November  2016, 36(11): 6539-6555. doi: 10.3934/dcds.2016083

Integrability of vector fields versus inverse Jacobian multipliers and normalizers

1. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

2. 

Department of Mathematics and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240

Received  April 2015 Revised  May 2016 Published  August 2016

In this paper we provide characterization of integrablity of a system of vector fields via inverse Jacobian multipliers (matrix) and normalizers of smooth (or holomorphic) vector fields. These results improve and extend some well known ones, including the classical holomorphic Frobenius integrability theorem. Here we obtain the exact expression of an integrable system of vector fields acting on a smooth function via their known common first integrals. Moreover we characterize the relations between the integrability and the existence of normalizers for a system of vector fields. In the case of integrability of a system of vector fields we not only prove the existence of normalizers but also provide their exact expressions.
Citation: Shiliang Weng, Xiang Zhang. Integrability of vector fields versus inverse Jacobian multipliers and normalizers. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6539-6555. doi: 10.3934/dcds.2016083
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechnics, The Benjamin/Cummings Pub., Massachusetts, 1978. Available from: http://zh.bookzz.org/book/458355/4ddae5.

[2]

V. I. Arnold, Mathmatical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. Available from: http://zh.bookzz.org/book/1132652/9217d1 doi: 10.1007/978-1-4757-2063-1.

[3]

L. R. Berrone and H. Giacomini, Inverse Jacobian multipliers, Rend. Circ. Mat. Palermo, LII (2003), 77-130. doi: 10.1007/BF02871926.

[4]

A. Buică, I. A. García and S. Maza, Existence of inverse Jacobian multipliers around Hopf points in $\mathbb R^3$: Emphasis on the center problem, J. Differential Equations, 252 (2012), 6324-6336. doi: 10.1016/j.jde.2012.03.009.

[5]

A. Buică, I. A. García and S. Maza, Multiple Hopf bifurcation in $\mathbb R^3$ and inverse Jacobi multipliers, J. Differential Equations, 256 (2014), 310-325. doi: 10.1016/j.jde.2013.09.006.

[6]

C. Camacho and A. Lins Neto, Geometric theory of foliations, translated from the Portuguese by Sue E. Goodman, Birkhäuser Boston, Inc., Boston, MA, 1985. Available from: http://zh.bookzz.org/book/837980/cd73ca doi: 10.1007/978-1-4612-5292-4.

[7]

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.

[8]

I. A. 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.

[9]

I. A. García, H. Giacomini and M. Grau, Generalized Hopf bifurcation for planar vector fields via the inverse integrating factor, J. Dynam. Differential Equations, 23 (2011), 251-281. doi: 10.1007/s10884-011-9209-2.

[10]

I. A. García and S. Maza, A new approach to center conditions for simple analytic monodromic singularities, J. Differential Equations, 248 (2010), 363-380. doi: 10.1016/j.jde.2009.09.002.

[11]

I. A. García and M. Grau, A survey on the inverse integrating factor, Qual. Theory Dyn. Syst., 9 (2010), 115-166. doi: 10.1007/s12346-010-0023-8.

[12]

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.

[13]

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.

[14]

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.

[15]

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.

[16]

A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific, New Jersey, 2001. Available from: http://zh.bookzz.org/book/503982/0ffa42 doi: 10.1142/9789812811943.

[17]

J. Harnad, P. Winternitz and G. Sabidussi, eds., Integrable Systems: From Classical to Quantum, American Mathematical Society, 2000.

[18]

Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, American Mathematical Society, Providence, RI, 2008. Available from: http://zh.bookzz.org/book/441977/381f05

[19]

J. Llibre and C. Valls, On the polynomial integrability of the Kirchoff equations, Physica D, 241 (2012), 1417-1420. doi: 10.1016/j.physd.2012.05.003.

[20]

J. Llibre and C. Valls, Analytic integrability of quadratic-linear polynomial differential systems, Ergodic Theory Dynam. Systems, 31 (2011), 245-258. doi: 10.1017/S0143385709000868.

