Superintegrable cases of four-dimensional dynamical systems. (English) Zbl 1344.34026
Summary: Degenerate tri-Hamiltonian structures of the Shivamoggi and generalized Raychaudhuri equations are studied. For certain specific values of the parameters, it is shown that hyperchaotic Lü and Qi systems are superintegrable and admit tri-Hamiltonian structures.
MSC:
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
Keywords:
first integrals; Darboux polynomials; Jacobi’s last multiplier; 4D Poisson structures; tri-Hamiltonian structures; Shivamoggi equations; generalized Raychaudhuri equations; Lü system and Qi systemReferences:
| [1] | Abraham, R. and Marsden, J.E., Foundations of Mechanics, Reading, Mass.: Benjamin/Cummings, 1978. · Zbl 0393.70001 |
| [2] | Abadoğlu, E. and Gümral, H., Bi-Hamiltonian Structure in Frenet-Serret Frame, Phys. D, 2009, vol. 238, no. 5, pp. 526-530. · Zbl 1157.37330 · doi:10.1016/j.physd.2008.11.013 |
| [3] | Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Hojman Construction and Hamiltonization of Nonholonomic Systems, SIGMA Symmetry Integrability Geom. Methods Appl., 2016, vol. 12, 012, 19 pp. · Zbl 1331.37087 |
| [4] | Blaszak, M. and Wojciechowski, S., Bi-Hamiltonian Dynamical Systems Related to Low-Dimensional Lie Algebras, Phys. A, 1989, vol. 155, no. 3, pp. 545-564. · Zbl 0706.58063 · doi:10.1016/0378-4371(89)90005-8 |
| [5] | del Castillo, G.T., The Hamiltonian Description of a Second-Order ODE, J. Phys. A, 2009, vol. 42, no. 26, 265202, 9 pp. · Zbl 1331.37075 · doi:10.1088/1751-8113/42/26/265202 |
| [6] | Chen, A., Lu, J., Lü, J., and Yu, S., Generating Hyperchaotic Lü Attractor via State Feedback Control, Phys. A, 2006, vol. 364, pp. 103-110. · doi:10.1016/j.physa.2005.09.039 |
| [7] | Darboux, G., Mémoire sur leséquations différentielles algébriques du premier ordre et du premier degré, Bulletin des Sciences Mathématiques et Astronomiques, Sér. 2, 1878, vol. 2, no. 1, pp. 60-96, 123-144, 151-200. · JFM 10.0214.01 |
| [8] | Dumortier, F., Llibre, J., and Artés, J.C., Qualitative Theory of Planar Differential Systems, Berlin: Springer, 2006. · Zbl 1110.34002 |
| [9] | Fernandes, R. L., Completely Integrable Bi-Hamiltonian Systems, J. Dynam. Differential Equations, 1994, vol. 6, no. 1, pp. 53-69. · Zbl 0796.58020 · doi:10.1007/BF02219188 |
| [10] | Gonera, C. and Nutku, Y., Super-Integrable Calogero-Type Systems Admit Maximal Number of Poisson Structures, Phys. Lett. A, 2001, vol. 285, no. 5, pp. 301-306. · Zbl 0969.37521 · doi:10.1016/S0375-9601(01)00365-6 |
| [11] | Goriely, A., Integrability and Nonintegrability of Dynamical Systems, Adv. Ser. Nonlinear Dynam., vol. 19, River Edge,N.J.: World Sci., 2001. · Zbl 1002.34001 |
| [12] | Choudhury, A. Gh., Guha, P., and Khanra, B., Determination of Elementary First Integrals of a Generalized Raychaudhuri Equation by the Darboux Integrability Method, J. Math. Phys., 2009, vol. 50, no. 10, 102502, 8 pp. · Zbl 1283.34001 · doi:10.1063/1.3243455 |
