# American Institute of Mathematical Sciences

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 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.
 [1] Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039 [2] Viktor L. Ginzburg, Başak Z. Gürel. On the generic existence of periodic orbits in Hamiltonian dynamics. Journal of Modern Dynamics, 2009, 3 (4) : 595-610. doi: 10.3934/jmd.2009.3.595 [3] Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 [4] Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097 [5] Alexandre Vidal. Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations. Conference Publications, 2007, 2007 (Special) : 1021-1030. doi: 10.3934/proc.2007.2007.1021 [6] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128 [7] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 [8] Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210 [9] Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778 [10] Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929 [11] Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Communications on Pure and Applied Analysis, 2019, 18 (1) : 425-434. doi: 10.3934/cpaa.2019021 [12] Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure and Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269 [13] Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353 [14] Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367 [15] Fatima Ezzahra Lembarki, Jaume Llibre. Periodic orbits for a generalized Friedmann-Robertson-Walker Hamiltonian system in dimension $6$. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1165-1211. doi: 10.3934/dcdss.2015.8.1165 [16] E. Canalias, Josep J. Masdemont. Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 261-279. doi: 10.3934/dcds.2006.14.261 [17] Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589 [18] Jianquan Li, Yanni Xiao, Yali Yang. Global analysis of a simple parasite-host model with homoclinic orbits. Mathematical Biosciences & Engineering, 2012, 9 (4) : 767-784. doi: 10.3934/mbe.2012.9.767 [19] Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109 [20] Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

2021 Impact Factor: 1.497