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Non-formally integrable centers admitting an algebraic inverse integrating factor
Department of Mathematics, Faculty of Experimental Sciences, Avda. Tres de Marzo s/n, 21071 Huelva, Spain |
We study the existence of a class of inverse integrating factor for a family of non-formally integrable systems whose lowest-degree quasi-homogeneous term is a Hamiltonian vector field. Once the existence of an inverse integrating factor is established, we study the systems having a center. Among others, we characterize the centers of the perturbations of the system $ -y^3\partial_x+x^3\partial_y$ having an algebraic inverse integrating factor.
References:
[1] |
A. Algaba, E. Freire, E. Gamero and C. García,
Quasihomogeneous normal forms, J. Comput. Appl. Math., 150 (2003), 193-216.
doi: 10.1016/S0377-0427(02)00660-X. |
[2] |
A. Algaba, N. Fuentes, C. García and M. Reyes,
A class of non-integrable systems admitting an inverse integrating factor, J. Math. Anal. Appl., 420 (2014), 1439-1454.
doi: 10.1016/j.jmaa.2014.06.047. |
[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 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. |
[5] |
A. Algaba, C. García and M. Reyes,
Nilpotent systems admitting an algebraic inverse integrating factor over $ \mathbb{C}((x,y))$, Qualitative Theory of Dynamical Systems, 10 (2011), 303-316.
doi: 10.1007/s12346-011-0046-9. |
[6] |
A. Algaba, C. García and M. Reyes,
Characterization of a monodromic singular point of a planar vector field, Nonlinear Analysis, 74 (2011), 5402-5414.
doi: 10.1016/j.na.2011.05.023. |
[7] |
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. |
[8] |
A. García, C. Algaba and M. Reyes,
Like-linearizations of vector fields, Bulletin des Sciences Mathématiques, 133 (2009), 806-816.
doi: 10.1016/j.bulsci.2009.09.006. |
[9] |
M. Berthier and R. Moussu,
Réversibilité et classification des centres nilpotents, Ann. Inst. Fourier, 44 (1994), 465-494.
doi: 10.5802/aif.1406. |
[10] |
A. V. Bolsinov and A. T. Fomenko,
Integrable Hamiltonian Systems; Geometry, Topology, Classification, Chapman and Hall, 2004. |
[11] |
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. |
[12] |
J. Chavarriga, H. Giacomini, J. Giné and J. Llibre,
Darboux integrability and the inverse integrating factor, J. Differential Equations, 194 (2003), 116-139.
doi: 10.1016/S0022-0396(03)00190-6. |
[13] |
C. J. Christopher and J. Llibre,
Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential Equations, 16 (2000), 5-19.
|
[14] |
C. Christopher, P. Mardesic and C. Rousseau,
Normalizable, integrable, and linealizable saddle points for complex quadratic systems in $ \mathbb{C}^2$, J. Dyn. Control Syst., 9 (2003), 311-363.
doi: 10.1023/A:1024643521094. |
[15] |
H. R. Dullin and A. Pelayo,
Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Science, 26 (2016), 787-811.
doi: 10.1007/s00332-016-9290-0. |
[16] |
H. Eliasson,
Normal forms for Hamiltonian systems with Poisson commuting, integrals -elliptic case, Comm. Math. Helv., 65 (1990), 4-35.
doi: 10.1007/BF02566590. |
[17] |
A. Enciso and D. Peralta-Salas,
Existence and vanishing set of inverse integrating factors for analytic vector fields, Bull. London Math. Soc., 41 (2009), 1112-1124.
doi: 10.1112/blms/bdp090. |
[18] |
I. García, H. Giacomini and M. Grau,
The inverse integrating factor and the Poincaré map, Trans. Amer. Math. Soc., 362 (2010), 3591-3612.
doi: 10.1090/S0002-9947-10-05014-2. |
[19] |
I. García, H. Giacomini and M. Grau,
Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor, J. Dyn. Differ. Equat., 23 (2011), 251-281.
doi: 10.1007/s10884-011-9209-2. |
[20] |
I. García and M. Grau,
A survey on the inverse integrating factor, Qual. Theory Dyn. Sist., 9 (2010), 115-166.
