doi: 10.3934/dcds.2019029
[17]

Roberto Castelli, Susanna Terracini. On the regularization of the collision solutions of the one-center problem with weak forces. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1197-1218. doi: 10.3934/dcds.2011.31.1197
[18]

Guowei Yu. Periodic solutions of the planar N-center problem with topological constraints. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5131-5162. doi: 10.3934/dcds.2016023
[19]

Yuika Kajihara, Misturu Shibayama. Variational proof of the existence of brake orbits in the planar 2-center problem. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5785-5797. doi: 10.3934/dcds.2019254
[20]

Min Hu, Tao Li, Xingwu Chen. Bi-center problem and Hopf cyclicity of a Cubic Liénard system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 401-414. doi: 10.3934/dcdsb.2019187

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (239)
  • HTML views (177)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy