phone +7 (3412) 91 60 92

Archive of Issues

Russia Izhevsk
Section Mechanics
Title Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: Examples and numerical study
Author(-s) Kuznetsov S.P.a
Affiliations Udmurt State Universitya
Abstract A system of $N$ rotators is investigated with a constraint given by the condition of vanishing sum of the cosines of the rotation angles. Equations of the dynamics are formulated and results of numerical simulation for the cases $N=3$, $4$, and $5$ are presented relating to the geodesic flows on a two-dimensional, three-dimensional, and four-dimensional manifold, respectively, in a compact region (due to the periodicity of the configuration space in angular variables). It is shown that a system of three rotators demonstrates chaos, characterized by one positive Lyapunov exponent, and for systems of four and five elements there are, respectively, two and three positive exponents (“hyperchaos”). An algorithm has been implemented that allows calculating the sectional curvature of a manifold in the course of numerical simulation of the dynamics at points of a trajectory. In the case of $N=3$, curvature of the two-dimensional manifold is negative (except for a finite number of points where it is zero), and Anosov's geodesic flow is realized. For $N=4$ and $5$, the computations show that the condition of negative sectional curvature is not fulfilled. Also the methodology is explained and applied for testing hyperbolicity based on numerical analysis of the angles between the subspaces of small perturbation vectors; in the case of $N=3$, the hyperbolicity is confirmed, and for $N=4$ and $5$ the hyperbolicity does not take place.
Keywords geodesic flow, chaos, Anosov dynamics, Lyapunov exponent
UDC 37D40, 37D20, 37D25, 34D08, 32Q05, 32Q10, 70F20
MSC 517.93
DOI 10.20537/vm180409
Received 8 October 2018
Language Russian
Citation Kuznetsov S.P. Chaos and hyperchaos of geodesic flows on curved manifolds corresponding to mechanically coupled rotators: Examples and numerical study, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 4, pp. 565-581.
  1. Paternain G.P. Geodesic flows, Birkhäuser, 1999, xiii+149 p. DOI: 10.1007/978-1-4612-1600-1
  2. Dubrovin B.A., Fomenko A.T., Novikov S.P. Modern geometry - methods and applications. Part II: The geometry and topology of manifolds, New York: Springer, 1985. xv+432 p. DOI: 10.1007/978-1-4612-1100-6
  3. Hadamard J. Les surfaces à courbures opposées et leurs lignes géodésiques, Journal de Mathématiques Pures et Appliquées $5$$\textit{e}$ série, 1898, vol. 4, pp. 27-74.
  4. Anosov D.V. Roughness of geodesic flows on compact Riemannian manifolds of negative curvature, Soviet Mathematics. Doklady, 1962, vol. 3, pp. 1068-1070.
  5. Anosov D.V. Geodesic flows on closed Riemann manifolds with negative curvature, Proceedings of the Steklov Institute of Mathematics, 1967, vol. 90, pp. 1-235.
  6. Anosov D.V., Sinai Ya.G. Some smooth ergodic systems, Russian Mathematical Surveys, 1967, vol. 22, no. 5, pp. 103-167. DOI: 10.1070/RM1967v022n05ABEH001228
  7. Anosov D.V., Aranson S.Kh., Grines V.Z., Plykin R.V., Sataev E.A., Safonov A.V., Solodov V.V., Starkov A.N., Stepin A.M., Shlyachkov S.V., Dynamical Systems with Hyperbolic Behaviour. Dynamical Systems IX, Encycl. Math. Sci., vol. 66, Berlin: Springer, 1995, 236 p.
  8. Thurston W.P., Weeks J.R. The mathematics of three-dimensional manifolds, Scientific American, 1984, vol. 251, no. 1, pp. 108-121.
  9. Hunt T.J., MacKay R.S. Anosov parameter values for the triple linkage and a physical system with a uniformly chaotic attractor, Nonlinearity, 2003, vol. 16, no. 4, pp. 1499-1510. DOI: 10.1088/0951-7715/16/4/318
  10. Kuznetsov S.P. Chaos in the system of three coupled rotators: from Anosov dynamics to hyperbolic attractor, Izv. Saratov Univ. (N. S.) Ser. Physics, 2015, vol. 15, issue 2, pp. 5-17 (in Russian).
  11. Kuznetsov S.P. Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories, Regular and Chaotic Dynamics, 2015, vol. 20, no. 6, pp. 649-666. DOI: 10.1134/S1560354715060027
  12. Meeks III W.H., Ros A., Rosenberg H. The global theory of minimal surfaces in flat spaces, Springer, 2002, 124 p. DOI: 10.1007/b83168
  13. Kuznetsov S.P. From Anosov's dynamics on a surface of negative curvature to electronic generator of robust chaos, Izv. Saratov Univ. (N. S.) Ser. Physics, 2016, vol. 16, issue 3, pp. 131-144 (in Russian). DOI: 10.18500/1817-3020-2016-16-3-131-144
  14. Kuznetsov S.P. From geodesic flow on a surface of negative curvature to electronic generator of robust chaos, International Journal of Bifurcation and Chaos, 2016, vol. 26, no. 14, 1650232. DOI: 10.1142/S0218127416502321
  15. Kuznetsov S.P. Chaos in three coupled rotators: From Anosov dynamics to hyperbolic attractors, Indian Academy of Sciences - Conference Series, 2017, vol. 1, no. 1, pp. 117-132. DOI: 10.29195/iascs.01.01.0017
  16. Fel'k E.V., Kuznetsov S.P., Savin A.V. Diffusion in the configuration space of a system of two coupled rotators, Proceedings of the 11th International School on “Chaotic Oscillations and Pattern Formation”, Saratov: Publishing Center “Science”, 2016, p. 110 (in Russian).
