Abstract
We use the Riccati equation method for the derivation of necessary conditions and a test for the stability of a system of two linear first-order ordinary differential equations. We consider an example in which our results are compared with the results obtained by the Lyapunov and Bogdanov methods by estimating the norms of solutions via Lozinskii logarithmic norms, and by the freezing method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Pontryagin, L.S., Obyknovennye differentsial’nye uravneniya (Ordinary Differential Equations), Moscow, 1974.
Cesari, L., Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Berlin, 1959. Translated under the title Asimptoticheskoe povedenie i ustoichivost’ reshenii obyknovennykh differentsial’nykh uravnenii, Moscow, 1964.
MacLachlan, N.W., Theory and Application of Mathieu Functions, Oxford: Clarendon Press, 1947. Translated under the title Teoriya i prilozheniya funktsii Mat’e, Moscow: Inostrannaya Literatura, 1953.
Andrianova, L.Ya., Vvedenie v teoriyu lineinykh sistem differentsial’nykh uravnenii (Introduction to the Theory of Linear Systems of Differential Equations), St.-Petersburg: St.-Peterburgsk. Gos. Univ., 1992.
Yakubovich, V.A. and Starzhinskii, V.M., Lineinye differentsial’nye uravneniya s periodicheskimi koeffitsientami i ikh prilozheniya (Linear Differential Equations with Periodic Coefficients and Their Applications), Moscow: Nauka, 1972.
Bellman, R., Stability Theory of Differential Equations, New-York, 1953. Translated under the title Teoriya ustoichivosti reshenii differentsial’nykh uravnenii, Moscow: Inostrannaya Literatura, 1954.
Burdina, V.I., On Boundedness of the Solutions of a System of Differential Equations, Dokl. Akad. Nauk SSSR, 1953, vol. 93, no. 4, pp. 603–606.
Sobol’, I.M., Investigation with the Aid of Polar Coordinates of the Asymptotic Behavior of Solutions of a Linear Differential Equation of the Second Order, Mat. Sb., 1951, vol. 28 (70), no. 3, pp. 707–714.
Fedoryuk, M.V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsial’nykh uravnenii (Asymptotic Methods for Linear Ordinary Differential Equations), Moscow: Nauka, 1983.
Hartman, Ph., Ordinary Differential Equations, New York, 1969. Translated under the title Obyknovennye differentsial’nye uravneniya, Moscow, 1970.
Grigoryan, G.A., On the Stability of Systems of Two First-Order Linear Ordinary Differential Equations, Differ. Uravn., 2015, vol. 51, no. 3, pp. 283–292.
Grigoryan, G.A., On Two Comparison Tests for Second-Order Linear Ordinary Differential Equations, Differ. Uravn., 2011, vol. 47, no. 9, pp. 1225–1240.
Grigoryan, G.A., On Some Properties of Solutions of the Riccati Equation, Izv. Nats. Akad. Nauk Armenii Mat., 2007, vol. 42, no. 4, pp. 11–26.
Grigoryan, G.A., Two Comparison Tests for Scalar Riccati Equations and Some of Their Applications, Izv. Vyssh. Uchebn. Zaved. Mat., 2012, no. 11, pp. 20–35.
Grigoryan, G.A., Global Solvability of Scalar Riccati Equations, Izv. Vyssh. Uchebn. Zaved. Mat., 2015, no. 3, pp. 35–48.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © G.A. Grigoryan, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 3, pp. 292–300.
Rights and permissions
About this article
Cite this article
Grigoryan, G.A. Necessary conditions and a test for the stability of a system of two linear ordinary differential equations of the first order. Diff Equat 52, 282–291 (2016). https://doi.org/10.1134/S0012266116030034
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266116030034