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

For citation:

Slepnev A. V., Vadivasova T. E. Period doubling bifurcations and noise excitation effects in a multistable self-sustained oscillatory medium. Izvestiya VUZ. Applied Nonlinear Dynamics, 2011, vol. 19, iss. 4, pp. 53-67. DOI: 10.18500/0869-6632-2011-19-4-53-67

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: 120)
Article type: 
537.86, 530.182

Period doubling bifurcations and noise excitation effects in a multistable self-sustained oscillatory medium

Slepnev Andrej Vjacheslavovich, Saratov State University
Vadivasova Tatjana Evgenevna, Saratov State University

The model of a self-oscillatory medium composed from the elements with complex self-oscillatory behavior is studied. Under periodic boundary conditions the stable selfoscillatory regimes in the form of traveling waves with different phase shifts are coexisted in medium. The study of mechanisms of the oscillations period doubling in time is performed for different coexisted modes. For all observed spatially-non-uniform regimes (traveling waves) the period doubling occurs through the appearance of time-quasiperiodic oscillations and their further evolution. The period doubling result in multistability development. For each mode with the given phase shift the different stable non-uniform structures, which are differed by the distribution of oscillations characteristics in space, emerge. The influence of a noise signal leads to the shift of doubling bifurcation in the direction of the control parameter increasing. When the value of control parameter is fixed the stochastic bifurcations of contingency, which are shown in reduction of extremes numbers in the probabilistic distribution, are observed with the increasing of noise intensity. When the noise is sufficiently great the spatially-non-uniform modes corresponding to nonzero phase shifts disappear.

  1. Gaponov-Grekhov AV, Rabinovich MI. Ginzburg-Landau equation and nonlinear dynamics of nonequilibrium media. Radiophys. Quantum Electron. 1987;30(2)93–102. DOI: 10.1007/BF01034485.
  2. Aranson IS, Kramer L. The world of the complex Ginzburg–Landau equation. Reviews of Modern Physics. 2002;74(1):99–143. DOI: 10.1103/RevModPhys.74.99.
  3. Osipov GV, Pikovsky AS, Rosenblum MG, Kurths J. Phase synchronization effects in a lattice of nonidentical Rossler oscillators. Physical Review E. 1997;55(3):2353–2361. DOI: 10.1103/PhysRevE.55.2353.
  4. Shabunin AV, Feudel U, Astakhov VV. Phase multistability, phase synchronization in an array of locally coupled period-doubling oscillators. Physical Review E. 2009;80(2):026211. DOI: 10.1103/PhysRevE.80.026211.
  5. Belykh VN, Verichev NN, Kocarev L, Chua LO. On chaotic synchronization in a linear array of Chua’s circuits. Journal of Circuits, Systems, Computers. 1993;3(2):325–335. DOI: 10.1142/9789812798855_0014.
  6. Shabunin AV, Astakhov VV, Anishchenko VS. Developing chaos on base of traveling waves in a chain of coupled oscillators with period-doubling: synchronization, hierarchy of multistability formation. International Journal of Bifurcation, Chaos. 2002;12(8):1895–1907. DOI: 10.1142/S021812740200556X.
  7. Anishchenko VS, Aranson IS, Postnov DE, Rabinovich MI. Spatial synchronization and bifurcations of the development of chaos in a chain of coupled generators. Proc. Acad. Sci. USSR. 1986;286(5):1120–1124 (in Russian).
  8. Kaneko K. Spatiotemporal chaos in one-, two-dimensional coupled map lattices. Physica D. 1989;37(1–3):60–82. DOI: 10.1016/0167-2789(89)90117-6.
  9. Kuznetsov AP, Kuznetsov SP. Critical dynamics of coupled-map lattices at onset of chaos (review). Radiophys. Quantum Electron. 1991;34(10–12):845–868. DOI:10.1007/BF01083617.
  10. Garca-Ojalvo J, Sancho JM. Noise in Spatially Extended Systems. New York: Springer; 1999. 307 p. DOI: 10.1007/978-1-4612-1536-3.
  11. Garca-Ojalvo J, Hernandez-Machado A, Sancho JM. Effects of external noise on the Swift–Hohenberg equation. Physical Review Letters. 1993;71(10):1542–1545. DOI: 10.1103/PhysRevLett.71.1542.
  12. Cross MC, Hohenberg PC. Pattern formation outside of equilibrium. Reviews of Modern Physics. 1993;65(3):851–1112. DOI: 10.1103/RevModPhys.65.851.
  13. Vinals J, Hernandez-Garca E, Miguel MS, Toral R. Numerical study of the dynamical aspects of pattern selection in the stochastic Swift-Hohenberg equation in one dimension. Physical Review A. 1991;44(2):1123–1133. DOI: 10.1103/PhysRevA.44.1123.
  14. Kuznetsov SP. Noise-induced absolute instability. Mathematics, Computers in Simulation. 2002;58(4–6):435–442. DOI: 10.1016/S0378-4754(01)00382-2.
  15. Anishchenko VS, Akopov AA, Vadivasova TE, Strelkova GI. Mechanisms of chaos onset in an inhomogeneous medium under cluster synchronization destruction. New Journal of Physics. 2006;8(6):84. DOI: 10.1088/1367-2630/8/6/084.
  16. Hramov AE, Koronovskii AA, Popov PV. Incomplete noise-induced synchronization of spatially extended systems. Physical Review E. 2008;77(3):036215. DOI: 10.1103/PhysRevE.77.036215.
  17. Shabunin AV, Akopov AA, Astahov VV, Vadivasova TE. Running waves in a discrete anharmonic self-oscillating medium. Izvestiya VUZ. Applied Nonlinear Dynamics. 2005;13(4):37–55 (in Russian). DOI: 10.18500/0869-6632-2005-13-4-37-55.
  18. Anishchenko VS, Astakhov VV. Experimental study of the mechanism of occurrence and structure of a strange attractor in a generator with inertial nonlinearity. Radio Engineering and Electronic Physics. 1983;28(6):1109–1115 (in Russian).
  19. Anishchenko VS. Complex Vibrations in Simple Systems. Moscow: Nauka; 1990. 312 p. (in Russian).
  20. Anishchenko VS, Vadivasova TE, Strelkova GI. Instantaneous phase method in studying chaotic, stochastic oscillations and its limitations. Fluctuation, Noise Letters. 2004;4(1):L219–L229. DOI: 10.1142/S0219477504001835.
  21. Pecora LM. Synchronization conditions, desynchronizing patterns in coupled limit-cycle, chaotic systems. Physical Review E. 1998;58(1):347–360. DOI: 10.1103/PhysRevE.58.347.
  22. Astakhov VV, Bezruchko BP, Gulyaev YV, Seleznev EP. Multistable states of dissipatively coupled Feigenbaum systems. Tech. Phys. Lett. 1988;15(3):60–65 (in Russian).
  23. Astakhov VV, Bezruchko BP, Ponomarenko VI. Multistability and isomer classification and evolution in coupled feigenbaum systems. Radiophys. Quantum Electron. 1991;34(1):28–33. DOI: 10.1007/BF01048411.
  24. Slepnev AV, Vadivasova TE, Listov AS. Multistability, period doubling and traveling waves suppression by noise excitation in a nonlinear self-oscillatory medium with periodic boundary conditions. Russian Journal of Nonlinear Dynamics 2010;6(4):755–767 (in Russian). DOI: 10.20537/nd1004004.
  25. Slepnev A.V. Phase multistability and the influence of a local noise source in a self-oscillating medium model. In: Nonlinear Days in Saratov for Young People - 2009: Collection of Materials of the Scientific School-Conference. Saratov: LLL RC «Nauka»; 2010. P. 94–97 (in Russian).
  26. Svensmark H, Samuelsen MR. Perturbed period-doubling bifurcation. I. Theory. Physical Review B. 1990;41(7):4181–4188. DOI: 10.1103/PhysRevB.41.4181.
  27. Arnold L. Random Dynamical Systems. Berlin: Springer; 2003. 586 p. DOI: 10.1007/978-3-662-12878-7.
Short text (in English):
(downloads: 74)