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On the Hamiltonian dynamics and geometry of the Rabinovich system
1. | The West University of Timişoara, Faculty of Mathematics and C.S., Department of Mathematics, B-dul. Vasile Pârvan, No. 4, 300223 - Timişoara, Romania |
2. | "Politehnica" University of Timişoara, Department of Mathematics, Piaţa Victoriei nr. 2, 300006 - Timişoara, Romania |
References:
[1] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics. Second Edition," Graduate Texts in Mathematics, Vol.60, Springer, 1989. |
[2] |
V. I. Arnold, On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid, Dokl. Akad. Nauk SSSR, 162 (1965), 975-978. |
[3] |
A. Ay, M. Gurses and K. Zheltukhin, Hamiltonian equations on $mathbb{R}^{3}$, J. Math. Phys., 44 (2003), 5688-5705.
doi: 10.1063/1.1619204. |
[4] |
O. Babelon, D. Bernard and M. Talon, "Introduction to Classical Integrable Systems," Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2003. |
[5] |
T. C. Bountis, A. Ramani, B. Grammaticos and B. Dorizzi, On the complete and partial integrability of non-Hamiltonian systems, Physica A, 128 (1984), 268-288.
doi: 10.1016/0378-4371(84)90091-8. |
[6] |
C. Chen, J. Cao and X. Zhang, The topological structure of the Rabinovich system having an invariant algebraic surface, Nonlinearity, 21 (2008), 211-220.
doi: 10.1088/0951-7715/21/2/002. |
[7] |
O. Chis and M. Puta, The dynamics of the Rabinovich system, preprint, (2007), 1-15. arXiv:0710.4583 |
[8] |
O. Chis and M. Puta, Geometrical and dynamical aspects in the theory of Rabinovich system, Int. J. Geom. Methods Mod. Phys., 5 (2008), 521-535.
doi: 10.1142/S0219887808002916. |
[9] |
R. H. Cushman and L. Bates, "Global Aspects Of Classical Integrable Systems," Basel: Birkhauser, 1977. |
[10] |
D. D. Holm, J. E. Marsden, T. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Physics Reports, 123 (1985), 1-116.
doi: 10.1016/0370-1573(85)90028-6. |
[11] |
J. Goedert, F. Haas, D. Hua, M. R. Feix and L. Cairo, Generalized Hamiltonian structures for systems in three dimensions with a rescalable constant of motion, J. Phys. A, 27 (1994), 6495.
doi: 10.1088/0305-4470/27/19/020. |
[12] |
B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics," Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2004. |
[13] |
J. Llibre, M. Messias and P. R. da Silva, On the global dynamics of the Rabinovich system, J. Phys. A, 41 (2008), 275210, 21 pp. |
[14] |
J. Llibre and C. Valls, Global analytic integrability of the Rabinovich system, J. Geom. Phys., 58 (2008), 1762-1771.
doi: 10.1016/j.geomphys.2008.08.009. |
[15] |
J. E. Marsden, "Lectures on Mechanics," London Mathematical Society Lecture Notes Series, vol. 174, Cambridge University Press, 1992. |
[16] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, vol. 17, Springer, Berlin, 1994. |
[17] |
M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces," Lecture Notes in Mathematics, vol. 510, Springer, Berlin, 2001. |
[18] |
A. S. Pikovskii and M. I. Rabinovich, Stochastic behavior of dissipative systems, Soc. Sci. Rev. C: Math. Phys. Rev., 2 (1981), 165-208. |
[19] |
A. S. Pikovskii, M. I. Rabinovich and V. Yu. Trakhtengerts, Onset of stochasticity in decay confinement of parametric instability, Soc. Phys. JETP, 47 (1978), 715-719. |
[20] |
T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra, Chapter II: A crash course in geometric mechanics, in "Geometric Mechanics and Symmetry: The Peyresq Lectures," London Mathematical Society Lecture Notes Series, vol. 306, Cambridge University Press, (2005), 23-156. |
