Self-sustained oscillations of dynamical and stochastic systems and their mathematical image --- an attractor
Russian journal of nonlinear dynamics, Tome 6 (2010) no. 1, pp. 107-126.

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In the present paper autonomous and nonautonomous oscillations of dynamical and stochastic systems are analyzed in the framework of common concepts. The definition of an attractor is introduced for a nonautonomous system. The definitions of self-sustained oscillations and a self-sustained oscillatory system is proposed, that generalize A. A. Andronov's concept introduced for autonomous systems with one degree of freedom.
Keywords: self-sustained oscillations, dynamical chaos, attractor
Mots-clés : fluctuations.
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V. S. Anishchenko; T. E. Vadivasova; G. I. Strelkova. Self-sustained oscillations of dynamical and stochastic systems and their mathematical image --- an attractor. Russian journal of nonlinear dynamics, Tome 6 (2010) no. 1, pp. 107-126. http://geodesic.mathdoc.fr/item/ND_2010_6_1_a6/

[1] Andronov A. A., Vitt A. A., Khaikin S. E., Teoriya kolebanii, Nauka, M., 1981 | MR | Zbl

[2] Afraimovich V. S., Shilnikov L. P., “Strange attractors and quasiattractors”, Nonlinear dynamics and turbulence, eds. G. I. Barenblatt, G. Iooss, D. D. Joseph, Pitman, Boston, 1983, 1–34 | MR

[3] Anischenko V. S., Slozhnye kolebaniya v prostykh sistemakh, Nauka, M., 1990 ; 2-е изд., УРСС, М., 2009 | MR

[4] Arnold V. I., Afraimovich V. S., Ilyashenko Yu. S., Shilnikov L. P., “Teoriya bifurkatsii”, Ser. Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, VINITI, M., 1986 | MR

[5] Arnold L., Random dynamical systems, Springer, Berlin, 2003 | MR

[6] Schmalfuss B., “The random attractor of the stochastic Lorenz system”, ZAMP, 48 (1997), 951–975 | DOI | MR | Zbl

[7] Crauell H., Debussche A., Flandoli F., “Random attractors”, J. Dynam. Differential Equations, 9 (1997), 307–341 | DOI | MR

[8] Stratonovich R. L., Izbrannye voprosy teorii fluktuatsii v radiotekhnike, Sov. radio, M., 1961 | Zbl

[9] Malakhov A. N., Fluktuatsii v avtokolebatelnykh sistemakh, Nauka, M., 1968

[10] Anischenko V. S., Astakhov V. V., Vadivasova T. E., Strelkova G. I., Sinkhronizatsiya regulyarnykh, khaoticheskikh i stokhasticheskikh kolebanii, NITs “Regulyarnaya i khaoticheskaya dinamika”, M.-Izhevsk, 2008

[11] Anischenko V. S., Astakhov V. V., Vadivasova T. E., Neiman A. B., Strelkova G. I., Shimanskii-Gaier L., Nelineinye effekty v khaoticheskikh i stokhasticheskikh sistemakh, Inst. kompyuter. issled., M.-Izhevsk, 2003

[12] Neiman A. B., “Synchronizationlike phenomena in coupled stochastic bistable systems”, Phys. Rev. E, 49 (1994), 3484–3488 | DOI

[13] Shulgin B. V., Neiman A. B., Anishchenko V. S., “Mean switching frequency locking in stochastic bistable systems driven by periodic force”, Phys. Rev. Lett., 75 (1995), 4157–4160 | DOI

[14] Pikovsky A., Kurths J., “Coherence resonance in a noisy driven excitable system”, Phys. Rev. Lett., 78 (1997), 775–778 | DOI | MR | Zbl

[15] Han S. K., Yim T. G., Postnov D. E., Sosnovtseva O. V., “Interacting coherence resonance oscillators”, Phys. Rev. Lett., 83:9 (1999), 1771–1774 | DOI

[16] FitzHugh R. A., “Impulses and physiological states in theoretical models of nerve membrane”, Biophys. J., 1 (1961), 445–466 | DOI

[17] Scott A. C., “The electrophysics of a nerve fiber”, Rev. Mod. Phys., 47 (1975), 487–533 | DOI

[18] Landa P. S., Nonlinear oscillation and waves in dynamical systems, Kluwer Academic, Dordrecht, 1996 | MR | Zbl