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

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Khovanov I. A., Khovanova N. A., McClintock P. Optimal control of fluctuations applied to the suppression of noise-induced failures of chaos stabilization. Izvestiya VUZ. Applied Nonlinear Dynamics, 2003, vol. 11, iss. 3, pp. 46-55. DOI: 10.18500/0869-6632-2003-11-3-46-55

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Language: 
English
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
519.6: 537.86:519.2
DOI: 
10.18500/0869-6632-2003-11-3-46-55

Optimal control of fluctuations applied to the suppression of noise-induced failures of chaos stabilization

Autors: 
Khovanov Igor Aleksandrovich, Saratov State University
Khovanova Natalia Aleksandrovna, Saratov State University
McClintock Peter Vaughan Elsmere, Lancaster University
Abstract: 

Double strategy of chaos and fluctuation controls is developed. Noise-induced failures in the stabilization of аn unstable orbit in the one-dimensional logistic mар are considered as large fluctuations from a stable state. The properties of the large fluctuations are examined by determination and analysis of the optimal path and the optimal fluctuational force corresponding to the stabilization failure. The problem of controlling noise-induced large fluctuations is discussed, and methods of control have been developed.

Key words: 
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Acknowledgments: 
We thank D.G. Luchinsky and V. Smelyansky for useful and stimulating discussions and help. The research was supported by the Engineering and Physical Sciences Research Council (UK) and INTAS (grant 01-867).
Reference: 
  1. Boccaletti S, Grebogi C, Lai Y-C, Mancini H, Maza D. The control of chaos: theory and applications. Phys. Rep. 2000;329(3):103–197. DOI: 10.1016/S0370-1573(99)00096-4.
  2. Ott E, Grebogi C, Yorke J. Controlling chaos. Phys. Rev. Lett. 1990;64(11):1196–1190. DOI: 10.1103/PhysRevLett.64.1196.
  3. Dykman MI, McClintock PVE, Smelyanski VN, Stein ND, Stocks NG. Optimal paths and the prehistory problem for large fluctuations in noise-driven systems. Phys. Rev. Lett. 1992;68(18):2718–2721. DOI: 10.1103/PhysRevLett.68.2718.
  4. Graham R, Hamm A, Tel Т. Nonequilibrium potentials for dynamical systems with fractal attractors or repellers. Phys. Rev. Lett. 1991;66(24):3089–3092. DOI: 10.1103/PhysRevLett.66.3089.
  5. Grassberger Р. Noise-induced escape from attractors. J. Phys. А. 1989;22(16):3283. DOI: 10.1088/0305-4470/22/16/018.
  6. Pontryagin LS. The Mathematical Theory of Optimal Processes. Macmillan; 1964. 382 p.
  7. Smelyanskiy VN, Dykman МI. Optimal control of large fluctuations. Phys. Rev. Е. 1997;55(3):2516–2521. DOI: 10.1103/PhysRevE.55.2516.
  8. Whittle P. Optimal Control: Basics and Beyond. Wiley; 1996. 474 p.
  9. Luchinsky DG. On the nature of large fluctuations in equilibrium systems: observation of an optimal force. J. Phys. А. 1997;30(16):L577. DOI: 10.1088/0305-4470/30/16/004.
  10. Khovanov LA, Luchinsky DG, Mannella R, McClintock PVE. Fluctuations and the energy-optimal control of chaos. Phys. Rev. Lett. 2000;85(10):2100–2103. DOI: 10.1103/PhysRevLett.85.2100.
  11. Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing. Cambridge: Cambridge University Press; 1989. 1256 p.
  12. Reimann Р, Talkner Р. Invariant densities for noisy maps. Phys. Rev. Е. 1991;44(10):6348–6363. DOI: 10.1103/PhysRevA.44.6348.
  13. Badii R, Brun E, Finardi M, Flepp L, Holzner R, Parisi J, Reyl C, Simonet J. Progress in the analysis of experimental chaos through periodic orbits. Rev. Mod. Phys. 1994;66(4):1389–1415. DOI: 10.1103/RevModPhys.66.1389.
Received: 
12.09.2003
Accepted: 
10.10.2003
Available online: 
23.11.2023
Published: 
31.12.2003
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