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 & Continuous Dynamical Systems, 2016, 36 (11) : 6539-6555. doi: 10.3934/dcds.2016083
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show all references

References:
[1]

The Benjamin/Cummings Pub., Massachusetts, 1978. Available from: http://zh.bookzz.org/book/458355/4ddae5.  Google Scholar

[2]

Springer-Verlag, New York, 1989. Available from: http://zh.bookzz.org/book/1132652/9217d1 doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

Rend. Circ. Mat. Palermo, LII (2003), 77-130. doi: 10.1007/BF02871926.  Google Scholar

[4]

J. Differential Equations, 252 (2012), 6324-6336. doi: 10.1016/j.jde.2012.03.009.  Google Scholar

[5]

J. Differential Equations, 256 (2014), 310-325. doi: 10.1016/j.jde.2013.09.006.  Google Scholar

[6]

Birkhäuser Boston, Inc., Boston, MA, 1985. Available from: http://zh.bookzz.org/book/837980/cd73ca doi: 10.1007/978-1-4612-5292-4.  Google Scholar

[7]

Bull. London Math. Soc., 41 (2009), 1112-1124. doi: 10.1112/blms/bdp090.  Google Scholar

[8]

Trans. Amer. Math. Soc., 362 (2010), 3591-3612. doi: 10.1090/S0002-9947-10-05014-2.  Google Scholar

[9]

J. Dynam. Differential Equations, 23 (2011), 251-281. doi: 10.1007/s10884-011-9209-2.  Google Scholar

[10]

J. Differential Equations, 248 (2010), 363-380. doi: 10.1016/j.jde.2009.09.002.  Google Scholar

[11]

Qual. Theory Dyn. Syst., 9 (2010), 115-166. doi: 10.1007/s12346-010-0023-8.  Google Scholar

[12]

Nonlinearity, 9 (1996), 501-516. doi: 10.1088/0951-7715/9/2/013.  Google Scholar

[13]

Z. Angew. Math. Phys., 55 (2004), 725-740. doi: 10.1007/s00033-004-1093-8.  Google Scholar

[14]

Discrete Contin. Dyn. Syst., 33 (2013), 4531-4547. doi: 10.3934/dcds.2013.33.4531.  Google Scholar

[15]

J. Differential Equations, 252 (2012), 344-357. doi: 10.1016/j.jde.2011.08.044.  Google Scholar

[16]

World Scientific, New Jersey, 2001. Available from: http://zh.bookzz.org/book/503982/0ffa42 doi: 10.1142/9789812811943.  Google Scholar

[17]

American Mathematical Society, 2000.  Google Scholar

[18]

American Mathematical Society, Providence, RI, 2008. Available from: http://zh.bookzz.org/book/441977/381f05  Google Scholar

[19]

Physica D, 241 (2012), 1417-1420. doi: 10.1016/j.physd.2012.05.003.  Google Scholar

[20]

Ergodic Theory Dynam. Systems, 31 (2011), 245-258. doi: 10.1017/S0143385709000868.  Google Scholar

[21]

North-Holland Mathematical Library 35, North-Holland Publishing Co., Amsterdam, 1985. Available from: http://zh.bookzz.org/book/574089/51f58e  Google Scholar

[22]

Springer-Verlag, New York, 1993. Available from: http://zh.bookzz.org/book/446638/759fc9 doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[23]

J. Differential Equations, 244 (2008), 1287-1303. doi: 10.1016/j.jde.2008.01.002.  Google Scholar

[24]

J. Differential Equations, 246 (2009), 3750-3753. doi: 10.1016/j.jde.2009.02.009.  Google Scholar

[25]

Physics Letter A, 373 (2009), 2445-2453. doi: 10.1016/j.physleta.2009.04.075.  Google Scholar

[26]

Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161-191; 11 (1897), 193-239. Google Scholar

[27]

Trans. Amer. Math. Soc., 279 (1983), 215-229. doi: 10.1090/S0002-9947-1983-0704611-X.  Google Scholar

[28]

Discrete Contin. Dyn. Syst., 33 (2013), 1645-1655. doi: 10.3934/dcds.2013.33.1645.  Google Scholar

[29]

Trans. Amer. Math. Soc., 333 (1992), 673-688. doi: 10.1090/S0002-9947-1992-1062869-X.  Google Scholar

[30]

Physica D, 250 (2013), 47-51. doi: 10.1016/j.physd.2013.01.011.  Google Scholar

[31]

J. Differential Equations, 254 (2013), 3000-3022. doi: 10.1016/j.jde.2013.01.016.  Google Scholar

[32]

J. Differential Equations, 256 (2014), 3278-3299. doi: 10.1016/j.jde.2014.02.002.  Google Scholar

[33]

Trans. Amer. Math. Soc., 368 (2016), no. 1, 607-620. doi: 10.1090/S0002-9947-2014-06387-3.  Google Scholar

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