| [13] | Guha, P. and Choudhury, A. Gh., On Planar and Non-planar Isochronous Systems and Poisson Structures, Int. J. Geom. Methods Mod. Phys., 2010, vol. 7, no. 7, pp. 1115-1131. · Zbl 1377.70034 · doi:10.1142/S0219887810004750 |
| [14] | Guha, P. and Choudhury, A. Gh., First Integrals and Hamiltonian Structure for a System of Ordinary Differential Equations occurring in Magnetohydrodynamics, in AIP Conf. Proc. (2014), vol. 1582, pp. 116-123. · doi:10.1063/1.4865350 |
| [15] | Gümral, H., Existence of Hamiltonian Structure in 3D, Adv. Dyn. Syst. Appl., 2010, vol. 5, no. 2, pp. 159-171. |
| [16] | Gümral, H. and Nutku, Y., Poisson Structure of Dynamical Systems with Three Degrees of Freedom, J. Math. Phys., 1993, vol. 34, no. 12, pp. 5691-5723. · Zbl 0783.58036 · doi:10.1063/1.530278 |
| [17] | Jacobi, C.G. J., Sul principio dell’ultimo moltiplicatore, e suo uso come nuovo principio generale di meccanica, Giornale Arcadico di Scienze, Lettere ed Arti, 1844, vol. 99, pp. 129-146. |
| [18] | Jacobi, C.G. J., Theoria novi multiplicatoris systemati aequationum differentialium vulgarium applicandi, J. Reine Angew. Math., 1844, vol. 27, pp. 199-268; Jacobi, C.G. J., Theoria novi multiplicatoris systemati aequationum differentialium vulgarium applicandi, J. Reine Angew. Math., 1845, vol. 29, pp. 213-279, 333-376. · ERAM 027.0793cj · doi:10.1515/crll.1844.27.199 |
| [19] | Juanolou, J.P., Équations de Pfaff algébriques, Lecture Notes in Math., vol. 708, New York: Springer, 1979. · Zbl 0477.58002 |
| [20] | Khimshiashvili, G. and Przybysz, R., On Certain Super-Integrable Hamiltonian Systems, J. Dynam. Control Systems, 2002, vol. 8, no. 2, pp. 217-244. · Zbl 1029.37035 · doi:10.1023/A:1015369510234 |
| [21] | Libermann, P. and Marle, Ch.-M., Symplectic Geometry and Analytical Mechanics, Math. Appl., vol. 35, Dordrecht: Reidel, 1987. · Zbl 0643.53002 · doi:10.1007/978-94-009-3807-6 |
| [22] | Magri, F. and Morosi, C., A Geometrical Characterization of Integrable Hamiltonian Systems through the Theory of Poisson-Nijenhuis Manifolds, Milan: Univ. of Milan, 1984. |
| [23] | Man, Y. K., First Integrals of Autonomous Systems of Differential Equations and the Prelle-Singer Procedure, J. Phys. A, 1994, vol. 27, no. 10, L329-L332. · Zbl 0830.34003 · doi:10.1088/0305-4470/27/10/005 |
| [24] | Nucci, M.C., Jacobi Last Multiplier and Lie Symmetries: A Novel Application of an Old Relationship, J. Nonlinear Math. Phys., 2005, vol. 12, no. 2, pp. 284-304. · Zbl 1080.34003 · doi:10.2991/jnmp.2005.12.2.9 |
| [25] | Nucci, M.C. and Leach, P.G. L., Jacobi’s LastMultiplier and Lagrangians for Multidimensional Systems, J. Math. Phys., 2008, vol. 49, no. 7, 073517, 8 pp. · Zbl 1152.81573 · doi:10.1063/1.2956486 |
| [26] | Laurent-Gengoux, C., Pichereau, A., and Vanhaecke, P., Poisson Structures, Grundlehren Math. Wiss., vol. 347, New York: Springer, 2013. · Zbl 1284.53001 · doi:10.1007/978-3-642-31090-4 |
| [27] | Lorenz, E.N., Deterministic Nonperiodic Flow, J. Atmos. Sci., 1963, vol. 20, no. 2, pp. 130-141. · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |
