Документ взят из кэша поисковой машины. Адрес оригинального документа : http://pont2008.cs.msu.ru/files/en/abstracts/KolesovMishchenkoRozov.pdf Дата изменения: Tue Feb 5 21:32:17 2008 Дата индексирования: Mon Oct 1 19:57:33 2012 Кодировка:

Buffer phenomenon in mathematical mo dels of natural sciences Mishchenko E. F. Steklov Institute of Mathematics, Russian Academy of Science e-mail: mishch@mi.ras.ru Kolesov A. Yu. Demidov Yaroslavl State University e-mail: kolesov@uniyar.ac.ru Rozov N. Kh. Lomonosov Moscow State University e-mail: rozov@rozov.mccme.ru Speak, that in mathematical model of a nonlinear distributed selfoscillatory system the buffer phenomenon (bufferness) is observed, if in this model there is any predetermined finite numbers of attractors of the same type (stable equilibrium states, stable periodic on time solutions, tori, etc.) for an appropriate choice of its parameters. On the given problem it is necessary to consider as the first publication the paper of A. A. Vitt [1], the employee and the colleague of A. A. Andronov, his co-author under the classical monograph "Theory of oscillations". A. A. Vitt investigated mathematical model of the auto-generator containing a segment of a long two-wire line with uniformly distributed inductance, capacitance, and resistance; at a heuristic level of rigor he had been specified conditions for existence of the appearance named afterwards by buffer phenomenon. Much later the fact of magnification of number of possible stable self-oscillatory regimes at a variation of parameters of the auto-generator managed to be noticed experimentally [2]. Strict theoretical research of the buffer phenomenon is lead in [3­6] (including system of Vitt [7]) by means of special infinite-dimensional analog of the asymptotic method of Krylov - Bogolyubov - Mitropolskii - Samoilenko. It has appeared, that the bufferness is typical for a wide class of mathematical models which adequately describe many nonlinear processes in natural sciences (radio-physics [8], mechanics [9], optics [10], theory of combustion [11], ecology [12]). Besides this, connection is traced of buffer phenomenon with such appearances, as origin of turbulence or birth of dynamic chaos [13]. Considered mathematical models represent a boundary value problems for systems of partial differential equations of hyperbolic or parabolic type, and the script of growth of number, for example, stable periodic on time solutions (cycles) is torn as some parameter tends to zero. Essentially, that


the concept "buffer phenomenon" assumes presence a certain bifurcation process as a result of which there is a unlimited growth of number of amount oexisting attractors.

References
[1] Vitt A.A., "Distributed Self-Oscillatory Systems", J. Tech. Phys., 4. No. 1, 144­157 (1934). [2] Azjan Yu.M., Migulin V.V., "About Self-Oscillations in Systems with Delayed Feedback", Radiotech. Electronics, 1, No. 4, 126­130 (1956) (in Russian). [3] Kolesov A.Yu., Mishchenko E.F., and Rozov N.Kh., "Asymptotic Methods of Investigation of Periodic Solutions of the Nonlinear Hyperbolic Equations", Proc. Steklov Inst. Math., 222, 1­189 (1998). [4] Kolesov A.Yu., Mishchenko E.F., and Rozov N.Kh., "The Buffer Property in Resonance Systems of Nonlinear Hyperbolic Equations", Russ. Math. Surv., 55, No. 2, 297­321 (2000). [5] Kolesov A.Yu., Mishchenko E.F., and Rozov N.Kh., "Buffer Phenomenon in Nonlinear Physics", Proc. Steklov Inst. Math., 250, 102­ 168 (2005). [6] Kolesov A.Yu., and Rozov N.Kh., "On Theoretical Explanation of the Diffusion Buffer Phenomenon", Comput. Math. Math. Phys., 44, No. 11, 1922­1941 (2004). [7] Kolesov A. Yu., and Rozov N. Kh., "Asymptotic Theory of Oscillations in Vitt Systems", J. Math. Sci., 105, No. 1, 1697­1737 (2001). [8] Kolesov A.Yu., and Rozov N.Kh., "The Buffer Phenomenon in an RCLG-Oscillator: Theoretical Analysis and Experimental Results", Proc. Steklov Inst. Math., 233, 143­196 (2001). [9] Kolesov A.Yu., and Rozov N.Kh., "The Bufferness Phenomenon in Distributed Mechanical Systems", J. Appl. Math. Mech., 65, No. 2, 179­193 (2001). [10] Kolesov A.Yu., Rozov N.Kh., "The Optical Buffering and Mechanisms for Its Occurrence", heor. Math. Phys., 140, No. 1, 14­28 (2004). [11] Kolesov A. Yu., and Rozov N. Kh., "The Buffer Phenomenon in Combustion Theory", Dokl. Math., 69, No. 3, 469­472 (2004).

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[12] Kolesov A.Yu., and Rozov N.Kh., "The Diffusion-Buffer Phenomenon in a Mathematical Model of Biology", Izv. Math., 62, 985­1012 (1998). [13] Mishchenko E.F., Sadovnichii V.A., Kolesov A.Yu., and Rozov N.Kh., Processes in Nonlinear Media with Diffusion, - Moscow, Fizmatlit, (2005) (in Russian). [14] Kolesov A.Yu., Rozov N.Kh., and Sadovnichii V.A. "Life on the Edge of Chaos", J. Math. Sci., 120, No. 3, 1372­1398 (2004). [15] Kolesov A.Yu., Mishchenko E.F., and Rozov N.Kh., "Buffer Phenomenon in Systems Close to Two-Dimensional Hamiltonian Ones", Proc. Steklov Inst. Math., No. 1, 117­150 (2006). [16] Kolesov A.Yu., and Rozov N.Kh., "The Nature of the Bufferness Phenomenon in Weakly Dissipative Systems", Theor. Math. Phys., 146, No. 3, 447­466 (2006).

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