[21]

R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland Mathematical Library 35, North-Holland Publishing Co., Amsterdam, 1985. Available from: http://zh.bookzz.org/book/574089/51f58e

[22]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 107, Springer-Verlag, New York, 1993. Available from: http://zh.bookzz.org/book/446638/759fc9 doi: 10.1007/978-1-4612-4350-2.

[23]

D. Peralta-Salas, Period function and normalizers of vector fields in $\mathbb R^n$ with $n-1$ first integrals, J. Differential Equations, 244 (2008), 1287-1303. doi: 10.1016/j.jde.2008.01.002.

[24]

G. E. Prince, Comment on "Period function and normalizers of vector fields in $\mathbb R^n$ with $n-1$ first integrals" by D.Peralta-Salas [J.Differential Equations 244(6)(2008) 1287-1303], J. Differential Equations, 246 (2009), 3750-3753. doi: 10.1016/j.jde.2009.02.009.

[25]

S. I. Popov, W. Respondek and J.-M. Strelcyn, On rational integrability of Euler equations on Lie algebra $so(4,\mathbb C)$, revisited, Physics Letter A, 373 (2009), 2445-2453. doi: 10.1016/j.physleta.2009.04.075.

[26]

H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I and II, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161-191; 11 (1897), 193-239.

[27]

M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229. doi: 10.1090/S0002-9947-1983-0704611-X.

[28]

S. Shi and W. Li, Non-integrability of generalized Yang-Mills Hamiltonian system, Discrete Contin. Dyn. Syst., 33 (2013), 1645-1655. doi: 10.3934/dcds.2013.33.1645.

[29]

M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333 (1992), 673-688. doi: 10.1090/S0002-9947-1992-1062869-X.

[30]

X. Zhang, Comment on "On the polynomial integrability of the Kirchoff equations, Physica D 241 (2012) 1417-1420", Physica D, 250 (2013), 47-51. doi: 10.1016/j.physd.2013.01.011.

[31]

X. Zhang, Analytic integrable systems: Analytic normalization and embedding flows, J. Differential Equations, 254 (2013), 3000-3022. doi: 10.1016/j.jde.2013.01.016.

[32]

X. Zhang, Inverse Jacobian multipliers and Hopf bifurcation on center manifolds, J. Differential Equations, 256 (2014), 3278-3299. doi: 10.1016/j.jde.2014.02.002.

[33]

X. Zhang, Liouvillian integrability of polynomial differential systems, Trans. Amer. Math. Soc., 368 (2016), no. 1, 607-620. doi: 10.1090/S0002-9947-2014-06387-3.

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechnics, The Benjamin/Cummings Pub., Massachusetts, 1978. Available from: http://zh.bookzz.org/book/458355/4ddae5.

[2]

V. I. Arnold, Mathmatical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. Available from: http://zh.bookzz.org/book/1132652/9217d1 doi: 10.1007/978-1-4757-2063-1.

[3]

L. R. Berrone and H. Giacomini, Inverse Jacobian multipliers, Rend. Circ. Mat. Palermo, LII (2003), 77-130. doi: 10.1007/BF02871926.

[4]

A. Buică, I. A. García and S. Maza, Existence of inverse Jacobian multipliers around Hopf points in $\mathbb R^3$: Emphasis on the center problem, J. Differential Equations, 252 (2012), 6324-6336. doi: 10.1016/j.jde.2012.03.009.

[5]

A. Buică, I. A. García and S. Maza, Multiple Hopf bifurcation in $\mathbb R^3$ and inverse Jacobi multipliers, J. Differential Equations, 256 (2014), 310-325. doi: 10.1016/j.jde.2013.09.006.

[6]

C. Camacho and A. Lins Neto, Geometric theory of foliations, translated from the Portuguese by Sue E. Goodman, Birkhäuser Boston, Inc., Boston, MA, 1985. Available from: http://zh.bookzz.org/book/837980/cd73ca doi: 10.1007/978-1-4612-5292-4.

[7]

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.

[8]

I. A. 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.

[9]

I. A. García, H. Giacomini and M. Grau, Generalized Hopf bifurcation for planar vector fields via the inverse integrating factor, J. Dynam. Differential Equations, 23 (2011), 251-281. doi: 10.1007/s10884-011-9209-2.

