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


For citation:

Adilova A. B., Kuznetsov A. P., Savin A. V. Complex dynamics in the system of two coupled discrete Rossler oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, iss. 5, pp. 108-119. DOI: 10.18500/0869-6632-2013-21-5-108-119

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text:
(downloads: 102)
Language: 
Russian
Article type: 
Article
UDC: 
517.9

Complex dynamics in the system of two coupled discrete Rossler oscillators

Autors: 
Adilova Asel Bauyrzhanovna, Saratov State University
Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Savin Aleksej Vladimirovich, Saratov State University
Abstract: 

We considered the discrete map with quasi-periodic dynamics in the wide band of the parameters and investigated the structure of the parameter plane of two coupled maps. We revealed the doublings of 3D-tori, the systems of 2D-tori and synchronization tongues and the resonance web. Also we revealed the attractors with complex structure and the largest Lyapunov exponent close to zero.

Reference: 
  1. Landa PS. Self-oscillations in systems with a finite number of degrees of freedom. Moscow: Nauka; 1980. 359 p. (In Russian).
  2. Rabinovich MI, Trubetskov DI. Introduction to the theory of vibrations and waves. Мoscow-Izhevsk: ICS; 2000. 560 p. (In Russian).
  3. Pikovsky A, Rosenblum M, Curts Yu. Synchronization is a fundamental nonlinear phenomenon. Moscow: Tehnosphera; 2003. 496 p. (In Russian).
  4. Anishchenko V.S., Astakhov V.V., Vadivasova T.E., Strelkova G.I. Synchronization of regular, chaotic and stochastic oscillations. Мoscow-Izhevsk: ICS; 2008. 136 p. (In Russian).
  5. Neimark YI, Landa PS. Stochastic and chaotic fluctuations. Moscow: Nauka; 1987. 424 p.(In Russian).
  6. Baesens C, Guckenheimer J, Kim S, MacKay RS. Three coupled oscillators: mode locking, global bifurcations and toroidal chaos. Physica D. 1991;49(3):387–475. DOI: 10.1016/0167-2789(91)90155-3.
  7. Anishchenko V, Astakhov S, Vadivasova T. Phase dynamics of two coupled oscillators under external periodic force. Europhysics Letters. 2009;86(3):30003. DOI: 10.1209/0295-5075/86/30003.
  8. Kuznetsov AP, Sataev IR, Turukina LV. Synchronization of quasi-periodic oscillations in coupled phase oscillators. Technical Physics Letters. 2010;36(5):478–481. DOI: 10.1134/S1063785010050263.
  9. Anishchenko VS, Nikolaev SM. Synchronization of two-frequency quasi-periodic oscillations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(2):69–86. DOI: 10.18500/0869-6632-2008-16-2-69-86.
  10. Kuznetsov AP, Pozdnyakov MV, Sedova JV. Coupled universal maps demonstrating Neimark–Saker bifurcation. Nelin. Dinam. 2012;8(3):473–482.
  11. Nishiuchi Y, Ueta T, Kawakami H. Stable torus and its bifurcation phenomena in a simple three-dimensional autonomous circuit. Chaos, Solutions & Fractals. 2006;27(4):941–951. DOI: 10.1016/j.chaos.2005.04.092.
  12. Anishchenko V, Nikolaev S, Kurths J. Bifurcational mechanisms of synchronization of a resonant limit cycle on a two-dimensional torus. CHAOS. 2008;18(3):037123. DOI: 10.1063/1.2949929.
  13. Kuznetsov AP, Kuznetsov SP, Stankevich NV. Autonomous generator of quasiperiodic oscillations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(2):51–61. DOI: 10.18500/0869-6632-2010-18-2-51-61.
  14. Zaslavsky G.M. Chaos physics in Hamiltonian systems. Мoscow-Izhevsk: ICS; 2004. 288 p. (In Russian).
  15. Morozov AD. Resonances, cycles and chaos in quasi-conservative systems. Мoscow-Izhevsk: ICS; 2005. 423 p. (In Russian).
  16. Kuznetsov AP, Savin AV, Sedova YV. Bogdanov–Takens bifurcation: from flows to discrete systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(6):139–158. DOI: 10.18500/0869-6632-2009-17-6-139-158.
  17. Rossler OE. An equation for continuous chaos. Phys. Lett. 1976;57(5):397–398. DOI: 10.1016/0375-9601(76)90101-8.
  18. Kuznetsov AP, Paksjutov VI. Dynamics of two nonidentical coupled self-sustained systems with period doublings on the example of Rossler oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(2):3–15. DOI: 10.18500/0869-6632-2006-14-2-3-15.
  19. Froeschle C, Lega E, Guzzo M. Analysis of the chaotic behaviour of orbits diffusing along the Arnold web. In book «Periodic, Quasi-Periodic and Chaotic Motions in Celestial Mechanics: Theory and Applications» 2006;95(1-4):141–153. DOI:10.1007/978-1-4020-5325-2_8.
  20. Guzzo M, Lega E, Froeschle C. Diffusion and stability in perturbed non-convex integrable systems. Nonlinearity. 2006;19(5):1049–1067. DOI:10.1088/0951-7715/19/5/003.
  21. Honjo S, Kaneko K. Is Arnold diffusion relevant to global diffusion? http://arxiv.org/abs/nlin/0307050.
  22. Vitolo R, Broer H, Simу C. Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms. Nonlinearity. 2010;23(8):1919–1947. DOI: 10.1088/0951-7715/23/8/007.
Received: 
06.02.2013
Accepted: 
06.02.2013
Published: 
31.12.2013
Short text (in English):
(downloads: 82)