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March  2018, 38(3): 967-988. doi: 10.3934/dcds.2018041

Non-formally integrable centers admitting an algebraic inverse integrating factor

Antonio Algaba , Natalia Fuentes , Cristóbal García and  Manuel Reyes ,

Department of Mathematics, Faculty of Experimental Sciences, Avda. Tres de Marzo s/n, 21071 Huelva, Spain

* Corresponding author: Manuel Reyes

Received  April 2016 Revised  October 2017 Published  December 2017

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.

Keywords: Nonlinear differential systems, integrability, inverse integrating factor, centers, normal form.
Mathematics Subject Classification: 34C05, 34C14, 34C20.
Citation: Antonio Algaba, Natalia Fuentes, Cristóbal García, Manuel Reyes. Non-formally integrable centers admitting an algebraic inverse integrating factor. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 967-988. doi: 10.3934/dcds.2018041
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. 

[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. 

[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. 

Table 1.  Range and co-range of operator $\ell_{j} $ for the system (8).
Range($\ell_{2}$)=span{$-bx^2-3y^2,3ax^2+2bxy$}.
If $a\ne 0,$ Cor($\ell_{2}$)=span{$xy$}. If $a=0$, Cor($\ell_{2}$)=span{$x^2$}.
Range($\ell_{3}$)=span{$-2bx^3-6xy^2,6ax^3+4bx^2y-3h,6ax^2y+4bxy^2$}.
Cor($\ell_{3}$)=span{$h$}.
Range($\ell_{4}$)=span{$3bx^4+9x^2y^2,-9ax^4-6bx^3y+6xh,$ $ -9ax^3y-6bx^2y^2+3yh$}.
Cor($\ell_{4}$)=span{$xh,yh$}.
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Range($\ell_{2}$)=span{$-bx^2-3y^2,3ax^2+2bxy$}.
If $a\ne 0,$ Cor($\ell_{2}$)=span{$xy$}. If $a=0$, Cor($\ell_{2}$)=span{$x^2$}.
Range($\ell_{3}$)=span{$-2bx^3-6xy^2,6ax^3+4bx^2y-3h,6ax^2y+4bxy^2$}.
Cor($\ell_{3}$)=span{$h$}.
Range($\ell_{4}$)=span{$3bx^4+9x^2y^2,-9ax^4-6bx^3y+6xh,$ $ -9ax^3y-6bx^2y^2+3yh$}.
Cor($\ell_{4}$)=span{$xh,yh$}.
Table 2.  Range and co-range of operator $\ell_{j} $ for the system (15)
Range($\ell_{6}$)=span{$0$}, Cor($\ell_{6}$)=span{$x^2$}
Range($\ell_{7}$)=span{$0$}, Cor($\ell_{7}$)=span{$xy$}
Range($\ell_{8}$)=span{$y^2$}, Cor($\ell_{8}$)=span{$0$}
Range($\ell_{9}$)=span{$x^3$}, Cor($\ell_{9}$)=span{$0$}
Range($\ell_{10}$)=span{$0$}, Cor($\ell_{10}$)=span{$x^2y$}
Range($\ell_{11}$)=span{$xy^2$}, Cor($\ell_{11}$)=span{$0$}
Range($\ell_{12}$)=span{$7x^4-12h$}, Cor($\ell_{12}$)=span{$h$}
Range($\ell_{13}$)=span{$x^3y$}, Cor($\ell_{13}$)={$0$}
Range($\ell_{14}$)=span{$x^2y^2$}, Cor($\ell_{14}$)={$0$}
Range($\ell_{15}$)=span{$x^3-6xh$}, Cor($\ell_{15}$)=span{$xh$}
Range($\ell_{16}$)=span{$11x^4y-12yh$}, Cor($\ell_{16}$)=span{$yh$}
Range($\ell_{17}$)=span{$x^3y^2$}, Cor($\ell_{17}$)={$0$}
Range($\ell_{18}$)=span{$13x^6-36x^2h$}, Cor($\ell_{18}$)=span{$x^2h$}
Range($\ell_{19}$)=span{$7x^5-12xyh$}, Cor($\ell_{19}$)=span{$xyh$}
Range($\ell_{22}$)=span{$17x^6y-9x^2yh$}, Cor($\ell_{22}$)=span{$x^2yh$}
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Range($\ell_{6}$)=span{$0$}, Cor($\ell_{6}$)=span{$x^2$}
Range($\ell_{7}$)=span{$0$}, Cor($\ell_{7}$)=span{$xy$}
Range($\ell_{8}$)=span{$y^2$}, Cor($\ell_{8}$)=span{$0$}
Range($\ell_{9}$)=span{$x^3$}, Cor($\ell_{9}$)=span{$0$}
Range($\ell_{10}$)=span{$0$}, Cor($\ell_{10}$)=span{$x^2y$}
Range($\ell_{11}$)=span{$xy^2$}, Cor($\ell_{11}$)=span{$0$}
Range($\ell_{12}$)=span{$7x^4-12h$}, Cor($\ell_{12}$)=span{$h$}
Range($\ell_{13}$)=span{$x^3y$}, Cor($\ell_{13}$)={$0$}
Range($\ell_{14}$)=span{$x^2y^2$}, Cor($\ell_{14}$)={$0$}
Range($\ell_{15}$)=span{$x^3-6xh$}, Cor($\ell_{15}$)=span{$xh$}
Range($\ell_{16}$)=span{$11x^4y-12yh$}, Cor($\ell_{16}$)=span{$yh$}
Range($\ell_{17}$)=span{$x^3y^2$}, Cor($\ell_{17}$)={$0$}
Range($\ell_{18}$)=span{$13x^6-36x^2h$}, Cor($\ell_{18}$)=span{$x^2h$}
Range($\ell_{19}$)=span{$7x^5-12xyh$}, Cor($\ell_{19}$)=span{$xyh$}
Range($\ell_{22}$)=span{$17x^6y-9x^2yh$}, Cor($\ell_{22}$)=span{$x^2yh$}
Table 3.  Range and co-range of operator $\ell_{j} $ for the system (21).
Range($\ell_{3}$)=span{$x^3,y^3$}.
Cor($\ell_{3}$)=span{$x^2y,xy^2$}.
Range($\ell_{4}$)=span{$xy^3,x^4+2h,x^3y$}.
Cor($\ell_{4}$)=span{$x^2y^2,h$}.
Range($\ell_{5}$)=span{$x^2y^3,3x^5+8xh,3x^4y+4yh,x^3y^2$}.
Cor($\ell_{5}$)=span{$xh,yh$}.
Range($\ell_{6}$)=span{$x^3y^3,x^6+3x^2h,x^5y+2xyh, x^4y^2+y^2h$}.
Cor($\ell_{6}$)=span{$x^2h,xyh,y^2h$}.
Table Options

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Range($\ell_{3}$)=span{$x^3,y^3$}.
Cor($\ell_{3}$)=span{$x^2y,xy^2$}.
Range($\ell_{4}$)=span{$xy^3,x^4+2h,x^3y$}.
Cor($\ell_{4}$)=span{$x^2y^2,h$}.
Range($\ell_{5}$)=span{$x^2y^3,3x^5+8xh,3x^4y+4yh,x^3y^2$}.
Cor($\ell_{5}$)=span{$xh,yh$}.
Range($\ell_{6}$)=span{$x^3y^3,x^6+3x^2h,x^5y+2xyh, x^4y^2+y^2h$}.
Cor($\ell_{6}$)=span{$x^2h,xyh,y^2h$}.
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