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Differential Equations

, Volume 54, Issue 1, pp 13–27 | Cite as

Dynamics of Delay Systems with Rapidly Oscillating Coefficients

  • S. A. Kashchenko
Ordinary Differential Equations
  • 20 Downloads

Abstract

The problems of generalization of the averaging principle to delay systems are considered. New effects are revealed in the study of bifurcation problems, as are new phenomena that arise in the case of rapid oscillations of the delay. As an application of the results, the dynamics of a logistic equation with rapidly oscillating coefficients is studied.

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References

  1. 1.
    Bogolyubov, N.N. and Mitropol’skii, Yu.A., Asimptoticheskie metody v teorii nelineinykh kolebanii (Asymptotic Methods in the Theory of Nonlinear Oscillations), Moscow: Gos. Izd. Fiz. Mat. Lit., 1958.zbMATHGoogle Scholar
  2. 2.
    Mitropol’skii, Yu.A., Averaging Method in Nonlinear Mechanics, Kiev, 1971.Google Scholar
  3. 3.
    Mitropol’skii, Yu.A., Nonstationary Processes in Nonlinear Oscillatory Systems, Kiev, 1955.Google Scholar
  4. 4.
    Volosov, V.M. and Morgunov, B.I., Metody osredneniya v teorii nelineinykh kolebatel’nykh sistem (Averaging Methods in the Theory of Nonlinear Oscillation Systems), Moscow: Mosk. Gos. Univ., 1971.zbMATHGoogle Scholar
  5. 5.
    Hale, J., Theory of Functional Differential Equations, Springer: New York, 1977.CrossRefzbMATHGoogle Scholar
  6. 6.
    Hartman, Ph., Ordinary Differential Equations, Wiley: New York, 1964.zbMATHGoogle Scholar
  7. 7.
    Wu, J., Theory and Applications of Partial Functional Differential Equations, New York, 1996.CrossRefzbMATHGoogle Scholar
  8. 8.
    Kolesov, Yu.S., Kolesov, V.S., and Fedik, I.I., Self-Excited Oscillations in Systems with Distributed Parameters, Kiev, 1979.Google Scholar
  9. 9.
    Kolesov, Yu.S. and Maiorov, V.V., A new method for studying stability of solutions of linear differential equations with nearly constant almost periodic coefficients, Differ. Uravn., 1974, vol. 10, no. 10, pp. 1778–1788.Google Scholar
  10. 10.
    Marsden, J.E. and McCracken, M., The Hopf Bifurcation and Its Applications, New York: Springer-Verlag, 1976. Translated under the title Bifurkatsiya rozhdeniya tsikla i ee prilozheniya, Moscow: Mir, 1980.CrossRefzbMATHGoogle Scholar
  11. 11.
    Bryuno, A.D., Local Method for Nonlinear Analysis of Differential Equations, Moscow, 1979.zbMATHGoogle Scholar
  12. 12.
    Grigorieva, E.V. and Kaschenko, S.A., Stability of equilibrium state in a laser with rapidly oscillating delay feedback, Physica D, 2015, vol. 291, pp. 1–7.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Samoilenko, A. and Petryshyn, R., Multifrequency Oscillations of Nonlinear Systems, in Mathematics and Its Applications, Vol. 567, Dordrecht, 2004.Google Scholar
  14. 14.
    Sanders, J.A., Verhulst, F., and Murdock, J., Averaging Methods in Nonlinear Dynamical Systems, New York, 2007.zbMATHGoogle Scholar
  15. 15.
    Leonov, G.A., Kuznetsov, N.V., Yuldashev, M.N., and Yuldashev, R.V., A characteristic of phase detector of classical system of phase self-tuning of frequency, Dokl. Akad. Nauk, 2015, vol. 461, no. 2, pp. 151–154.zbMATHGoogle Scholar
  16. 16.
    Kaschenko, S.A. and Maiorov, V.V., Algorithm for studying stability of solutions of linear differential equations with aftereffect and rapidly oscillating coefficients, in Studies of Stability and Theory of Oscillations, Yaroslavl, 1977, pp. 70–81.Google Scholar
  17. 17.
    Jones, G.S., The existence of periodic solutions of f’(x) = −αf(x − 1)[1 + f(x)], J. Math. Anal. Appl., 1962, vol. 5, pp. 435–450.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Vasil’eva, A.B., and Butuzov, V.F., Asimptoticheskie metody v teorii singuliarnykh vozmushchenii (Asymptotic Methods in Singular Perturbation Theory), Moscow: Vysshaya Shkola, 1990.zbMATHGoogle Scholar
  19. 19.
    Kakutani, S. and Markus, L., On the non-linear difference-differential equation y’(t) = [A−By(t−τ)]y(t), in Contributions to the Theory of Nonlinear Oscillations, Vol. 4, Ed. by Lefschetz, S., New Jersey, 1958, pp. 1–18.MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kaschenko, S.A., Asymptotics of solutions of generalized Hutchinson equation, Modelirovanie i Analiz Inform. Systems, 2012, vol. 19, no. 3, pp. 32–62.MathSciNetGoogle Scholar
  21. 21.
    Kaschenko, S.A., Asymptotics of steady-state modes of parabolic equations with coefficients rapidly oscillating in time and variable domain, Ukrain. Mat. Zh., 1987, vol. 39, no. 5, pp. 578–582.MathSciNetGoogle Scholar
  22. 22.
    Kaschenko, S.A., Study of stability of solutions of linear parabolic equations with nearly constant coefficients and small diffusion, Tr. Semin. im. I.G. Petrovskogo, 1991, no. 15, pp. 128–155.Google Scholar
  23. 23.
    Kaschenko, S.A. and Polst’yanov, A.S., Asymptotics of periodic solutions of autonomous parabolic equations with rapidly oscillating coefficients and equations with large diffusion coefficients, Modelirovanie i Analiz Inform. Systems, 2012, vol. 19, no. 1, pp. 7–23.Google Scholar
  24. 24.
    Akhmanov, S.A. and Vorontsov, M.A., Instabilities and structures in coherent nonlinearly optical systems with two-dimensional feedback, in Nonlinear Waves. Dynamics and Evolution, Moscow, 1989, pp. 228–238.Google Scholar
  25. 25.
    Grigorieva, E.V., Haken, H., Kaschenko, S.A., and Pelster, A., Travelling wave dynamics in a nonlinear interferometer with spatial field transformer in feedback, Physica D, 1999, vol. 125, no. 1–2, pp. 123–141.CrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Demidov Yaroslavl State UniversityYaroslavlRussia
  2. 2.National Research Nuclear University “MEPhI,”MoscowRussia

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