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IssuesArchive of Issues2002-6pp.34-41

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M. A. Kagan and M. I. Feigin, "To the theory of bifurcation memory effects," Mech. Solids. 37 (6), 34-41 (2002)
Year 2002 Volume 37 Number 6 Pages 34-41
Title To the theory of bifurcation memory effects
Author(s) M. A. Kagan (Nizhni Novgorod)
M. I. Feigin (Nizhni Novgorod)
Abstract We consider a general approach to the analysis of the behavior of dynamical systems depending on a parameter the variation of which eventually leads to the loss of stability or disappearance of a steady-state mode. Quantitative characteristics of such effects are calculated for a self-sustained oscillation system with dry friction. Along with the cases of instantaneous and quasistatic change of the parameter, we consider a scenario referred to as diagnostic. This scenario involves additional specific perturbations of the state variables, which enables one to assess the closeness of the system to a bifurcation state and to formulate a bifurcation prediction criterion convenient for practical utilization.
References
1.  M. I. Feigin, "To the theory of a trigger," in In Memory of A. A. Andronov [in Russian], pp. 300-333, Izd-vo AN SSSR, Moscow, 1955.
2.  M. A. Shishkova, "Consideration of a system of differential equations with a small parameter at the highest-order derivative," Doklady AN SSSR, Vol. 209, No. 3, pp. 576-579, 1973.
3.  M. I. Feigin, "To the theory of motion of a ship unstable with respect to the course angle," Izv. AN. SSSR. MTT [Mechanics of Solids], No. 1, pp. 66-72, 1982.
4.  M. I. Feigin and M. M. Chirkova, "On the existence of a reduced controllability domain for ships unstable with respect to the course angle," Izv. AN. SSSR. MTT [Mechanics of Solids], No. 2, pp. 73-78, 1985.
5.  M. I. Feigin and M. M. Chirkova, "Dynamics of ships unstable with respect to the course angle," Sudostroenie, No. 7, pp. 23-25, 1987.
6.  M. I. Feigin, Forced Oscillations in Discontinuous Nonlinear Systems [in Russian], Nauka, Moscow, 1994.
7.  M. I. Feigin, "On the initial uncontrollability of a dynamical system," in Problems of the Theory of Oscillations [in Russian], pp. 184-197, Izd-vo Nizhegorod. Un-ta, Nizhni Novgorod, 1995.
8.  Ya. A. Kuryakov and M. I. Feigin, "Analysis of a mathematical model of an automatic steersman with a block of identification and suppression of the initial uncontrollability," in Modeling and Optimization of Complex Systems. Issue 273. Part 1 [in Russian], pp. 79-82, Izd-vo Volzhsk. Akademii Vodnogo Transporta, Nizhni Novgorod, 1997.
9.  V. F. Zhuravlev and D. M. Klimov, Applied Methods in the Theory of Oscillations [in Russian], Nauka, Moscow, 1988.
10.  A. I. Neishtadt, "Asymptotic analysis of the loss of stability by an equilibrium in the case of slow passage of a pair of eigenvalues through the imaginary axis," Uspekhi Matem. Nauk, Vol. 40, No. 5, 1985, pp. 300-301.
11.  A. I. Neishtadt, "On the delayed loss of stability in the case of dynamic bifurcations. Parts 1 & 2," Differentsial'nye Uravneniya, Vol. 23, No. 12, pp. 2060-2067, 1987; Vol. 24, No. 2, pp. 226-233, 1988.
12.  C. Beasens, "Gevery series and Dynamic bifurcations for analytic slow-fast mappings," Nonlinearity, Vol. 8, No. 2, pp. 179-201, 1995.
13.  A. I. Neishtadt, C. Simo, and D. V. Treshchev, "On stability loss delay for a periodic trajectory," in Nonlinear Dynamical Systems and Chaos. Volume 19 [in Russian], pp. 253-278, Birkhauser, Boston, 1996.
14.  A. M. Feigin and I. V. Konovalov, "On the possibility of complicated dynamic behavior of atmospheric photochemical systems: Instability of the Antarctic photochemistry during the ozone hole formation," J. Geophys. Res., Vol. 101, No. D20, pp. 26023-26038, 1996.
15.  I. V. Konovalov, A. M. Feigin, and A. Y. Mikhina, "Toward understanding of the nonlinear nature of atmospheric photochemistry: multiple equilibrium states in the high-latitude lower stratospheric photochemical system," J. Geophys. Res., Vol. 104, No. D3, pp. 3669-3689, 1999.
16.  A. I. Neishtadt and V. V. Sidorenko, "Delayed loss of stability in Ziegler's system," PMM [Applied Mathematics and Mechanics], Vol. 61, No. 1, pp. 18-29, 1997.
17.  M. I. Feigin, "Investigation of bifurcation memory effects in behavior of nonlinear controlled systems," in Proc. of Intern. Conf. "Control of Oscillations and Chaos." Volume 3 [in Russian], pp. 474-477, St. Petersburg, 1997.
17.  M. I. Feigin and M. M. Chirkova, "A method of control of motion of a ship," Certificate of Authorship No. 1066896 SSSR, Byull. Izobretenii, No. 2, p 77. 1984; Certificate of Authorship No. 1178652 SSSR, Byull. Izobretenii, No. 34, p 63. 1985.
18.  N. V. Butenin, "Fundamentals of the Theory of Nonlinear Oscillations," Sudpromgiz, Leningrad, 1962.
Received 24 July 2000
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