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

S. A. Kashchenko ; D. O. Loginov

Matematičeskie zametki, Tome 108 (2020) no. 1, pp. 47-63.

Résumé

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.

Détail

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

Bibliographie

[1] N. N. Bogolyubov, Yu. A. Mitropolskii, Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR | Zbl

[2] Yu. A. Mitropolskii, Metod usredneniya v nelineinoi mekhanike, Naukova dumka, Kiev, 1971 | MR

[3] Yu. A. Mitropolskii, Nestatsionarnye protsessy v nelineinykh kolebatelnykh sistemakh, AN USSR, Kiev, 1955 | MR

[4] V. M. Volosov, B. I. Morgunov, Metod osredneniya v teorii nelineinykh kolebatelnykh sistem, Izd-vo Mosk. un-ta, M., 1971

[5] Yu. S. Kolesov, V. S. Kolesov, I. I. Fedik, Avtokolebaniya v sistemakh s raspredelennymi parametrami, Naukova dumka, Kiev, 1979

[6] Yu. S. Kolesov, V. V. Maiorov, “Novyi metod issledovaniya ustoichivosti reshenii lineinykh differentsialnykh uravnenii s blizkimi k postoyannym pochti periodicheskimi koeffitsientami”, Differents. uravneniya, 10:10 (1974), 1778–1788 | MR | Zbl

[7] S. A. Kaschenko, “Dinamika sistem s zapazdyvaenim i bystro ostsilliruyuschimi koeffitsientami”, Differents. uravneniya, 54:1 (2018), 15–29 | MR

[8] S. A. Kaschenko, “Primenenie printsipa usredneniya k issledovaniyu dinamiki logisticheskogo uravneniya s zapazdyvaniem”, Matem. zametki, 104:2 (2018), 216–230 | DOI

[9] J. Wu, Theory and Applications of Partial Functional-Differential Equations, Appl. Math. Sci., 119, Springer-Verlag, New York, 1996 | MR | Zbl

[10] A. Bensoussan, J. L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl., 5, North-Holland Publ., Amsterdam, 1978 | MR

[11] M. L. Kleptsina, A. L. Pyatnitskii, “Usrednenie sluchainoi nestatsionarnoi zadachi konvektsii-diffuzii”, UMN, 57:4 (346) (2002), 95–118 | DOI | MR | Zbl

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

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

[14] V. B. Levenshtam, “Asimptoticheskoe integrirovanie parabolicheskikh zadach s bolshimi vysokochastotnymi slagaemymi”, Sib. matem. zhurn., 46:4 (2005), 805–821 | MR | Zbl

[15] S. A. Kaschenko, “Asimptotika ustanovivshikhsya rezhimov parabolicheskikh uravnenii s bystro ostsilliruyuschimi po vremeni koeffitsientami i peremennoi oblastyu opredeleniya”, Ukr. matem. zhurn., 39:5 (1987), 578–582 | MR

[16] S. A. Gurli, Dzh. V.-Kh. Sou, Dzh. Kh. Vu, “Nelokalnye uravneniya reaktsii-diffuzii s zapazdyvaniem: biologicheskie modeli i nelineinaya dinamika”, SMFN, 1, MAI, M., 2003, 84–120 | MR | Zbl

[17] A. B. Vasileva, V. F. Butuzov, Asimptoticheskie razlozheniya reshenii singulyarno vozmuschennykh uravnenii, Nauka, M., 1973 | MR | Zbl

[18] A. B. Vasileva, V. F. Butuzov, Singulyarno vozmuschennye uravneniya v kriticheskikh sluchayakh, Izd-vo Mosk. un-ta, M., 1978

[19] V. F. Butuzov, N. T. Levashova, “O sisteme tipa reaktsiya-diffuziya-perenos v sluchae maloi diffuzii i bystrykh reaktsii”, Zh. vychisl. matem. i matem. fiz., 43:7 (2003), 1005–1017 | MR | Zbl

[20] S. A. Kaschenko, “Asimptotika periodicheskikh reshenii avtonomnykh parabolicheskikh uravnenii s maloi diffuziei”, Sib. matem. zhurn., 27:6 (1986), 116–127 | MR

[21] A. N. Tikhonov, A. A. Samarskii, Uravneniya matematicheskoi fiziki, Izd-vo Mosk. un-ta, M., 1977

[22] C. A. Kaschenko, D. O. Loginov, “Bifurkatsii pri varirovanii granichnykh uslovii v logisticheskom uravnenii s zapazdyvaniem i diffuziei”, Matem. zametki, 106:1 (2019), 138–143 | DOI

[23] P. L. Kapitsa, “Dinamicheskaya ustoichivost mayatnika pri koleblyuscheisya tochke podvesa”, ZhETF, 21:5 (1951), 588