Differential Equations

, Volume 54, Issue 4, pp 427–449 | Cite as

Stability of Steady-State Solutions of Systems of Nonlinear Nonautonomous Delay Differential Equations

  • I. V. Boikov
Ordinary Differential Equations


Sufficient conditions for the stability of steady-state solutions of systems of nonautonomous linear and nonlinear differential equations with time-dependent delay are obtained in terms of coefficients. These sufficient conditions are written as inequalities relating quantities that can be calculated directly from the right-hand side of the system of equations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Volterra, V., Leçons sur la théorie mathématique de la lutte pour la vie, Paris: Gauthier-Villars, 1931. Translated under the title Matematicheskaya teoriya bor’by za sushchestvovanie, Moscow: Nauka, 1976.zbMATHGoogle Scholar
  2. 2.
    Volterra, V., Theory of Functionals and of Integral and Integro-Differential Equations, New York: Dover, 1959. Translated under the title Teoriya funktsionalov, integral’nykh i integro-differentsial’nykh uravnenii, Moscow: Nauka, 1982.zbMATHGoogle Scholar
  3. 3.
    Krasovskii, N.N., Nekotorye zadachi teorii ustoichivosti dvizheniya (Several Problems of the Theory of Stability of Motion), Moscow: Nauka, 1959.Google Scholar
  4. 4.
    Bellman, R. and Cooke, K., Differential-Difference Equations, New York: Academic, 1963. Translated under the title Differentsial’no-raznostnye uravneniya, Moscow: Mir, 1967.zbMATHGoogle Scholar
  5. 5.
    Myshkis, A.D., Lineinye differentsial’nye uravneniya s zapazdyvayushchim argumentom (Linear Differential Equations with a Retarded Argument), Moscow: Nauka, 1972.zbMATHGoogle Scholar
  6. 6.
    Hale, J., Theory of Functional Differential Equations, New York: Springer, 1977. Translated under the title Teoriya funktsional’no-differentsial’nykh uravnenii, Moscow: Mir, 1984.CrossRefzbMATHGoogle Scholar
  7. 7.
    Azbelev, N.V., Maksimov, V.P., and Rakhmatullina, L.F., Vvedenie v teoriyu funktsional’no-differentsial’nykh uravnenii (Introduction to the Theory of Functional-Differential Equations), Moscow: Nauka, 1991.zbMATHGoogle Scholar
  8. 8.
    Boikov, I.V., On stability of motion in a system with aftereffect, Prikl. Mat. Mekh., 1997, vol. 61, no. 3, pp. 398–402.MathSciNetGoogle Scholar
  9. 9.
    Boikov, I.V., On the stability of solutions of differential equations with aftereffect, Differ. Equations, 1998, vol. 34, no. 8, pp. 1138–1141.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Boikov, I.V., Ustoichivost’ reshenii differentsial’nykh uravnenii (Stability of Solutions of Differential Equations), Penza: Penz. Gos. Univ., 2008.Google Scholar
  11. 11.
    Marchuk, G.I., Matematicheskie modeli v immunologii. Vychislitel’nye metody i eksperimenty (Mathematical Models in Immunology. Computational Methods and Experiments), Moscow: Nauka, 1991.zbMATHGoogle Scholar
  12. 12.
    Bratus’, A.S., Novozhilov, A.S., and Platonov, A.P., Dinamicheskie modeli i modeli biologii (Dynamical Systems and Models in Biology), Moscow: Fizmatlit, 2010.Google Scholar
  13. 13.
    Paramonov, I.V., Construction of asymptotic expansion of the solution of the equation of neuron described by the differential equation with variable delay, Modelir. Anal. Inform. Sist., 2007, vol. 14, no. 2, pp. 36–39.Google Scholar
  14. 14.
    Hopfield, J.J., Neurons with graded response have collective computations properties like those of twostate neurons, Proc. Natl. Acad. Sci. USA, 1984, vol. 81, no. 10, pp. 3088–3092.CrossRefzbMATHGoogle Scholar
  15. 15.
    Cichocki, A. and Unbehauen, R., Neural Networks for Optimization and Signal Processing, New York: Wiley, 1993.zbMATHGoogle Scholar
  16. 16.
    Akca, H., Alassar, R., and Covachev, V., Stability of neural networks with time varying delays in the presence of impulses, Adv. Dynam. Syst. Appl., 2006, vol. 1, no. 1, pp. 1–15.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Norkin, S.B., On a case of retardation dependence on the sought function, Zh. Vychisl. Mat. Mat. Fiz., 1962, vol. 2, no. 2, pp. 343–348.Google Scholar
  18. 18.
    Daletskii, Yu.L. and Krein, M.G., Ustoichivost’ reshenii differentsial’nykh uravnenii v banakhovom prostranstve (Stability of Solutions of Differential Equations in a Banach Space), Moscow: Nauka, 1970.Google Scholar
  19. 19.
    Bylov, B.F., Vinograd, R.E., Grobman, D.M., and Nemytskii, V.V., Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti (Theory of Lyapunov Exponents and Its Applications to Stability Problems), Moscow: Nauka, 1966.Google Scholar
  20. 20.
    Dekker, K. and Verwer, J.G., Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations, Amsterdam: North Holland, 1984. Translated under the title Ustoichivost’ metodov Runge-Kutty dlya zhestkikh nelineinykh differentsial’nykh uravnenii, Moscow: Mir, 1988.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Penza State UniversityPenzaRussia

Personalised recommendations