Publication: Dynamical systems on infinitely sheeted Riemann surfaces
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2007-03-15
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Elsevier
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This paper is part of a program that aims to understand the connection between the emergence of chaotic behaviour in dynamical systems in relation with the multi-valuedness of the solutions as functions of complex time tau. In this work we consider a family of systems whose solutions can be expressed as the inversion of a single hyperelliptic integral. The associated Riemann surface R -> C = {tau} is known to be an infinitely sheeted covering of the complex time plane, ramified at an infinite set of points whose projection in the tau-plane is dense. The main novelty of this paper is that the geometrical structure of these infinitely sheeted Riemann surfaces is described in great detail, which allows us to study global properties of the flow such as asymptotic behaviour of the solutions, periodic orbits and their stability or sensitive dependence on initial conditions. The results are then compared with a numerical integration of the equations of motion. Following the recent approach of Calogero, the real time trajectories of the system are given by paths on R that are projected to a circle on the complex plane tau. Due to the branching of R, the solutions may have different periods or may be aperiodic.
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© 2007 Elsevier B.V. All rights reserved.
We thank L. Gavrilov, P. Santini, and V. Enolski for discussions and valuable remarks. Our research was partially supported by the Spanish Ministry of Science and Technology under grant BFM 2003-09504-C02-02.
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[1] S. Abenda and Yu. Fedorov. On the weak Kowalevski-Painlevé property for hyperelliptically separable systems. Acta Appl. Math., 60(2):137–178, 2000.
[2] S. Abenda, V. Marinakis, and T. Bountis. On the connection between hyperelliptic separability and Painlev´e integrability. J. Phys. A, 34(17):3521– 3539, 2001.
[3] Mark S. Alber and Yuri N. Fedorov. Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians. Inverse Problems, 17(4):1017–1042, 2001. Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000).
[4] Larry Bates and Richard Cushman. Complete integrability beyond liouville-arnold. Rep. Math. Phys., 56(1):77–91, 2005.
[5] E. D. Belokolos and V. Z. Enolskii. Reduction of abelian functions and algebraically integrable systems. I. J. Math. Sci. (New York), 106(6):3395– 3486, 2001. Complex analysis and representation theory, 2.
[6] E.D. Belokolos, A.I. Bobenko, V.Z. Enol’ski, A.R. Its, and V.B. Matveev. Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer Series in Nonlinear Dynamics. Springer– Verlag, Berlin, 1994.
[7] C. M. Bender, J.-H. Chen, D. W. Darg, and Milton K. A. Classical trajectories for complex hamiltonians. J. Phys. A, 39:4219–4238, 2006.
[8] Christina Birkenhake and Herbert Lange. Complex abelian varieties, volume 302 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2004.
[9] Oskar Bolza. On Binary Sextics with Linear Transformations into Themselves. Amer. J. Math., 10(1):47–70, 1887.
[10] T. Bountis, L. Drossos, and I. C. Percival. Nonintegrable systems with algebraic singularities in complex time. J. Phys. A, 24(14):3217–3236, 1991.
[11] F. Calogero and J.-P. Fran¸coise. Periodic motions galore: how to modify nonlinear evolution equations so that they feature a lot of periodic solutions. J. Nonlinear Math. Phys., 9(1):99–125, 2002.
[12] F. Calogero and J.-P. Fran¸coise. Periodic motions galore: how to modify nonlinear evolution equations so that they feature a lot of periodic solutions. J. Nonlinear Math. Phys., 9(1):99–125, 2002.
[13] F. Calogero, J.-P. Fran¸coise, and M. Sommacal. Periodic solutions of a many-rotator problem in the plane. II. Analysis of various motions. J. Nonlinear Math. Phys., 10(2):157–214, 2003.
[14] F. Calogero, D. Gómez-Ullate, P. M. Santini, and M. Sommacal. The transition from regular to irregular motions, explained as travel on Riemann surfaces. J. Phys. A, 38(41):8873–8896, 2005.
[15] F. Calogero and M. Sommacal. Periodic solutions of a system of complex ODEs. II. Higher periods. J. Nonlinear Math. Phys., 9(4):483–516, 2002.
[16] Y. F. Chang and G. Corliss. Ratio-like and recurrence relation tests for convergence of series. J. Inst. Math. Appl., 25(4):349–359, 1980.
[17] Y. F. Chang, J. M. Greene, M. Tabor, and J. Weiss. The analytic structure of dynamical systems and self-similar natural boundaries. Phys. D, 8(1- 2):183–207, 1983.
[18] Y. F. Chang, M. Tabor, and J. Weiss. Analytic structure of the H´enonHeiles Hamiltonian in integrable and nonintegrable regimes. J. Math. Phys., 23(4):531–538, 1982.
[19] V. Z. Enolskii, M. Pronine, and P. H. Richter. Double pendulum and θ-divisor. J. Nonlinear Sci., 13(2):157–174, 2003.
[20] H. Flaschka. A remark on integrable Hamiltonian systems. Phys. Lett. A, 131(9):505–508, 1988.
[21] A. S. Fokas and T. Bountis. Order and the ubiquitous occurrence of chaos. Phys. A, 228(1- 4):236–244, 1996.
[22] B. Gambier. Sur les équations différentielles du second ordre et du premier degré don’t lintégrale générale est à points critiques fixes. Acta Mathematica, 33:1–55, 1910.
[23] S. Kowalewskaya. Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Mathematica, 12:177–232, 1889.
[24] S. Kowalewskaya. Sur une propriété du système d’équations différentielles qui définit la rotation d’un corps solide autour d’un point fixe. Acta Mathematica, 14:81–93, 1890.
[25] M. D. Kruskal, A. Ramani, and B. Grammaticos. Singularity analysis and its relation to complete, partial and nonintegrability. In Partially integrable evolution equations in physics (Les Houches, 1989), volume 310 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 321– 372. Kluwer Acad. Publ., Dordrecht, 1990.
[26] G. Levine and M. Tabor. Integrating the nonintegrable: analytic structure of the Lorenz system revisited. Phys. D, 33(1-3):189–210, 1988. Progress in chaotic dynamics.
[27] A. I. Markushevich. Introduction to the classical theory of abelian functions, volume 96 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1992. Translated from the 1979 Russian original by G. Bluher.
[28] S. P. Novikov. Topology of the generic hamiltonian foliations on the riemann surface. arXiv:math.GT/0505342.
[29] P. Painlevé. Mémoire sue les équations différentielles du premier ordre. Acta Mathematica, 14:81–93, 1890.
[30] Ramani, A., Dorizzi, B. and Grammaticos, B.: Painlevé conjecture revisited, Phys. Rev. Lett. 49(21) (1982), 15391541.
[31] A. Ramani, B. Grammaticos, and T. Bountis. The Painlev´e property and singularity analysis of integrable and nonintegrable systems. Phys. Rep., 180(3):159–245, 1989.
[32] M. Tabor and J. Weiss. Analytic structure of the lorenz system. Phys. Rev. A, 24(4):2157– 2167, 1981.
[33] Pol Vanhaecke. Integrable systems and symmetric products of curves. Math. Z., 227(1):93–127, 1998.