[21] |
A. Weinstein, Normal modes for non-linear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.
doi: 10.1007/BF01405263. |
[22] |
F. Xie and X. Zhang, Invariant algebraic surfaces of the Rabinovich system, J. Phys. A, 36 (2003), 499-516.
doi: 10.1088/0305-4470/36/2/314. |
[23] |
X. Zhang, Integrals of motion of the Rabinovich system, J. Phys. A, 33 (2000), 5137-5155.
doi: 10.1088/0305-4470/33/28/315. |
show all references
References:
[1] |
V. I. Arnold, "Mathematical Methods of Classical Mechanics. Second Edition," Graduate Texts in Mathematics, Vol.60, Springer, 1989. |
[2] |
V. I. Arnold, On conditions for non-linear stability of plane stationary curvilinear flows of an ideal fluid, Dokl. Akad. Nauk SSSR, 162 (1965), 975-978. |
[3] |
A. Ay, M. Gurses and K. Zheltukhin, Hamiltonian equations on $mathbb{R}^{3}$, J. Math. Phys., 44 (2003), 5688-5705.
doi: 10.1063/1.1619204. |
[4] |
O. Babelon, D. Bernard and M. Talon, "Introduction to Classical Integrable Systems," Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2003. |
[5] |
T. C. Bountis, A. Ramani, B. Grammaticos and B. Dorizzi, On the complete and partial integrability of non-Hamiltonian systems, Physica A, 128 (1984), 268-288.
doi: 10.1016/0378-4371(84)90091-8. |
[6] |
C. Chen, J. Cao and X. Zhang, The topological structure of the Rabinovich system having an invariant algebraic surface, Nonlinearity, 21 (2008), 211-220.
doi: 10.1088/0951-7715/21/2/002. |
[7] |
O. Chis and M. Puta, The dynamics of the Rabinovich system, preprint, (2007), 1-15. arXiv:0710.4583 |
[8] |
O. Chis and M. Puta, Geometrical and dynamical aspects in the theory of Rabinovich system, Int. J. Geom. Methods Mod. Phys., 5 (2008), 521-535.
doi: 10.1142/S0219887808002916. |
[9] |
R. H. Cushman and L. Bates, "Global Aspects Of Classical Integrable Systems," Basel: Birkhauser, 1977. |
[10] |
D. D. Holm, J. E. Marsden, T. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Physics Reports, 123 (1985), 1-116.
doi: 10.1016/0370-1573(85)90028-6. |
[11] |
J. Goedert, F. Haas, D. Hua, M. R. Feix and L. Cairo, Generalized Hamiltonian structures for systems in three dimensions with a rescalable constant of motion, J. Phys. A, 27 (1994), 6495.
doi: 10.1088/0305-4470/27/19/020. |
[12] |
B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics," Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2004. |
[13] |
J. Llibre, M. Messias and P. R. da Silva, On the global dynamics of the Rabinovich system, J. Phys. A, 41 (2008), 275210, 21 pp. |
[14] |
J. Llibre and C. Valls, Global analytic integrability of the Rabinovich system, J. Geom. Phys., 58 (2008), 1762-1771.
doi: 10.1016/j.geomphys.2008.08.009. |
[15] |
J. E. Marsden, "Lectures on Mechanics," London Mathematical Society Lecture Notes Series, vol. 174, Cambridge University Press, 1992. |
[16] |
J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry," Texts in Applied Mathematics, vol. 17, Springer, Berlin, 1994. |
[17] |
M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces," Lecture Notes in Mathematics, vol. 510, Springer, Berlin, 2001. |
[18] |
A. S. Pikovskii and M. I. Rabinovich, Stochastic behavior of dissipative systems, Soc. Sci. Rev. C: Math. Phys. Rev., 2 (1981), 165-208. |
[19] |
A. S. Pikovskii, M. I. Rabinovich and V. Yu. Trakhtengerts, Onset of stochasticity in decay confinement of parametric instability, Soc. Phys. JETP, 47 (1978), 715-719. |
[20] |
T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra, Chapter II: A crash course in geometric mechanics, in "Geometric Mechanics and Symmetry: The Peyresq Lectures," London Mathematical Society Lecture Notes Series, vol. 306, Cambridge University Press, (2005), 23-156. |
[21] |
A. Weinstein, Normal modes for non-linear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.
doi: 10.1007/BF01405263. |
[22] |
F. Xie and X. Zhang, Invariant algebraic surfaces of the Rabinovich system, J. Phys. A, 36 (2003), 499-516.
doi: 10.1088/0305-4470/36/2/314. |
[23] |
X. Zhang, Integrals of motion of the Rabinovich system, J. Phys. A, 33 (2000), 5137-5155.
doi: 10.1088/0305-4470/33/28/315. |
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