stability of solutions of differential and integrodifferential equations; periodic solutions; dynamics of rigid body; mathematical physics.
Subject:
The investigation of exponential stability of zero solution is given by Lyapunov's First Method for integrodifferential equations of the Volterra type with nonlinear terms depending on functional (in particular in the form of Fre'chet's series). The structure of the general solution is determined in a neighborhood of zero. The method of estimation of the region of attraction is offered; it is based on the method of majorizing equations using Lyapunov majorants. In the critical cases of one zero and pair of pure imaginary roots for the Volterra integrodifferential equations with holomorphic nonlinear terms the method of determination of the Lyapunov constants is given. The conditions on this constants for instability (stability) of zero are obtained. In the dynamics of a heavy rigid body with a fixed point the class of periodic solutions near the rapid regular Lagrange precession is found. Those solutions are represented by convergent power series depending on entire or fractional power of the small parameter (inversely proportional angular velocity of rotation of the body). Some periodic motions of a rigid body are investigated using the method of normal forms and action-angle variables. The general method of estimation of a small parameter for convergence of the series representing periodic solutions of Poincare' is given for autonomous systems of differential equations possessing by first integrals.
Biography
Graduated from the Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University in 1963 (department of Theoretical mechanics). Ph.D. thesis ("Some problem of a rigid body motion about a fixed point") was defended in 1969. D.Sci. thesis ("Stability in systems with aftereffect described by integrodifferential equations of the Volterra type") was defended in 2000. A list of my scientific works contains more than 60 titles.
Main publications:
Sergeev B. C. O neustoichivosti v kriticheskom sluchae pary chisto mnimykh kornei dlya odnogo klassa sistem s posledeistviem // Prikladnaya matematika i mekhanika (PMM), t. 62, vyp. 1, 1998, 79–86.
Sergeev B. C. Ob asimptoticheskoi ustoichivosti i otsenke oblasti prityazheniya v nekotorykh sistemakh s posledeistviem // PMM, t .60, vyp. 5, 1996, 744–751.
Sergeev B. C. O neustoichivosti nulevogo resheniya odnogo klassa integrodifferentsialnykh uravnenii // Differents. uravneniya, t. 24, # 8, 1988, 1443–1454.
Sergeev B. C. O periodicheskikh resheniyakh uravnenii dvizheniya tyazhelogo tverdogo tela vokrug nepodvizhnoi tochki // Vestn. Mosk. un-ta, matem., mekh., # 1, 1969, 40–51.
V. S. Sergeev, “On the Limiting Periodic Solutions of Integral Differential Volterra Equations and their Stability”, Avtomat. i Telemekh., 2013, no. 8, 148–159; Autom. Remote Control, 74:8 (2013), 1356–1365
2011
2.
V. S. Sergeev, “Maximum periodic solutions of the Volterra integrodifferential equations in the critical case of a pair of pure imaginary roots”, Avtomat. i Telemekh., 2011, no. 9, 87–98; Autom. Remote Control, 72:9 (2011), 1876–1886
V. S. Sergeev, “A case of motional stability of railway wheel pair”, Avtomat. i Telemekh., 2009, no. 9, 157–161; Autom. Remote Control, 70:9 (2009), 1579–1583
V. S. Sergeev, “Instability of the zero solution of a class of integro-differential equations”, Differ. Uravn., 24:8 (1988), 1443–1454; Differ. Equ., 24:8 (1988), 949–957
V. S. Sergeev, “Stability of solutions for a class of integro-differential equations”, Differ. Uravn., 22:3 (1986), 518–523
1978
7.
V. S. Sergeev, “A method for obtaining estimates for regions of attraction with the aid of Ljapunov functions constructed by a numerical method”, Zh. Vychisl. Mat. Mat. Fiz., 18:5 (1978), 1154–1161; U.S.S.R. Comput. Math. Math. Phys., 18:5 (1978), 80–87
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