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

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

Received  April 2010 Revised  September 2010 Published  February 2011

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.
Citation: Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & 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, 60 (1989). Google Scholar

[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. Google Scholar

[3]

A. Ay, M. Gurses and K. Zheltukhin, Hamiltonian equations on $\R^3$,, J. Math. Phys., 44 (2003), 5688. doi: 10.1063/1.1619204. Google Scholar

[4]

O. Babelon, D. Bernard and M. Talon, "Introduction to Classical Integrable Systems,", Cambridge Monographs on Mathematical Physics, (2003). Google Scholar

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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. doi: 10.1016/0378-4371(84)90091-8. Google Scholar

[6]

C. Chen, J. Cao and X. Zhang, The topological structure of the Rabinovich system having an invariant algebraic surface,, Nonlinearity, 21 (2008), 211. doi: 10.1088/0951-7715/21/2/002. Google Scholar

[7]

O. Chis and M. Puta, The dynamics of the Rabinovich system,, preprint, (2007), 1. Google Scholar

[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. doi: 10.1142/S0219887808002916. Google Scholar

[9]

R. H. Cushman and L. Bates, "Global Aspects Of Classical Integrable Systems,", Basel: Birkhauser, (1977). Google Scholar

[10]

D. D. Holm, J. E. Marsden, T. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria,, Physics Reports, 123 (1985), 1. doi: 10.1016/0370-1573(85)90028-6. Google Scholar

[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). doi: 10.1088/0305-4470/27/19/020. Google Scholar

[12]

B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics,", Cambridge Monographs on Applied and Computational Mathematics, (2004). Google Scholar

[13]

J. Llibre, M. Messias and P. R. da Silva, On the global dynamics of the Rabinovich system,, J. Phys. A, 41 (2008). Google Scholar

[14]

J. Llibre and C. Valls, Global analytic integrability of the Rabinovich system,, J. Geom. Phys., 58 (2008), 1762. doi: 10.1016/j.geomphys.2008.08.009. Google Scholar

[15]

J. E. Marsden, "Lectures on Mechanics,", London Mathematical Society Lecture Notes Series, 174 (1992). Google Scholar

[16]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Texts in Applied Mathematics, 17 (1994). Google Scholar

[17]

M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces,", Lecture Notes in Mathematics, 510 (2001). Google Scholar

[18]

A. S. Pikovskii and M. I. Rabinovich, Stochastic behavior of dissipative systems,, Soc. Sci. Rev. C: Math. Phys. Rev., 2 (1981), 165. Google Scholar

[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. Google Scholar

[20]

T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra, Chapter II: A crash course in geometric mechanics,, in, 306 (2005), 23. Google Scholar

[21]

A. Weinstein, Normal modes for non-linear Hamiltonian systems,, Invent. Math., 20 (1973), 47. doi: 10.1007/BF01405263. Google Scholar

[22]

F. Xie and X. Zhang, Invariant algebraic surfaces of the Rabinovich system,, J. Phys. A, 36 (2003), 499. doi: 10.1088/0305-4470/36/2/314. Google Scholar

[23]

X. Zhang, Integrals of motion of the Rabinovich system,, J. Phys. A, 33 (2000), 5137. doi: 10.1088/0305-4470/33/28/315. Google Scholar

show all references

References:
[1]

V. I. Arnold, "Mathematical Methods of Classical Mechanics. Second Edition,", Graduate Texts in Mathematics, 60 (1989). Google Scholar

[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. Google Scholar

[3]

A. Ay, M. Gurses and K. Zheltukhin, Hamiltonian equations on $\R^3$,, J. Math. Phys., 44 (2003), 5688. doi: 10.1063/1.1619204. Google Scholar

[4]

O. Babelon, D. Bernard and M. Talon, "Introduction to Classical Integrable Systems,", Cambridge Monographs on Mathematical Physics, (2003). Google Scholar

[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. doi: 10.1016/0378-4371(84)90091-8. Google Scholar

[6]

C. Chen, J. Cao and X. Zhang, The topological structure of the Rabinovich system having an invariant algebraic surface,, Nonlinearity, 21 (2008), 211. doi: 10.1088/0951-7715/21/2/002. Google Scholar

[7]

O. Chis and M. Puta, The dynamics of the Rabinovich system,, preprint, (2007), 1. Google Scholar

[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. doi: 10.1142/S0219887808002916. Google Scholar

[9]

R. H. Cushman and L. Bates, "Global Aspects Of Classical Integrable Systems,", Basel: Birkhauser, (1977). Google Scholar

[10]

D. D. Holm, J. E. Marsden, T. Ratiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria,, Physics Reports, 123 (1985), 1. doi: 10.1016/0370-1573(85)90028-6. Google Scholar

[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). doi: 10.1088/0305-4470/27/19/020. Google Scholar

[12]

B. Leimkuhler and S. Reich, "Simulating Hamiltonian Dynamics,", Cambridge Monographs on Applied and Computational Mathematics, (2004). Google Scholar

[13]

J. Llibre, M. Messias and P. R. da Silva, On the global dynamics of the Rabinovich system,, J. Phys. A, 41 (2008). Google Scholar

[14]

J. Llibre and C. Valls, Global analytic integrability of the Rabinovich system,, J. Geom. Phys., 58 (2008), 1762. doi: 10.1016/j.geomphys.2008.08.009. Google Scholar

[15]

J. E. Marsden, "Lectures on Mechanics,", London Mathematical Society Lecture Notes Series, 174 (1992). Google Scholar

[16]

J. E. Marsden and T. S. Ratiu, "Introduction to Mechanics and Symmetry,", Texts in Applied Mathematics, 17 (1994). Google Scholar

[17]

M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces,", Lecture Notes in Mathematics, 510 (2001). Google Scholar

[18]

A. S. Pikovskii and M. I. Rabinovich, Stochastic behavior of dissipative systems,, Soc. Sci. Rev. C: Math. Phys. Rev., 2 (1981), 165. Google Scholar

[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. Google Scholar

[20]

T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra, Chapter II: A crash course in geometric mechanics,, in, 306 (2005), 23. Google Scholar

[21]

A. Weinstein, Normal modes for non-linear Hamiltonian systems,, Invent. Math., 20 (1973), 47. doi: 10.1007/BF01405263. Google Scholar

[22]

F. Xie and X. Zhang, Invariant algebraic surfaces of the Rabinovich system,, J. Phys. A, 36 (2003), 499. doi: 10.1088/0305-4470/36/2/314. Google Scholar

[23]

X. Zhang, Integrals of motion of the Rabinovich system,, J. Phys. A, 33 (2000), 5137. doi: 10.1088/0305-4470/33/28/315. Google Scholar

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