 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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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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