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 Mat. Zametki, 2020, Volume 108, Issue 1, Pages 47–63 (Mi mz12484)

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

S. A. Kashchenko, D. O. Loginov

P.G. Demidov Yaroslavl State University

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.

Keywords: averaging, logistic equation, delay, boundary conditions, bifurcations, stability.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-29-10043 Ministry of Science and Higher Education of the Russian Federation 1.13560.2019/13.1 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).

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DOI: https://doi.org/10.4213/mzm12484

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English version:
Mathematical Notes, 2020, 108:1, 50–63

UDC: 517
Revised: 10.01.2020

Citation: S. A. Kashchenko, D. O. Loginov, “Andronov–Hopf Bifurcation in Logistic Delay Equations with Diffusion and Rapidly Oscillating Coefficients”, Mat. Zametki, 108:1 (2020), 47–63; Math. Notes, 108:1 (2020), 50–63

Citation in format AMSBIB
\Bibitem{KasLog20} \by S.~A.~Kashchenko, D.~O.~Loginov \paper Andronov--Hopf Bifurcation in Logistic Delay Equations with Diffusion and Rapidly Oscillating Coefficients \jour Mat. Zametki \yr 2020 \vol 108 \issue 1 \pages 47--63 \mathnet{http://mi.mathnet.ru/mz12484} \crossref{https://doi.org/10.4213/mzm12484} \transl \jour Math. Notes \yr 2020 \vol 108 \issue 1 \pages 50--63 \crossref{https://doi.org/0.1134/S0001434620070056}