ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Kuznetsov A. P., Rahmanova A. Z., Savin A. V. The effect of symmetry breaking on reversible systems with myxed dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics, 2018, vol. 26, iss. 6, pp. 20-31. DOI: 10.18500/0869-6632-2018-26-6-20-31

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 538)
Article type: 
530.182, 517.9

The effect of symmetry breaking on reversible systems with myxed dynamics

Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Rahmanova A. Zh., Saratov State University
Savin Aleksej Vladimirovich, Saratov State University

Theme – the effect of symmetry violation on the structure of the phase space of invertible systems. Aim – to study the changes in the phase space structure of invertible systems caused by the violation of symmetry, in particular, the possibility of multistability and the types of coexisting attractors. The peculiarities in comparison with the similar regimes in the systems with fixed constant dissipation also studied. Methods – the numerical simulation of the system of coupled phase equations for four oscillators with weak coupling with different coupling functions both with symmetry and without it. The methods of phase portraits and attractors plotting, the calculation of Lyapunov exponents spectra, the search for stable and unstable cycles and the manifolds of saddle cycles are used. Results. It was shown that the violation of symmetry results in the destruction of conservative dynamics and the attractors occur. Unlike the systems with constant weak dissipation the number of coexisting attractors is small but both periodic and chaotic attractors occur. The heteroclinic structures also are revealed. Discussion – the results are rather common because of the simple nature of used system which is the model system for the wide class of systems – the chains of oscillating systems with weak coupling.  

