Non-integrability of ABC flow

Abstract We investigate the ABC flow when A 2 = B 2 , and we prove that this system does not admit a meromorphic first integral when ABC ≠0. Moreover, for C 2 / A 2

[1]  P. Kenneth Seidelmann,et al.  Dynamics of Natural and Artificial Celestial Bodies , 2001 .

[2]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[3]  Jean-François Pommaret.  Differential Galois Theory , 1983 .

[4]  M. Audin.  Les systèmes hamiltoniens et leur intégrabilité, Cours Spécialisés, ,. , 2001 .

[5]  S. L. Ziglin.  Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics. I , 1982 .

[6]  Xiaohua Zhao,et al.  Chaotic and Resonant Streamlines in the ABC Flow , 1993, SIAM J. Appl. Math..

[7]  Andy R. Magid,et al.  Lectures on differential Galois theory , 1994 .

[8]  Jerald J. Kovacic,et al.  An Algorithm for Solving Second Order Linear Homogeneous Differential Equations , 1986, J. Symb. Comput..

[9]  Uriel Frisch,et al.  Chaotic streamlines in the ABC flows , 1986, Journal of Fluid Mechanics.

[10]  S. L. Ziglin.  The absence of an additional real-analytic first integral in some problems of dynamics , 1997 .

[11]  G. Beck,et al.  Actualités scientifiques et industrielles , 1933 .

[12]  M. Audin.  Les systèmes hamiltoniens et leur intégrabilité , 2001 .

[14]  I. Kaplansky.  An introduction to differential algebra , 1957 .

[15]  J. I. Ramos.  Computer Algebra and Differential Equations : E. Tournier, editor Academic Press, New York, 1991, £35 , 1993 .

[16]  Juan José Morales Ruiz,et al.  Differential Galois Theory and Non-Integrability of Hamiltonian Systems , 1999 .

[17]  S. L. Ziglin.  On the absence of a real-analytic first integral for ABC flow when A=B. , 1998, Chaos.

[18]  TheABC-flow is not integrable forA=B , 1996 .

[19]  The Restless Universe , 1935 .