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Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion
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 |
Recently, in [
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. |
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