doi: 10.1007/s12346-010-0023-8. |
[21] |
I. García and D. Shafer,
Integral invariants and limit sets of planar vector fields, J. Differential Equations, 217 (2005), 363-376.
doi: 10.1016/j.jde.2005.06.022. |
[22] |
A. Gasull and J. Torregrosa,
Center problem for several differential equations via Cherkas' method, J. Math. Anal. Appl., 228 (1998), 322-343.
doi: 10.1006/jmaa.1998.6112. |
[23] |
H. Giacomini, J. Llibre and M. Viano,
On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity, 9 (1996), 501-516.
doi: 10.1088/0951-7715/9/2/013. |
[24] |
J. Giné and D. Peralta-Salas,
Existence of inverse integrating factors and Lie symmetries for degenerate planar centers, J. Differential Equations, 252 (2012), 344-357.
doi: 10.1016/j.jde.2011.08.044. |
[25] |
R. E. Kooij and C. J. Christopher,
Algebraic invariant curves and the integrability of polynomial systems, Appl. Math. Lett., 6 (1993), 51-53.
doi: 10.1016/0893-9659(93)90123-5. |
[26] |
L. Mazzi and M. Sabatini,
A characterization of centers via first integrals, J. Differential Equations, 76 (1988), 222-237.
doi: 10.1016/0022-0396(88)90072-1. |
[27] |
R. Moussu,
Symétrie et forme normaledes centres et foyers dégénérés, Ergodic Theory Dynam. Sys., 2 (1982), 241-251.
|
[28] |
H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math., 37 (1881), 375-422. Google Scholar |
[29] |
M. J. Prelle and M. F. Singer,
Elementary first integrals of of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229.
doi: 10.1090/S0002-9947-1983-0704611-X. |
[30] |
A. P. Sadovskii,
Problem of distinguishing a center and a focus for a system with a nonvanishing linear part, Diff. Urav., 12 (1976), 1238-1246 (in Russian).
|
[31] |
S. Walcher,
On the Poincaré problem, J. Differential Equations, 166 (2000), 51-78.
doi: 10.1006/jdeq.2000.3801. |
[32] |
S. Walcher,
Local integrating factors, J. Lie Theory, 13 (2003), 279-289.
|
show all references
References:
[1] |
A. Algaba, E. Freire, E. Gamero and C. García,
Quasihomogeneous normal forms, J. Comput. Appl. Math., 150 (2003), 193-216.
doi: 10.1016/S0377-0427(02)00660-X. |
[2] |
A. Algaba, N. Fuentes, C. García and M. Reyes,
A class of non-integrable systems admitting an inverse integrating factor, J. Math. Anal. Appl., 420 (2014), 1439-1454.
doi: 10.1016/j.jmaa.2014.06.047. |
[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 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. |
[5] |
A. Algaba, C. García and M. Reyes,
Nilpotent systems admitting an algebraic inverse integrating factor over $ \mathbb{C}((x,y))$, Qualitative Theory of Dynamical Systems, 10 (2011), 303-316.
doi: 10.1007/s12346-011-0046-9. |
[6] |
A. Algaba, C. García and M. Reyes,
Characterization of a monodromic singular point of a planar vector field, Nonlinear Analysis, 74 (2011), 5402-5414.
doi: 10.1016/j.na.2011.05.023. |
[7] |
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. |
[8] |
A. García, C. Algaba and M. Reyes,
Like-linearizations of vector fields, Bulletin des Sciences Mathématiques, 133 (2009), 806-816.
doi: 10.1016/j.bulsci.2009.09.006. |
[9] |
M. Berthier and R. Moussu,
Réversibilité et classification des centres nilpotents, Ann. Inst. Fourier, 44 (1994), 465-494.
doi: 10.5802/aif.1406. |
[10] |
A. V. Bolsinov and A. T. Fomenko,
Integrable Hamiltonian Systems; Geometry, Topology, Classification, Chapman and Hall, 2004. |
[11] |
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. |
[12] |
J. Chavarriga, H. Giacomini, J. Giné and J. Llibre,
Darboux integrability and the inverse integrating factor, J. Differential Equations, 194 (2003), 116-139.