  1. 1. Lichtenberg A., Lieberman M. Regular and Chaotic Dynamics. Springer-Verlag, 1983.
  2. Schuster H.G. Deterministic Chaos. Physik-Verlag, Weinheim, 1984.
  3. Gonchenko S.V., Turaev D.V. On three types of dynamics and the notion of attractor. Proc. of Steklov Inst. of Math., 2017, vol. 297, iss. 1, pp. 116–137.
  4. Lamb J.S.W., Roberts J.A.G. Time-reversal symmetry in dynamical systems: A survey. Physica D, 1998, vol. 112, pp. 1–39.
  5. Lamb J.S.W., Sten’kin O.V. Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits. Nonlinearity, 2004, vol. 17, pp. 1217–1244.
  6. Delshams A., Gonchenko S.V., Gonchenko V. S., Lazaro J. T., Sten’kin O. Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps. Nonlinearity, 2013, vol. 26, pp. 1–33.
  7. Gonchenko S.V., Lamb J.S.V., Rios I., Turaev D. Attractors and repellers near generic elliptic points of reversible systems. Doklady Mathematics, 2014, vol. 89, no. 1, p. 65.
  8. Leviatan A., Whelan N.D. Partial dynamical symmetry and mixed dynamics. Phys. Rew. Lett., 1996, vol.77, no. 26, pp. 5202–5205.
  9. Gonchenko A.S., Gonchenko S.V., Kazakov A.O., Turaev D.V. On the phenomenon of mixed dynamics in Pikovsky–Topaj system of coupled rotators. Physica D, 2017, vol. 350, pp. 45–57.
  10. Kazakov A.O. Strange atractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane. Regular and Chaotic Dynamics, 2013, vol. 18, no. 5, pp. 508–520.
  11. Feudel U., Grebogi C., Hunt B.R., Yorke J.A.. Map with more than 100 coexisting low-period attractors. Phys. Rev. E, 1996, vol.71, pp. 71–81.
  12. Feudel U., Grebogi C. Why are chaotic attractors rare in multistable systems? Phys. Rev. Lett., 2003, vol. 91, no. 13, 134102.
  13. Kolesov A.Yu., Rozov N.Kh. The nature of the bufferness phenomenon in weakly dissipative systems. Theoretical and Mathematical systems, 2006, vol. 146, no. 3, pp. 376–392.
  14. Martins L.С., Gallas J.A.C. Multistability, phase diagrams and statistical properties of the kicked rotor: A map with many coexisting attractors. International Journal of Bifurcation and Chaos, 2008, vol. 18, no. 6, pp. 1705–1717.
  15. Feudel U. Complex dynamics in multistable systems. International Journal of Bifurcation and Chaos, 2008, vol. 18, no. 6, pp. 1607–1626.
  16. Blazejczyk-Okolewska B., Kapitaniak T. Coexisting attractors of impact oscillator. Chaos, Solitons & Fractals, 1998, vol. 9, pp. 1439–1443.
  17. Feudel U., Grebogi C. Multistability and the control of complexity. Chaos, 1997, vol. 7, no. 4, pp. 597–604.
  18. Rech P., Beims M., Gallas J. Basin size evolution between dissipative and conservative limits. Physical Review E, 2005, vol. 71, no. 1, 017202.
  19. Jousseph C.F., Kruger T.S., Manchein C., Lopes S.R., Beims M.W. Weak dissipative effects on trajectories from the edge of basins of attraction. Physica A, 2016, vol. 456, pp. 68–74.
  20. Sabarathinam S., Thamilmaran K. Transient chaos in a globally coupled system of nearly conservative Hamiltonian–Duffing oscillators. Chaos, Solitons & Fractals, 2015, vol. 73, pp. 129–140.
  21. Erdogan M.B., Marzuola J.L., Newhall K., Tsirakis N. The structure of global attractors for dissipative zakharov systems with forcing on the torus. SIAM J. Applied Dynamical Systems, 2015, vol. 14, no. 4, pp. 1978–1990.
  22. Shrimali M.D., Prasad A., Ramaswami R., Feudel U. The nature of attractor basins in multistable systems. Int. J. of Bif. & Chaos, 2008, vol. 18, pp. 1675–1688.
  23. de Oliveira J.A., Leonel E.D. The effect of weak dissipation in two-dimensional mapping. Int. J. of Bif. & Chaos, 2012, vol. 22, no. 10, 1250248.
  24. Sendina-Nadal I., Letellier C. Synchronizability of nonidentical weakly dissipative systems. Chaos, 2017, vol. 27, 103118.
  25. Kovaleva A. Energy localization in weakly dissipative resonant chains. Phys. Rev. E, 2016, vol. 94, 022208.
  26. Yamagishi T. Effect of weak dissipation on a drift orbit mapping. J. of Physical Society of Japan, 2000, vol. 69, no. 9, pp. 2889–2894.
  27. Celletti A., Froeschle C., Lega E.. Dissipative and weakly-dissipative regimes in nearly-integrable mappings. Discrete and Continuous Dynamical Systems, 2006, vol. 16, no. 4, pp. 757–781.
  28. Felk E.V., Savin A.V., Kuznetsov A.P. Transient chaos in multidimensional Hamiltonian system with weak dissipation. European Physical Journal. Special Topics, 2017, vol. 226, no. 9, pp. 1777– 1784.
  29. Kuznetsov A.P., Savin A.V., Savin D.V. On some properties of nearly conservative dynamics of Ikeda map and its relation with the conservative case. Physica A, 2008, vol. 387, no. 7, pp. 1464–1474.
  30. Felk E.V., Savin A.V., Kuznetsov A.P. Effect of weak dissipation on the dynamics of multidimensional Hamiltonian systems. Nonlinear Phenomena in Complex Systems, 2015, vol. 18, no. 2, pp. 259–265.
  31. Felk E.V., Kuznetsov A.P., Savin A.V. Multistability and transition to chaos in the degenerate Hamiltonian system with weak nonlinear dissipative perturbation. Physica A, 2014, vol. 410, pp. 561–572.
  32. Kuznetsov A.P., Savin A.V., Savin D.V. Features in the dynamics of an almost conservative Ikeda map. Technical Physics Letters, 2007, vol.33, no. 2, p. 122.
  33. Pikovsky A., Topaj D. Reversibility vs. synchronization in oscillator lattices. Physica D, 2002, vol. 170, pp. 118–130.
  34. Pikovsky A., Rosenblum M., Kurts J. Synchronization: A Universal Concept in Nonlinear Science. Cambridge University Press, 2001.
Short text (in English):
(downloads: 101)