Abstract
Systems with aftereffect are considered, which are described by integrodifferential equations of the Volterra type in the presence of a small perturbation prescribed by the periodic (or maximum periodic) time function. In the critical case of a pair of pure imaginary roots of a characteristic equation, the question is solved as to existence in the system of maximum periodic motions (i.e., motions tending at the unlimited increase of time to periodic modes) provided that the frequency of the periodic part of disturbance coincides with the natural frequency of a linearized homogeneous system. It is shown that in the analytic case the equations of motion of the system possess a set of the maximum periodic solutions representable by power series in a small parameter specifying a value of the disturbance, and in small arbitrary initial values of uncritical variables of the problem. The conditions of the existence of such solutions are indicated, which are defined by the terms up to the third order of equations.
Similar content being viewed by others
References
Bykov, Ya.V. and Ruzikulov, D., Periodicheskie resheniya differentsial’nykh i integrodifferentsial’nykh uravnenii i ikh asimptotiki (Periodic Solutions of Differential and Integrodifferential Equations and Their Asymptotics), Frunze: Ilim, 1986.
Burton, T.A., Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Orlando: Academic, 1985.
Ryabov, Yu.A. and Khysanov, D.Kh., Periodic Solutions of the Integrodifferential Equation of Second Order in the Nonresonant Case, Ukr. Mat. Zh., 1982, vol. 34, no. 5, pp. 644–664.
Khusanov, D.Kh., Periodic Solutions of Quasilinear Integrodifferential Equations of Second Order in the Resonant Case, Dokl. Akad. Uz.SSR, 1983, no. 7, pp. 8–11.
Khusanov, D.Kh., K konstruktivnoi i kachestvennoi teorii funktsional’no-differentsial’nykh uravnenii (To Constructive and Qulitative Theory of Functionally Differential Equations), Tashkent: Fan, 2002.
Grebenikov, E.S. and Ryabov, Yu.A., Konstruktivnye metody analiza nelineinykh system (Contructive Methods of Analysis of Nonlinear Systems), Moscow: Nauka, 1979.
Lika, D.K. and Ryabov, Yu.A., Metody iteratsii i mazhoriruyushchee uravnenie Lyapunova v teorii nelineinykh kolebanii (Methods of Iterations and the Majorizing Lyapunov Equation in the Theory of Nonlinear Vibrations), Kishinev: Shtiintsa, 1974.
Malkin, I.G., Teoriya ustoichivosti dvizheniya (Theory of Motion Stability), Moscow: Nauka, 1966.
Sergeev, V.S., On Instability in the Critical Case of the Pair of Pure Imaginary Roots for One Class of Systems with Aftereffect, Prikl. Mat. Mekh., 1998, vol. 62, no. 1, pp. 79–86.
Sergeev, V.S., On Maximum Periodic Movements in Some Systems with Aftereffect, Prikl. Mat. Mekh., 2004, vol. 68, no. 5, pp. 857–869.
Sergeev, V.S., On Instability of Solutions of Volterra Integrodifferential Equations in the Critical Case of the Pair of Pure Imaginary Roots, in Zadachi issledovaniya ustoichivosti i stabilizatsii dvizheniya (Problems of Investigation of Stability and Stabilization of Motion), Moscow: Vychisl. Tsentr Akad. Nauk SSSR, 1987, pp. 38–56.
Lyapunov, A.M., Obshchaya zadacha ob ustoichivosti dvizheniya. Sobr. soch. T. 2 (The Common Problem on Motion Stability. Collect. Articles, vol. 2), Moscow: Akad. Nauk SSSR, 1956.
Bykov, Ya.V., O nekotorykh zadachakh teorii integrodifferentsial’nykh uravnenii (On Some Problems of the Theory of Integrodifferential Equations), Frunze: Kirgiz. Gos. Univ., 1857.
Sergeev, V.S., On Instability of Zero Solution of One Class of Integrodifferential Equations, Diff. Uravn., 1988, vol. 24, no. 8, pp. 1443–1454.
Author information
Authors and Affiliations
Additional information
Original Russian Text © V.S. Sergeev, 2011, published in Avtomatika i Telemekhanika, 2011, No. 9, pp. 87–98.
Rights and permissions
About this article
Cite this article
Sergeev, V.S. Maximum periodic solutions of the Volterra integrodifferential equations in the critical case of a pair of pure imaginary roots. Autom Remote Control 72, 1876–1886 (2011). https://doi.org/10.1134/S0005117911090098
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117911090098