Topaj–Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators

Abstract

We discuss the Hamiltonian model of an oscillator lattice with local coupling. The Hamiltonian model describes localized spatial modes of nonlinear the Schrödinger equation with periodic tilted potential. The Hamiltonian system manifests reversibility of the Topaj–Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with asymptotic dynamics exactly equivalent to the Topaj–Pikovsky model. We examine the stability of trajectories belonging to invariant manifolds by means of numerical evaluation of Lyapunov exponents. We show that there is no contradiction between asymptotic dynamics on invariant manifolds and conservation of phase volume of the Hamiltonian system. We demonstrate the complexity of dynamics with results of numerical simulations.

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References

  1. 1.

    Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence, Springer Ser. Synergetics, vol. 19, Berlin: Springer, 1984.

    Google Scholar 

  2. 2.

    Pikovsky, A., Rosenblum, M., and Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Sci. Ser., vol. 12, New York: Cambridge Univ. Press, 2001.

    Google Scholar 

  3. 3.

    Pikovsky, A. and Politi, A., Lyapunov Exponents: A Tool to Explore Complex Dynamics, Cambridge: Cambridge Univ. Press, 2016.

    Google Scholar 

  4. 4.

    Strogatz, S. H., From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators. Bifurcations, Patterns and Symmetry, Phys. D, 2000, vol. 143, nos. 1–4, pp. 1–20.

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Topaj, D. and Pikovsky, A., Reversibility vs. Synchronization in Oscillator Lattices, Phys. D, 2002, vol. 170, no. 2, pp. 118–130.

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Gonchenko, A. S., Gonchenko, S. V., Kazakov, A. O., and Turaev, D. V., On the Phenomenon of Mixed Dynamics in Pikovsky–Topaj System of Coupled Rotators, Phys. D, 2017, vol. 350, pp. 45–57.

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Kuznetsov, A. P., Rahmanova, A. Z., and Savin, A. V., The Effect of Symmetry Breaking on Reversible Systems with Mixed Dynamics, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 2018, vol. 26, no. 6, pp. 20–31 (Russian).

    Google Scholar 

  8. 8.

    Tsang, K. Y., Mirollo, R. E., Strogatz, S. H., and Wiesenfeld, K., Reversibility and Noise Sensitivity of Josephson Arrays, Phys. Rev. Lett., 1991, vol. 66, no. 8, pp. 1094–1097.

    Article  Google Scholar 

  9. 9.

    Roberts, J. A. G. and Quispel, G. R. W., Chaos and Time-Reversal Symmetry. Order and Chaos in Reversible Dynamical Systems, Phys. Rep., 1992, vol. 216, nos. 2–3, pp. 63–177.

    MathSciNet  Article  Google Scholar 

  10. 10.

    Lamb, J. S. W. and Roberts, J. A. G., Time-Reversal Symmetry in Dynamical Systems: A Survey, Phys. D, 1998, vol. 112, nos. 1–2, pp. 1–39.

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Lerman, L. M. and Turaev, D. V., Breakdown of Symmetry in Reversible Systems, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3–4, pp. 318–336.

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Delshams, A., Gonchenko, S.V., Gonchenko, A. S., Lázaro, J. T., and Sten’kin, O., Abundance of Attracting, Repelling and Elliptic Periodic Orbits in Two-Dimensional Reversible Maps, Nonlinearity, 2013, vol. 26, no. 1, pp. 1–33.

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Moser, J. K. and Webster, S. M., Normal Forms for Real Surfaces in C 2 near Complex Tangents and Hyperbolic Surface Transformations, Acta Math., 1983, vol. 150, no. 1, pp. 255–296.

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Scheurle, J., Bifurcation of Quasi-Periodic Solutions from Equilibrium Points of Reversible Dynamical Systems, Arch. Rational Mech. Anal., 1987, vol. 97, no. 2, pp. 103–139.

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Bibikov, Yu. N. and Pliss, V. A., On the Existence of Invariant Tori in a Neighborhood of the Zero Solution of a System of Ordinary Differential Equations, Differ. Equ., 1967, vol. 3, no. 11, pp. 967–976; see also: Differ. Uravn., 1967, vol. 3, no. 11, pp. 1864–1881.