  17. Zaslavskiî G.M., Sagdeev R.Z., Usikov D.A., Chernikov A.A. Weak chaos and quasi-regular patterns, Cambridge University Press, 1991, 268 p.
  18. Dubrovin B.A., Fomenko A.T., Novikov S.P. Modern geometry - Methods and applications. Part I: The geometry of surfaces, transformation groups, and fields, 2nd edition, Springer, 1991, xvi+470 p.
  19. Sveshnikov A.A. Applied methods of the theory of random functions, Pergamon Press, 1966, 332 p. DOI: 10.1016/C2013-0-07845-0
  20. Jenkins G.M., Watts D.G. Spectral analysis and its applications, Holden-Day, 1969, 525 p.
  21. Benettin G., Galgani L., Giorgilli A., Strelcyn J.-M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part I: Theory, Meccanica, 1980, vol. 15, issue 1, pp. 9-20. DOI: 10.1007/BF02128236
  22. Shimada I., Nagashima T. A numerical approach to ergodic problem of dissipative dynamical systems, Progress of Theoretical Physics, 1979, vol. 61, issue 6, pp. 1605-1616. DOI: 10.1143/PTP.61.1605
  23. Kuznetsov S.P. Hyperbolic chaos: a physicist's view, Berlin-Heidelberg: Springer-Verlag, 2012, xvi, 320 p. DOI: 10.1007/978-3-642-23666-2
  24. Pikovsky A., Politi A. Lyapunov exponents: a tool to explore complex dynamics, Cambridge: Cambridge University Press, 2016, 295 p. DOI: 10.1017/CBO9781139343473
  25. Kuptsov P.V. Computation of Lyapunov exponents for spatially extended systems: advantages and limitations of various numerical methods, Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, issue 5, pp. 91-110 (in Russian). DOI: 10.18500/0869-6632-2010-18-5-91-110
  26. Rössler O.E. An equation for hyperchaos, Physics Letters A, 1979, vol. 71, issues 2-3, pp. 155-157. DOI: 10.1016/0375-9601(79)90150-6
  27. Letellier C., Rössler O.E. Hyperchaos, Scholarpedia, 2007, vol. 2, issue 8, p. 1936. DOI: 10.4249/scholarpedia.1936
  28. Lai Y.-C., Grebogi C., Yorke J.A., Kan I. How often are chaotic saddles nonhyperbolic?, Nonlinearity, 1993, vol. 6, no. 5, pp. 779-798. DOI: 10.1088/0951-7715/6/5/007
  29. Anishchenko V.S., Kopeikin A.S., Kurths J., Vadivasova T.E., Strelkova G.I. Studying hyperbolicity in chaotic systems, Physics Letters A, 2000, vol. 270, no. 6, pp. 301-307. DOI: 10.1016/S0375-9601(00)00338-8
  30. Kuznetsov S.P. Dynamical chaos and uniformly hyperbolic attractors: from mathematics to physics, Physics-Uspekhi, 2011, vol. 54, no. 2, pp. 119-144. DOI: 10.3367/UFNe.0181.201102a.0121
  31. Kuznetsov S.P., Kruglov V.P. On some simple examples of mechanical systems with hyperbolic chaos, Proceedings of the Steklov Institute of Mathematics, 2017, vol. 297, issue 1, pp. 208-234. DOI: 10.1134/S0081543817040137
  32. Kuptsov P.V. Fast numerical test of hyperbolic chaos, Physical Review E, 2012, vol. 85, no. 1, 015203. DOI: 10.1103/PhysRevE.85.015203
  33. Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity of chaotic dynamics in time-delay systems, Physical Review E, 2016, vol. 94, no. 1, 010201. DOI: 10.1103/PhysRevE.94.010201
  34. Kuptsov P.V., Kuznetsov S.P. Numerical test for hyperbolicity in chaotic systems with multiple time delays, Communications in Nonlinear Science and Numerical Simulation, 2018, vol. 56, pp. 227-239. DOI: 10.1016/j.cnsns.2017.08.016
  35. Kuptsov P.V., Kuznetsov S.P. Lyapunov analysis of strange pseudohyperbolic attractors: angles between tangent subspaces, local volume expansion and contraction, 2018, arXiv: 1805.06644 [nlin.CD].
Full text
<< Previous article
Next article >>