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


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Zulpukarov M. M., Malinetskii G. G., Podlazov A. V. Bifurcation theory inverse problem in a noisy dynamical system. Example solution. Izvestiya VUZ. Applied Nonlinear Dynamics, 2005, vol. 13, iss. 6, pp. 3-23. DOI: 10.18500/0869-6632-2005-13-5-3-23

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Language: 
Russian
Article type: 
Article
UDC: 
530.18

Bifurcation theory inverse problem in a noisy dynamical system. Example solution

Autors: 
Zulpukarov Magomed-Gerej Medzhidovich, Keldysh Institute of Applied Mathematics (Russian Academy of Sciences)
Malinetskii Georgij Gennadevich, Keldysh Institute of Applied Mathematics (Russian Academy of Sciences)
Podlazov Andrej Viktorovich, Keldysh Institute of Applied Mathematics (Russian Academy of Sciences)
Abstract: 

Bifurcations in nonlinear systems with weak noise are considered. The local bifurcations are discussed: the saddle-node bifurcation, the transcritical bifurcation, the supercritical and subcritical pitchfork bifurcations. Basing on the known prebifurcational noise rise and saturation phenomenon, the inverse problem is introduced: the problem of the bifurcation detection and determining it’s type by the observed noise change (noise deviation growth fashion, saturation level, probability density). The inverse problem solution algorithm is suggested.

Key words: 
Reference: 
  1. Malinetskiy GG, Potapov AB. Modern Problems of Nonlinear Dynamics. Moscow: URSS; 2002. 336 p. (in Russian).
  2. Kravtsov YA, Bilchinskaya SG, Butkovskii OY et al. Prebifurcational noise rise in nonlinear systems. J. Exp. Theor. Phys. 2001;93(6):1323–1329. DOI: 10.1134/1.1435756.
  3. Kuznetsov IV, Malinetski GG, Podlazov AV. About national system of scientific monitoring. Preprint Inst. Appl. Math. the Russian Academy of Science. No. 47. Moscow: Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences; 2004. 30 p. (in Russian).
  4. Nicolis G, Prigogine I. Self-Organization in Nonequilibrium Systems. From Dissipative Structures to Order through Fluctuations. New York, London, Sydney, Toronto: J. Wiley & Sons; 1977. 491 p.
  5. Kapitsa SP, Kurdyumov SP, Malinetskiy GG. Synergetics and Future Forecasts. 3d edition. Moscow: URSS; 2003. 288 p. (in Russian).
  6. Wiesenfeld K. Virtual Hopf phenomenon: A new precursor of period-doubling bifurcations. Phys. Rev. A. 1985;32(3):1744–1751. DOI: 10.1103/PhysRevA.32.1744.
  7. Kravtsov YA, Surovyatkina ED. Nonlinear saturation of prebifurcation noise amplification. Phys. Lett. A. 2003;319(3–4):348–351. DOI: 10.1016/j.physleta.2003.10.034.
  8. Surovyatkina E. Prebifurcation noise amplification and noise-dependent hysteresis as indicators of bifurcations in nonlinear geophysical systems. Nonlinear Processes in Geophysics. 2005;12(1):25–29. DOI: 10.5194/npg-12-25-2005.
  9. Juel A, Darbyshire AG, Mullin T. The effect of noise on pitchfork and Hopf bifurcations. Proc. R. Soc. Lond. A. 1997;453(1967):2627–2647. DOI: 10.1098/rspa.1997.0140.
  10. Anishchenko VS, Neiman AB. Structure and properties of chaos in the presence of noise. In: Sagdeev RZ, Frisch U, Hussain F, Moiseev SS, Erokhin NS. Nonlinear Dynamics of Structures. Singapore–New Jersey–London–Hong Kong: World Scientific; 1991. P. 21–48.
  11. Surovyatkina ED. Rise and saturation of the correlation time near bifurcation threshold. Phys. Lett. A. 2004;329(3):169–172. DOI: 10.1016/j.physleta.2004.06.092.
  12. Surovyatkina E, Kurths J. Pre-bifurcational noise-dependent phenomena as diagnostic instrument for revealing bifurcations in geophysical systems. Geophysical Research Abstracts. 2005;7:00462.
  13. Malinetskiy GG. Chaos. Structures. Computational Experiment. Introduction to Nonlinear Dynamics. Moscow: Nauka; 1997. 255 p. (in Russian).
  14. Iooss J, Joseph D. Elementary Stability and Bifurcation Theory. New York: Springer; 1990. 324 p. DOI: 10.1007/978-1-4612-0997-3.
  15. Zeldovich YB, Myshkis AD. Elements of Mathematical Physics. Non-Interacting Particle Environment. Moscow: Nauka; 1973. 352 p. (in Russian).
  16. Fedoryuk MV. Pass Method. Moscow: Nauka; 1977. 368 p. (in Russian).
  17. Tikhonov AN, Arsenin VY. Methods for Solving Ill-Posed Problems. New York: Wiley; 1979. 288 p.
Received: 
02.06.2005
Accepted: 
23.09.2005
Published: 
28.02.2006
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