doi: 10.1016/S0022-0396(03)00190-6. |
[13] |
C. J. Christopher and J. Llibre,
Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential Equations, 16 (2000), 5-19.
|
[14] |
C. Christopher, P. Mardesic and C. Rousseau,
Normalizable, integrable, and linealizable saddle points for complex quadratic systems in $ \mathbb{C}^2$, J. Dyn. Control Syst., 9 (2003), 311-363.
doi: 10.1023/A:1024643521094. |
[15] |
H. R. Dullin and A. Pelayo,
Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Science, 26 (2016), 787-811.
doi: 10.1007/s00332-016-9290-0. |
[16] |
H. Eliasson,
Normal forms for Hamiltonian systems with Poisson commuting, integrals -elliptic case, Comm. Math. Helv., 65 (1990), 4-35.
doi: 10.1007/BF02566590. |
[17] |
A. Enciso and D. Peralta-Salas,
Existence and vanishing set of inverse integrating factors for analytic vector fields, Bull. London Math. Soc., 41 (2009), 1112-1124.
doi: 10.1112/blms/bdp090. |
[18] |
I. García, H. Giacomini and M. Grau,
The inverse integrating factor and the Poincaré map, Trans. Amer. Math. Soc., 362 (2010), 3591-3612.
doi: 10.1090/S0002-9947-10-05014-2. |
[19] |
I. García, H. Giacomini and M. Grau,
Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor, J. Dyn. Differ. Equat., 23 (2011), 251-281.
doi: 10.1007/s10884-011-9209-2. |
[20] |
I. García and M. Grau,
A survey on the inverse integrating factor, Qual. Theory Dyn. Sist., 9 (2010), 115-166.
doi: 10.1007/s12346-010-0023-8. |
[21] |
I. García and D. Shafer,
Integral invariants and limit sets of planar vector fields, J. Differential Equations, 217 (2005), 363-376.
doi: 10.1016/j.jde.2005.06.022. |
[22] |
A. Gasull and J. Torregrosa,
Center problem for several differential equations via Cherkas' method, J. Math. Anal. Appl., 228 (1998), 322-343.
doi: 10.1006/jmaa.1998.6112. |
[23] |
H. Giacomini, J. Llibre and M. Viano,
On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity, 9 (1996), 501-516.
doi: 10.1088/0951-7715/9/2/013. |
[24] |
J. Giné and D. Peralta-Salas,
Existence of inverse integrating factors and Lie symmetries for degenerate planar centers, J. Differential Equations, 252 (2012), 344-357.
doi: 10.1016/j.jde.2011.08.044. |
[25] |
R. E. Kooij and C. J. Christopher,
Algebraic invariant curves and the integrability of polynomial systems, Appl. Math. Lett., 6 (1993), 51-53.
doi: 10.1016/0893-9659(93)90123-5. |
[26] |
L. Mazzi and M. Sabatini,
A characterization of centers via first integrals, J. Differential Equations, 76 (1988), 222-237.
doi: 10.1016/0022-0396(88)90072-1. |
[27] |
R. Moussu,
Symétrie et forme normaledes centres et foyers dégénérés, Ergodic Theory Dynam. Sys., 2 (1982), 241-251.
|
[28] |
H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math., 37 (1881), 375-422. Google Scholar |
[29] |
M. J. Prelle and M. F. Singer,
Elementary first integrals of of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229.
doi: 10.1090/S0002-9947-1983-0704611-X. |
[30] |
A. P. Sadovskii,
Problem of distinguishing a center and a focus for a system with a nonvanishing linear part, Diff. Urav., 12 (1976), 1238-1246 (in Russian).
|
[31] |
S. Walcher,
On the Poincaré problem, J. Differential Equations, 166 (2000), 51-78.
doi: 10.1006/jdeq.2000.3801. |
[32] |
S. Walcher,
Local integrating factors, J. Lie Theory, 13 (2003), 279-289.
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