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Lamb, J. S. W. and Stenkin, O. V., Newhouse Regions for Reversible Systems with Infinitely Many Stable, Unstable and Elliptic Periodic Orbits, Nonlinearity, 2004, vol. 17, no. 4, pp. 1217–1244.

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Gonchenko, S. V. and Turaev, D. V., On Three Types of Dynamics and the Notion of Attractor, Proc. Steklov Inst. Math., 2017, vol. 297, no. 1, pp. 116–137; see also: Tr. Mat. Inst. Steklova, 2017, vol. 297, pp. 133–157.

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Politi, A., Oppo, G. L., and Badii, R., Coexistence of Conservative and Dissipative Behavior in Reversible Dynamical Systems, Phys. Rev. A, 1986, vol. 33, no. 6, pp. 4055–4060.

    Article  Google Scholar 

  19. 19.

    Gonchar, V. Yu., Ostapchuk, P. N., Tur, A. V., and Yanovsky, V. V., Dynamics and Stochasticity in a Reversible System Describing Interaction of Point Vortices with a Potential Wave, Phys. Lett. A, 1991, vol. 152, nos. 5–6, pp. 287–292.

    MathSciNet  Article  Google Scholar 

  20. 20.

    Vetchanin, E. V. and Kazakov, A. O., Bifurcations and Chaos in the Dynamics of Two Point Vortices in an Acoustic Wave, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2016, vol. 26, no. 4, 1650063, 13 pp.

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Vetchanin, E. V. and Mamaev, I. S., Dynamics of Two Point Vortices in an External Compressible Shear Flow, Regul. Chaotic Dyn., 2017, vol. 22, no. 8, pp. 893–908.

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Borisov, A. V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393–403; see also: Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 407–418.

    Article  Google Scholar 

  23. 23.

    Borisov, A. V., Jalnine, A. Yu., Kuznetsov, S. P., Sataev, I. R., and Sedova, J. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512–532.

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Gonchenko, A. S., Gonchenko, S. V., and Kazakov, A. O., Richness of Chaotic Dynamics in Nonholonomic Models of a Celtic Stone, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 521–538.

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Borisov, A. V., Kazakov, A. O., and Kuznetsov, S. P., Nonlinear Dynamics of the Rattleback: A Nonholonomic Model, Physics-Uspekhi, 2014, vol. 57, no. 5, pp. 453–460; see also: Uspekhi Fiz. Nauk, 2014, vol. 184, no. 5, pp. 493–500.

    Article  Google Scholar 

  26. 26.

    Borisov, A. V., Kazakov, A. O., and Sataev, I. R., Spiral Chaos in the Nonholonomic Model of a Chaplygin Top, Regul. Chaotic Dyn., 2016, vol. 21, nos. 7–8, pp. 939–954.

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Bizyaev, I. A., Borisov, A. V., and Kazakov, A. O., Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors, Regul. Chaotic Dyn., 2015, vol. 20, no. 5, pp. 605–626.

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Hamiltonicity and Integrability of the Suslov Problem, Regul. Chaotic Dyn., 2011, vol. 16, nos. 1–2, pp. 104–116.

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Non-Holonomic Systems in the Neighborhood of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 443–464.

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Borisov, A. V., Mamaev, I. S., and Tsyganov, A. V., Nonholonomic Dynamics and Poisson Geometry, Russian Math. Surveys, 2014, vol. 69, no. 3, pp. 481–538; see also: Uspekhi Mat. Nauk, 2014, vol. 69, no. 3(417), pp. 87–144.

    MathSciNet  Article  Google Scholar 

  31. 31.

    Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Elementary Nonholonomic Systems, Russ. J. Math. Phys., 2015, vol. 22, no. 4, pp. 444–453.

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., The Hess–Appelrot Case and Quantization of the Rotation Number, Regul. Chaotic Dyn., 2017, vol. 22, no. 2, pp. 180–196.

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Thommen, Q., Garreau, J. C., and Zehnlé, V., Classical Chaos with Bose–Einstein Condensates in Tilted Optical Lattices, Phys. Rev. Lett., 2003, vol. 91, no. 21, 210405, 4 pp.

    Article  Google Scholar 

  34. 34.

    Witthaut, D. and Timme, M., Kuramoto Dynamics in Hamiltonian Systems, Phys. Rev. E, 2014, vol. 90, no. 3, 032917, 8 pp.

    Article  Google Scholar 

  35. 35.

