Advertisement

Computational Mathematics and Mathematical Physics

, Volume 58, Issue 12, pp 1926–1936 | Cite as

Dynamics of a Delay Logistic Equation with Slowly Varying Coefficients

  • S. A. KashchenkoEmail author
Article
  • 2 Downloads

Abstract

The existence, stability, and asymptotic behavior of steady modes for a delay logistic equation with slowly varying coefficients are analyzed.

Keywords:

bifurcations stability normal forms singular perturbations dynamics 

Notes

REFERENCES

  1. 1.
    Wu Jianhong, Theory and Applications of Partial Functional Differential Equations (Springer-Verlag, New York, 1996).Google Scholar
  2. 2.
    S. A. Gourley, J. W.-H. So, and J. H. Wu, “Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics”, J. Math. Sci. 124 (4), 5119–5153 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics (Springer-Verlag, Heidelberg, 1977).CrossRefzbMATHGoogle Scholar
  4. 4.
    S. A. Kashchenko, “Asymptotics of a periodic solution to the generalized Hutchinson equation,” Research into Stability and Oscillation Theory (Yaroslav. Gos. Univ., Yaroslavl, 1981), p. 22 [in Russian].Google Scholar
  5. 5.
    S. A. Kashchenko, “Asymptotics of the solutions of the generalized Hutchinson equation,” Autom. Control Comput. Sci. 47 (7), 470–494 (2013).CrossRefGoogle Scholar
  6. 6.
    A. N. Tikhonov, “Systems of differential equations containing small parameters in the derivatives,” Mat. Sb. 31 (3), 575–586 (1952).MathSciNetGoogle Scholar
  7. 7.
    A. V. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of Solutions to Singularly Perturbed Equations (Nauka, Moscow, 1973) [in Russian].zbMATHGoogle Scholar
  8. 8.
    A. B. Vasil’eva and V. F. Butuzov, Singularly Perturbed Equations in Critical Cases (Mosk. Univ., Moscow, 1978) [in Russian].Google Scholar
  9. 9.
    Yu. S. Kolesov, V. S. Kolesov, and I. I. Fedik, Self-Exciting Oscillations in Distributed-Parameter Systems (Naukova Dumka, Kiev, 1979) [in Russian].Google Scholar
  10. 10.
    A. I. Klimushev and I. N. Krasovskii, “Uniform asymptotic stability of systems of differential equations with a small parameter multiplying derivatives,” Prikl. Mat. Mekh. 25 (4), 680–690 (1961).Google Scholar
  11. 11.
    B. S. Razumikhin, “On the stability of solutions to systems of differential equations with a small parameter multiplying derivatives,” Sib. Mat. Zh. 4 (1), 206–211 (1963).MathSciNetGoogle Scholar
  12. 12.
    V. F. Chaplygin, “Exponential dichotomy of solutions of a linear almost periodic equation with aftereffect and slow time,” Vestn. Yaroslav. Univ., No. 13, 159–167 (1975).Google Scholar
  13. 13.
    S. A. Kashchenko, “On the stability of solutions to linear singularly perturbed differential equations with almost periodic coefficients in the resonance case,” Research into Stability and Oscillation Theory (Yaroslavl, 1980), pp. 25–34 [in Russian].zbMATHGoogle Scholar
  14. 14.
    S. A. Kashchenko, “Stability analysis of solutions to linear parabolic equations with nearly constant coefficients and weak diffusion,” Trudy Semin. Petrovskogo, No. 15, 128–155 (1991).Google Scholar
  15. 15.
    J. K. Hale, Theory of Functional Differential Equations (Springer-Verlag, New York, 1977).CrossRefzbMATHGoogle Scholar
  16. 16.
    P. Hartman, Ordinary Differential Equations (Wiley, New York, 1964).zbMATHGoogle Scholar
  17. 17.
    A. D. Bruno, Local Methods in Nonlinear Differential Equations (Nauka, Moscow, 1979; Springer-Verlag, Berlin, 1989).Google Scholar
  18. 18.
    N. D. Bykova, S. D. Glyzin, and S. A. Kashchenko, “Parametric resonance in the logistic equation with delay under a two-frequency perturbation,” Model. Anal. Inf. Sis. 20 (3), 86–98 (2013).CrossRefGoogle Scholar
  19. 19.
    S. A. Kashchenko and Yu. S. Kolesov, “Parametric resonance in systems with delay under a two-frequency perturbation,” Sib. Math. J. 21 (2), 231–235 (1980).CrossRefzbMATHGoogle Scholar
  20. 20.
    R. Edwards, Functional Analysis: Theory and Applications (Holt Rinehart and Winston, New York, 1965).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

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

Personalised recommendations