| [28] | Nambu, Y., Generalized Hamiltonian Dynamics, Phys. Rev. D, 1973, vol. 7, no. 8, pp. 2405-2412. · Zbl 1027.70503 · doi:10.1103/PhysRevD.7.2405 |
| [29] | Nutku, Y., Hamiltonian Structure of the Lotka-Volterra Equations, Phys. Lett. A, 1990, vol. 145, no. 1, pp. 27-28. · doi:10.1016/0375-9601(90)90270-X |
| [30] | Olver, P. J., Canonical Forms and Integrability of Bi-Hamiltonian Systems, Phys. Lett. A, 1990, vol. 148, no. 3, pp. 177-187. · doi:10.1016/0375-9601(90)90775-J |
| [31] | Olver, P. J., Applications of Lie Groups to Differential Equations, 2nd ed., Grad. Texts in Math., vol. 107, New York: Springer, 1993. · Zbl 0785.58003 · doi:10.1007/978-1-4612-4350-2 |
| [32] | Qi, G., van Wyk, M.A., van Wyk, B. J., and Chen, G., A New Hyperchaotic System and Its Circuit Implementation, Chaos Solitons Fractals, 2009, vol. 40, no. 5, pp. 2544-2549. · Zbl 1197.82017 · doi:10.1016/j.chaos.2007.10.053 |
| [33] | Prelle, M. J. and Singer, M. F., Elementary First Integrals of Differential Equations, Trans. Amer. Math. Soc., 1983, vol. 279, no. 1, pp. 215-229. · Zbl 0527.12016 · doi:10.1090/S0002-9947-1983-0704611-X |
| [34] | Raychaudhuri, A., Relativistic Cosmology: 1, Phys. Rev., 1955, vol. 98, no. 2, pp. 1123-1126. · Zbl 0066.22204 · doi:10.1103/PhysRev.98.1123 |
| [35] | Singer, M. F., Liouvillian First Integrals of Differential Equations, Trans. Amer. Math. Soc., 1992, vol. 333, no. 2, pp. 673-688. · Zbl 0756.12006 · doi:10.1090/S0002-9947-1992-1062869-X |
| [36] | Shivamoggi, B.K., Current-Sheet Formation Near a Hyperbolic Magnetic Neutral Line in the Presence of a Plasma Flow with a Uniform Shear-Strain Rate: An Exact Solution, Phys. Lett. A, 1999, vol. 258, no. 2, pp. 131-134. · doi:10.1016/S0375-9601(99)00274-1 |
| [37] | Valls, C., Darbouxian Integrals for Generalized Raychaudhuri Equations, J. Math. Phys., 2011, vol. 52, no. 3, 032703, 13 pp. · Zbl 1315.83013 · doi:10.1063/1.3559065 |
| [38] | Weinstein, A., The Local Structure of Poisson Manifolds, J. Differential Geom., 1983, vol. 18, no. 3, pp. 523-557. · Zbl 0524.58011 |
| [39] | Whittaker, E.T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies, Cambridge: Cambridge Univ. Press, 1988. · Zbl 0665.70002 · doi:10.1017/CBO9780511608797 |
| [40] | Tabor, M. and Weiss, J., Analytic Structure of the Lorenz System, Phys. Rev. A, 1981, vol. 24, no. 4, pp. 2157-2167. · doi:10.1103/PhysRevA.24.2157 |
| [41] | Kuś, M., Integrals of Motion for the Lorenz System, J. Phys. A, 1983, vol. 16, no. 18, L689-L691. · Zbl 0542.34013 · doi:10.1088/0305-4470/16/18/002 |
| [42] | Sen, T. and Tabor, M., Lie Symmetries of the Lorenz Model, Phys. D, 1990, vol. 44, no. 3, pp. 313-339. · Zbl 0716.58016 · doi:10.1016/0167-2789(90)90152-F |
| [43] | Steeb, W.-H. and Euler, N., Nonlinear Evolution Equations and Painlevé Test, Singapore: World Sci., 1988. · Zbl 0723.34001 · doi:10.1142/0723 |
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