    Witthaut, D., Werder, M., Mossmann, S., and Korsch, H. J., Bloch Oscillations of Bose–Einstein Condensates: Breakdown and Revival, Phys. Rev. E, 2005, vol. 71, no. 3, 036625, 9 pp.

    Article  Google Scholar 

  36. 36.

    Adler, R., A Study of Locking Phenomena in Oscillators, Proc. IRE, 1946, vol. 34, no. 6, pp. 351–357.

    Article  Google Scholar 

  37. 37.

    Bloch, A. M., Asymptotic Hamiltonian Dynamics: The Toda Lattice, the Three-Wave Interaction and the Non-Holonomic Chaplygin Sleigh, Phys. D, 2000, vol. 141, nos. 3–4, pp. 297–315.

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 1: Theory, Meccanica, 1980, vol. 15, no. 1, pp. 9–20.

    MATH  Article  Google Scholar 

  39. 39.

    Shimada, I. and Nagashima, T., A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems, Progr. Theoret. Phys., 1979, vol. 61, no. 6, pp. 1605–1616.

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Arnol’d, V. I., Instability of Dynamical Systems with Many Degrees of Freedom, Dokl. Akad. Nauk SSSR, 1964, vol. 156, no. 1, pp. 9–12 (Russian).

    MathSciNet  Google Scholar 

  41. 41.

    Watanabe, Sh. and Strogatz, S. H., Integrability of a Globally Coupled Oscillator Array, Phys. Rev. Lett., 1993, vol. 70, no. 16, pp. 2391–2394.

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Watanabe, S. and Strogatz, S. H., Constants of Motion for Superconducting Josephson Arrays, Phys. D, 1994, vol. 74, nos. 3–4, pp. 197–253.

    MATH  Article  Google Scholar 

  43. 43.

    Marvel, S., Mirollo, R., and Strogatz, S., Identical Phase Oscillators with Global Sinusoidal Coupling Evolve by Möbius Group Action, Chaos, 2009, vol. 19, no. 4, 043104, 11 pp.

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Vlasov, V., Rosenblum, M., and Pikovsky, A., Dynamics of Weakly Inhomogeneous Oscillator Populations: Perturbation Theory on Top of Watanabe–Strogatz Integrability, J. Phys. A, 2016, vol. 49, no. 31, 31LT02, 8 pp.

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    Lohe, M. A., Higher-Dimensional Generalizations of the Watanabe–Strogatz Transform for Vector Models of Synchronization, J. Phys. A, 2018, vol. 51, no. 22, 225101, 24 pp.

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Witthaut, D., Wimberger, S., Burioni, R., and Timme, M., Classical Synchronization Indicates Persistent Entanglement in Isolated Quantum Systems, Nat. Commun., 2017, vol. 8, Art. 14829, 7 pp.

  47. 47.

    Ahnert, K. and Mulansky, M., Odeint — Solving Ordinary Differential Equations in C++, AIP Conf. Proc., 2011, vol. 1389, no. 1, pp. 1586–1589.

    Article  Google Scholar 

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Acknowledgments

The authors thank Prof. A. V. Borisov, who inspired them to study the Topaj–Pikovsky problem and its possible generalizations. The authors thank A. R. Safin for bringing the paper [34] to their attention. The authors thank Prof. A. Pikovsky, L. A. Smirnov, I. R. Sataev and A. O. Kazakov for useful discussions.

Funding

The work of V. P. Kruglov and S. P. Kuznetsov was supported by the grant of the Russian Science Foundation (project no. 15-12-20035) (formulation of the problem, analytical calculations and research of related topics (Sections 1 and 2)). The work of V. P. Kruglov was supported by the grant of the Russian Science Foundation (project no. 19-71-30012) (analytical and numerical calculations and interpretation of obtained results (Sections 3–6)).

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Correspondence to Vyacheslav P. Kruglov or Sergey P. Kuznetsov.

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Kruglov, V.P., Kuznetsov, S.P. Topaj–Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators. Regul. Chaot. Dyn. 24, 725–738 (2019). https://doi.org/10.1134/S1560354719060108

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Keywords

  • reversibility
  • involution
  • Hamiltonian system
  • Topaj–Pikovsky model
  • phase oscillator lattice

MSC2010 numbers

  • 37J15
  • 37C10
  • 37C70
  • 34D08
  • 34C15
  • 34C60