[10]

I. A. García and S. Maza, A new approach to center conditions for simple analytic monodromic singularities, J. Differential Equations, 248 (2010), 363-380. doi: 10.1016/j.jde.2009.09.002.

[11]

I. A. García and M. Grau, A survey on the inverse integrating factor, Qual. Theory Dyn. Syst., 9 (2010), 115-166. doi: 10.1007/s12346-010-0023-8.

[12]

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.

[13]

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.

[14]

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.

[15]

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.

[16]

A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific, New Jersey, 2001. Available from: http://zh.bookzz.org/book/503982/0ffa42 doi: 10.1142/9789812811943.

[17]

J. Harnad, P. Winternitz and G. Sabidussi, eds., Integrable Systems: From Classical to Quantum, American Mathematical Society, 2000.

[18]

Y. Ilyashenko and S. Yakovenko, Lectures on Analytic Differential Equations, American Mathematical Society, Providence, RI, 2008. Available from: http://zh.bookzz.org/book/441977/381f05

[19]

J. Llibre and C. Valls, On the polynomial integrability of the Kirchoff equations, Physica D, 241 (2012), 1417-1420. doi: 10.1016/j.physd.2012.05.003.

[20]

J. Llibre and C. Valls, Analytic integrability of quadratic-linear polynomial differential systems, Ergodic Theory Dynam. Systems, 31 (2011), 245-258. doi: 10.1017/S0143385709000868.

[21]

R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland Mathematical Library 35, North-Holland Publishing Co., Amsterdam, 1985. Available from: http://zh.bookzz.org/book/574089/51f58e

[22]

P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics 107, Springer-Verlag, New York, 1993. Available from: http://zh.bookzz.org/book/446638/759fc9 doi: 10.1007/978-1-4612-4350-2.

[23]

D. Peralta-Salas, Period function and normalizers of vector fields in $\mathbb R^n$ with $n-1$ first integrals, J. Differential Equations, 244 (2008), 1287-1303. doi: 10.1016/j.jde.2008.01.002.

[24]

G. E. Prince, Comment on "Period function and normalizers of vector fields in $\mathbb R^n$ with $n-1$ first integrals" by D.Peralta-Salas [J.Differential Equations 244(6)(2008) 1287-1303], J. Differential Equations, 246 (2009), 3750-3753. doi: 10.1016/j.jde.2009.02.009.

[25]

S. I. Popov, W. Respondek and J.-M. Strelcyn, On rational integrability of Euler equations on Lie algebra $so(4,\mathbb C)$, revisited, Physics Letter A, 373 (2009), 2445-2453. doi: 10.1016/j.physleta.2009.04.075.

[26]

H. Poincaré, Sur l'intégration des équations différentielles du premier ordre et du premier degré I and II, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161-191; 11 (1897), 193-239.

[27]

M. J. Prelle and M. F. Singer, Elementary first integrals of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229. doi: 10.1090/S0002-9947-1983-0704611-X.

[28]

S. Shi and W. Li, Non-integrability of generalized Yang-Mills Hamiltonian system, Discrete Contin. Dyn. Syst., 33 (2013), 1645-1655. doi: 10.3934/dcds.2013.33.1645.

[29]

M. F. Singer, Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc., 333 (1992), 673-688. doi: 10.1090/S0002-9947-1992-1062869-X.

[30]

X. Zhang, Comment on "On the polynomial integrability of the Kirchoff equations, Physica D 241 (2012) 1417-1420", Physica D, 250 (2013), 47-51. doi: 10.1016/j.physd.2013.01.011.

[31]

X. Zhang, Analytic integrable systems: Analytic normalization and embedding flows, J. Differential Equations, 254 (2013), 3000-3022. doi: 10.1016/j.jde.2013.01.016.

[32]

X. Zhang, Inverse Jacobian multipliers and Hopf bifurcation on center manifolds, J. Differential Equations, 256 (2014), 3278-3299. doi: 10.1016/j.jde.2014.02.002.

[33]

X. Zhang, Liouvillian integrability of polynomial differential systems, Trans. Amer. Math. Soc., 368 (2016), no. 1, 607-620. doi: 10.1090/S0002-9947-2014-06387-3.

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