Skip to main content
SpringerLink
Log in

Andronov–Hopf Bifurcation in Logistic Delay Equations with Diffusion and Rapidly Oscillating Coefficients

  • Research Articles
  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

A logistic delay equation with diffusion, which is important in applications, is studied. It is assumed that all of its coefficients, as well as the coefficients in the boundary conditions, are rapidly oscillating functions of time. An averaged equation is constructed, and the relation between its solutions and the solutions of the original equation is studied. A result on the stability of the solutions is formulated, and the problem of local dynamics in the critical case is studied. An algorithm for constructing the asymptotics of the solutions and an algorithm for studying their stability are proposed. It is important to note that the corresponding algorithm contains both a regular and a boundary layer component. Meaningful examples are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Figure 8.

Similar content being viewed by others

REFERENCES

  1. N. N. Bogolyubov and Yu. A. Mitropol’skii, Asymptotic Methods in the Theory of Nonlinear Oscillations (Nauka, Moscow, 1974) [in Russian].

    MATH  Google Scholar 

  2. Yu. A. Mitropol’skii, The Method of Averaging in Nonlinear Mechanics (Naukova Dumka, Kiev, 1971) [in Russian].

    Google Scholar 

  3. Yu. A. Mitropol’skii, Transient Processes in Nonlinear Oscillatory Systems (Izd. Akad. Nauk Ukrain. SSR, Kiev, 1955) [in Russian].

    Google Scholar 

  4. V. M. Volosov and B. I. Morgunov, The Method of Averaging in the Theory of Nonlinear Oscillatory Systems (Izd. Mosk. Univ., Moscow, 1971) [in Russian].

    MATH  Google Scholar 

  5. Yu. S. Kolesov, V. S. Kolesov and and I. I. Fedik, Self-Oscillations in Distributed-Parameter Systems (Naukova Dumka, Kiev, 1979) [in Russian].

    Google Scholar 

  6. Yu. S. Kolesov and V. V. Maiorov, “A new method of investigation of the stability of the solutions of linear differential equations with nearly constant almost-periodic coefficients,” Differ. Uravn. 10 (10), 1778–1788 (1974).

    MathSciNet  Google Scholar 

  7. S. A. Kashchenko, “Dynamics of delay systems with rapidly oscillating coefficients,” Differ. Equ. 54 (1), 13–27 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  8. S. A. Kashchenko, “Application of the averaging principle to the study of the dynamics of the delay logistic equation,” Math. Notes 104 (2), 231–243 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Wu, Theory and Applications of Partial Functional-Differential Equations, in Appl. Math. Sci. (Springer-Verlag, New York, 1996), Vol. 119.

    Book  MATH  Google Scholar 

  10. A. Bensoussan, J. L. Lions and and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, in Stud. Math. Appl. (North-Holland Publ., Amsterdam, 1978), Vol. 5.

    MATH  Google Scholar 

  11. M. L. Kleptsina and A. L. Pyatnitskii, “Homogenization of a random non-stationary convection-diffusion problem,” Russian Math. Surveys 57 (4), 729–751 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  12. E. Marǔsić-Paloka and A. L. Piatnitski, “Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection,” J. London Math. Soc. (2) 72 (2), 391–409 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Allaire, I. Pankratova and and A. Piatnitski, “Homogenization of a nonstationary convection-diffusion equation in a thin rod and in a layer,” SeMA J. 58 (1), 53–95 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. B. Levenshtam, “Asymptotic integration of parabolic problems with large high-frequency summands,” Siberian Math. J. 46 (4), 637–651 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  15. S. A. Kashchenko, “Asymptotic behavior of steady-state regimes of parabolic equations with coefficients rapidly oscillating with respect to time and with a variable domain of definition,” Ukrain. Mat. Zh. 39 (5), 578–582 (1987).

    MathSciNet  Google Scholar 

  16. S. A. Gourley, J. W.-H. So and and Wu Jian Hong, “Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics,” J. Math. Sci. 124 (4), 5119–5153 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  17. A. B. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of the Solutions of Singularly Perturbed Equations (Nauka, Moscow, 1973) [in Russian].

    MATH  Google Scholar 

  18. A. B. Vasil’eva and V. F. Butuzov, Singularly Perturbed Equations in Critical Cases (Izd. Mosk. Univ., Moscow, 1978) [in Russian].

    Google Scholar 

  19. V. F. Butuzov and N. T. Levashova, “On a system of reaction-diffusion-transfer type in the case of small diffusion and fast reactions,” Comput. Math. Math. Phys. 43 (7), 962–974 (2003).

    MathSciNet  MATH  Google Scholar 

  20. S. A. Kashchenko, “Asymptotic behavior of periodic solutions of autonomous parabolic equations with small diffusion,” Sibirsk. Mat. Zh. 27 (6), 116–127 (1986).

    MathSciNet  Google Scholar 

  21. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Izd. Mosk. Univ., Moscow, 1977) [in Russian].

    Google Scholar 

  22. C. A. Kashchenko and D. O. Loginov, “Bifurcations due to the variation of boundary conditions in the logistic equation with delay and diffusion,” Math. Notes 106 (1), 136–141 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  23. P. L. Kapitsa, “Dynamical stability of a pendulum in the case of an oscillating suspension point,” Zh. Éxper. Teoret. Fiz. 21 (5), 588 (1951).

    MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research under grant 18-29-10043 and by the Ministry of Science and Higher Education of the Russian Federation (project RNOMTs no. 1.13560.2019/13.1).

Author information

Authors and Affiliations

  1. Demidov Yaroslavl State University, 150003, Yaroslavl, Russia

    S. A. Kashchenko & D. O. Loginov

Authors
  1. S. A. Kashchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. O. Loginov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. A. Kashchenko.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kashchenko, S.A., Loginov, D.O. Andronov–Hopf Bifurcation in Logistic Delay Equations with Diffusion and Rapidly Oscillating Coefficients. Math Notes 108, 50–63 (2020). https://doi.org/10.1134/S0001434620070056

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434620070056

Keywords

Search

Navigation