On the Hamiltonian dynamics and geometry of the Rabinovich system

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December 2011, 15(3): 789-823. doi: 10.3934/dcdsb.2011.15.789

On the Hamiltonian dynamics and geometry of the Rabinovich system

Răzvan M. Tudoran ^1, and Anania Gîrban ^2,

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

Received April 2010 Revised September 2010 Published February 2011

Abstract
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In this paper, we describe some relevant dynamical and geometrical properties of the Rabinovich system from the Poisson geometry and the dynamics point of view. Starting with a Lie-Poisson realization of the Rabinovich system, we determine and then completely analyze the Lyapunov stability of all the equilibrium states of the system, study the existence of periodic orbits, and then using a new geometrical approach, we provide the complete dynamical behavior of the Rabinovich system, in terms of the geometric semialgebraic properties of a two-dimensional geometric figure, associated with the problem. Moreover, in tight connection with the dynamical behavior, by using this approach, we also recover all the dynamical objects of the system (e.g. equilibrium states, periodic orbits, homoclinic and heteroclinic connections). Next, we integrate the Rabinovich system by Jacoby elliptic functions, and give some Lax formulations of the system. The last part of the article discusses some numerics associated with the Poisson geometrical structure of the Rabinovich system.

Keywords: Rabinovich system, stability analysis, periodic orbits, Hamiltonian dynamics, homoclinic and heteroclinic orbits..

Mathematics Subject Classification: Primary: 70H05; Secondary: 70H14, 70K4.

Citation: Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789

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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