Regular and Chaotic Dynamics

, Volume 24, Issue 6, pp 725–738 | Cite as

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

  • Vyacheslav P. KruglovEmail author
  • Sergey P. KuznetsovEmail author


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.


reversibility involution Hamiltonian system Topaj–Pikovsky model phase oscillator lattice 

MSC2010 numbers

37J15 37C10 37C70 34D08 34C15 34C60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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.


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)).

Conflict of Interest

The authors declare that they have no conflicts of interest.


  1. 1.
    Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence, Springer Ser. Synergetics, vol. 19, Berlin: Springer, 1984.zbMATHCrossRefGoogle 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.zbMATHCrossRefGoogle Scholar
  3. 3.
    Pikovsky, A. and Politi, A., Lyapunov Exponents: A Tool to Explore Complex Dynamics, Cambridge: Cambridge Univ. Press, 2016.zbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Topaj, D. and Pikovsky, A., Reversibility vs. Synchronization in Oscillator Lattices, Phys. D, 2002, vol. 170, no. 2, pp. 118–130.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.CrossRefGoogle 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.MathSciNetCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.CrossRefGoogle 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.MathSciNetCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.CrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.CrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.CrossRefGoogle Scholar
  34. 34.
    Witthaut, D. and Timme, M., Kuramoto Dynamics in Hamiltonian Systems, Phys. Rev. E, 2014, vol. 90, no. 3, 032917, 8 pp.CrossRefGoogle 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.CrossRefGoogle Scholar
  36. 36.
    Adler, R., A Study of Locking Phenomena in Oscillators, Proc. IRE, 1946, vol. 34, no. 6, pp. 351–357.CrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.zbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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).MathSciNetGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.zbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.MathSciNetzbMATHCrossRefGoogle 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.Google Scholar
  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.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Kotelnikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov BranchSaratovRussia
  3. 3.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations