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

For citation:

Kuznetsov A. P., Stankevich N. V., Chernyshov N. Y. Stabilization of chaos in the rossler system by pulsed or harmonic signal. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 4, pp. 3-16. DOI: 10.18500/0869-6632-2010-18-4-3-16

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: 128)
Article type: 

Stabilization of chaos in the rossler system by pulsed or harmonic signal

Kuznetsov Aleksandr Petrovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Stankevich Natalija Vladimirovna, National Research University "Higher School of Economics"
Chernyshov Nikolaj Yurevich, Saratov State University

The stabilization of chaos in the Rossler system by external signal is investigated. Different types of external action are considered: both of pulsed and harmonic signal. There are illustrations: charts of dynamical regimes, phase porters, stroboscopic section of Poincare, spectrum of Lyapunov exponents. Comparative analysis of efficiency of stabilization of band chaos and spiral chaos by different signal is carried out. The dependence of synchronization picture on direction of acting pulses is shown.

  1. Ott E. Chaos in Dynamical Systems. Cambridge University press; 1993.
  2. Anishchenko VS, Vadivasova TE, Astakhov VV. Nonlinear dynamics of chaotic and stochastic systems. Fundamental principles and selected issues. Saratov: Saratov Univ. Publ.; 1999. 368 p. (in Russian).
  3. Pikovsky A, Rosenblum M, Kurts Yu. Synchronization: A fundamental nonlinear phenomenon. Moscow: Tehnosphera; 2003. 493 p.
  4. Schuster HG. Handbook of Chaos Control. Weinheim: Wiley-VCH; 1999.
  5. Boccaletti S, Grebogi C, Lai YC, Mancini H, Maza D. The control of chaos: theory and applications. Physics Reports – Review Section of Physics Letters. 2000;329:103–197. DOI: 10.1016/S0370-1573(99)00096-4.
  6. Gauthier D, Hall GM, Olivier RA, Dixon-Tulloch EG, Wolf PD, Bahar S. Progress toward controlling in vivo fibrillating sheep atria using a nonlinear dynamics based closed loop feedback method. CHAOS. 2002;12(3):952–961. DOI: 10.1063/1.1494155.
  7. Ott E, Grebogi C, Yorke J.A. Controlling chaos. Phys. Rev. Lett. 1990;64(11):1196–1199. DOI: 10.1103/PhysRevLett.64.1196.
  8. Mori H, Kuramoto Y. Dissipative Structures and Chaos. Springer; 1998.
  9. Stone EF. Frequency entrainment of phase coherent attractor. Physics Letters A. 1992;163(5-6):367–374. DOI: 10.1016/0375-9601(92)90841-9.
  10. Rossler OE. An equation for continuous chaos. Physics Letters A. 1976;57(5):397–398. DOI: 10.1016/0375-9601(76)90101-8.
  11. Rossler OE. Chaos in abstract kinetics: Two prototypes. Bulletin of Mathematical Biology. 1977;39(2):275–289. DOI: 10.1007/BF02462866.
  12. Kuznetsov SP. Dynamical Chaos: Course of Lectures. Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  13. Kuznetsov SP, Sataev IR. Universality and scaling for the breakup of phase synchronization at the onset of chaos in a periodically driven Rössler oscillator. Phys. Rev. E. 2001;64:046214. DOI: 10.1103/PhysRevE.64.046214.
  14. Kuznetsov AP, Stankevich NV, Turukina LV. Features of the synchronization picture by the pulses in the system with 3-dimensional phase space by the example of the Ressler system. Izvestiya VUZ. Applied Nonlinear Dynamics. 2006;14(6):43–53. DOI: 10.18500/0869-6632-2006-14-6-43-53.
  15. Kuznetsov AP, Stankevich NV, Turukina LV. Stabilizing the Rössler system by external pulses on a runaway trajectory. Technical Physics Letters. 2008;34:618–621. DOI: 10.1134/S1063785008070250.
  16. Ding EJ. Structure of parameter space for a prototype nonlinear oscillator. Phys. Rev. A. 1987;36(3):1488–1491. DOI: 10.1103/PhysRevA.36.1488.
  17. Ding EJ. Structure of the parameter space for the van der Pol oscillator. Physica Scripta. 1988;38(1):9–17. DOI: 10.1088/0031-8949/38/1/001.
  18. Glass L, Sun J. Periodic forcing of a limit-cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations. Phys. Rev. E. 1994;50(6):5077–5084. DOI: 10.1103/physreve.50.5077.
  19. Carcasses J, Mira C, Bosch M, Simo C, Tatjer JC. «Crossroad area – spring area» transition (I) Parameter plane representation. Int. J. Bif. and Chaos. 1991;1(1):183–196. DOI: 10.1142/S0218127491000117.
  20. Carcasses J, Mira C, Bosch M, Simo C, Tatjer JC. «Crossroad area – spring area» transition (II) Foliated parametric representation. Int. J. Bif. and Chaos. 1991;1(2):339–348.
Short text (in English):
(downloads: 81)