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


For citation:

Koronovskii A. A., Hramov A. E. Detection of unstable periodical spatio-temporal states of spatial extended chaotic systems dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics, 2007, vol. 15, iss. 4, pp. 26-33. DOI: 10.18500/0869-6632-2007-15-4-26-33

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)
Language: 
Russian
Heading: 
Article type: 
Article
UDC: 
517.9

Detection of unstable periodical spatio-temporal states of spatial extended chaotic systems dynamics

Autors: 
Koronovskii Aleksei Aleksandrovich, Saratov State University
Hramov Aleksandr Evgenevich, Immanuel Kant Baltic Federal University
Abstract: 

The method of detection of the unstable periodic spatio-temporal states of spatial extended chaotic systems dynamics is proposed. The application of this method is illustrated by the consideration of the fluid model of Pierce diode which is one of the base system of plasma physics and of microwave electronics.  

Key words: 
Reference: 
  1. Cvitanovic P. Periodic orbits as the skeleton of classical and quantum chaos. Physica D. 1991;51:138–151. DOI: 10.1016/0167-2789(91)90227-Z.
  2. Barreto E, Hunt BR, Grebogi C, Yorke JA. From high dimensional chaos to stable periodic orbits: The structure of parameter space. Phys. Rev. Lett. 1997;78(24):4561–4564. DOI: 10.1103/PhysRevLett.78.4561.
  3. Carroll TL. Approximating chaotic time series through unstable periodic orbits. Phys. Rev. E. 1999;59(2):1615.
  4. Pikovsky AS, Grassberger P. Symmetry breaking bifurcation for coupled chaotic attractors. J. Phys. A. 1991;24:4587–4597. DOI: 10.1088/0305-4470/24/19/022.
  5. Pikovsky AS, Zaks M, Rosenblum MG, Osipov GV, Kurths J. Phase synchronization of chaotic oscillators in terms of periodic orbits. Chaos. 1997;7(4):680–687. DOI: 10.1063/1.166265.
  6. Hramov AE, Koronovskii AA, Kurovskaya MK, Moskalenko OI. Synchronization of spectral components and its regularities in chaotic dynamical systems. Phys. Rev. E. 2005;71(5):056204. DOI: 10.1103/PhysRevE.71.056204.
  7. Pyragas K. Continuous control of chaos, by self-controlling feedback. Phys. Lett. A. 1992;170:421–428. DOI: 10.1016/0375-9601(92)90745-8.
  8. Lathrop DP, Kostelich EJ. Characterization of an experimental strange attractor by periodic orbits. Phys. Rev. A. 1989;40(7):4028–4031. DOI: 10.1103/physreva.40.4028.
  9. Schmelcher P, Diakonos FK. Detecting unstable periodic orbits of chaotic dynamical systems. Phys. Rev. Lett. 1997;79(25):4733–4736. DOI: 10.1103/PhysRevLett.78.4733.
  10. Pingel D, Schmelcher P, Diakonos FK. Detecting unstable periodic orbits in chaotic continuous-time dynamical systems. Phys. Rev. E. 2001;64(2):026214. DOI: 10.1103/PhysRevE.64.026214.
  11. Koronovsky AA, Rempen IS, Hramov AE. Investigation of unstable periodic spatiotemporal states in distributed active system with supercritical current. Bulletin of the Russian Academy of Sciences: Physics. 2003;67(12):1705–1708 (in Russian).
  12. Rempen IS, Hramov AE. Stabilization of unstable periodic states of chaotic dynamics in Pierce diode. Bulletin of the Russian Academy of Sciences: Physics. 2004;68(12):1781–1785 (in Russian).
  13. Franceschini G, Bose S, Scholl E. Control of chaotic spatiotemporal spiking by time-delay autosynchronization. Phys. Rev. E. 1999;60(5):5426–5434. DOI: 10.1103/physreve.60.5426.
  14. Hramov AE, Koronovskii AA, Rempen IS. Controlling chaos in spatially extended beam-plasma system by the continuous delayed feedback. Chaos. 2006;16(1):013123. DOI: 10.1063/1.2168394.
  15. Friedel H, Grauer R, Spatschek HK. Controlling chaotic states of a Pierce diode. Physics of plasmas. 1998;5(9):3187–3194. DOI: 10.1063/1.872987.
  16. Anfinogentov V. G. Chaotic oscillation in the electron beam with virtual cathode. Izvestiya VUZ. Applied Nonlinear Dynamics. 1994;2(5):69–83 (in Russian).
  17. Trubetskov DI, Khramov AE. Lectures on Microwave Electronics for Physicists. Vol. 1. Moscow: Fizmatlit; 2003. (in Russian)
  18. Godfrey BB. Oscillatory nonlinear electron flow in a Pierce diode. Phys. Fluids. 1987;30:1553–1560. DOI: 10.1063/1.866217.
  19. Anfinogentov VG, Trubetskov DI. Chaotic oscillations in the hydrodynamic model of the Pierce's diode. Journal of Communications Technology and Electronics. 1992;37:2251 (in Russian).
  20. Matsumoto H, Yokoyama H, Summers D. Computer simulations of the chaotic dynamics of the Pierce beam–plasma system. Phys. of Plasmas. 1996;3(1):177–191. DOI: 10.1063/1.871844.
  21. Hramov AE, Rempen IS. Investigation of the complex dynamics and regime control in Pierce diode with the delay feedback. Int. J. Electronics. 2004;91(1):1–12. DOI: 10.1080/00207210310001658932.
  22. Pecora LM, Carroll TL, Heagy JF. Statistics for mathematical properties of maps between time series embeddings. Phys. Rev. E. 1995;52(4):3420–3439. DOI: 10.1103/physreve.52.3420.
Received: 
10.01.2007
Accepted: 
10.01.2007
Published: 
31.07.2007
Short text (in English):
(downloads: 81)