About Bifurcations at Small Perturbations in a Logistic Equation with Delay
https://doi.org/10.18255/1818-1015-2017-2-168-185
Abstract
The article considers bifurcation problems for a logistic equation with delay at small perturbations. The most interesting results are for the case when small perturbations contain a large delay. The main results are special nonlinear equations of evolution in the normal form. Their nonlocal dynamics defines the behaviour of the solutions of the original equation in a small neigbourhood of the balance state or the cycle. It turns out that the order of large delay magnitude is principal. For the simplest case, when this order is congruent with the magnitude inverse to the small parameter appearing in the equation, the normal form is a complex equation with delay. In the case when the order of the delay coefficient is even higher, the normal form is presented by a multiparameter family of special boundary-value problems of degenerate-parabolic type. All these things allow to make a conclusion about the fact that in the considered problems with large delay the multistability is typical.
About the Author
Sergey A. Kashchenko
P.G. Demidov Yaroslavl State University
Russian Federation
Russian Federation
professor
14 Sovetskaya str., Yaroslavl 150003, Russia
References
1. Wright E. M., “A non-linear difference-differential equation”, Journal fuЁr die reine und angewandte Mathematik, 194 (1955), 66–87.
2. Kakutani S., Markus L., “On the non-linear difference-differential equation y(t) = (a − by(t − τ))y(t)”, Contributions to the Theory of Nonlinear Oscillations, 4, ed. S. Lefschetz, Princeton University Press, Princeton, 1958, 1–18, Annals of Mathematical Studies (AM-41).
3. К вопросу об оценке в пространстве параметров области глобальной устойчивости уравнения Хатчинсона, Нелинейные колебания в задачах экологии, ЯрГУ, Ярославль, 1985, 55–62; [Kashchenko S. A., “K voprosu ob otsenke v prostranstve parametrov oblasti globalnoy ustoychivosti uravneniya Khatchinsona”, Nelineynye kolebaniya v zadachakh ekologii, YarGU, Yaroslavl, 1985, 55–62, (in Russian).]
4. Jones G. S., “The existence of periodic solutions of f(x) = −αf(x−1)[1+f(x)]”, Journal of Contemporary Mathematical Analysis, 5 (1962), 435–450.
5. Кащенко С.А., “Сложные стационарные режимы одного дифференциально-разностного уравнения, обобщающего уравнение Хатчинсона”, Исследования по устойчивости и теории колебаний, ЯрГУ, Ярославль, 1983, 8; [Kashchenko S.A., “Slozhnye statsionarnye rezhimy odnogo differentsialno-raznostnogo uravneniya, obobshchayushchego uravnenie Khatchinsona”, Issledovaniya po ustoychivosti i teorii kolebaniy, YarGU, Yaroslavl, 1983, 8, (in Russian).]
6. Кащенко С.А., “О периодических решениях уравнения x(t) = −lx(t − 1)[1 + x(t)]”, Исследования по устойчивости и теории колебаний, ЯрГУ, Ярославль, 1978, 110–117; [Kashchenko S. A., “O periodicheskikh resheniyakh uravneniya x(t) = −lx(t−1)[1+x(t)]”, Issledovaniya po ustoychivosti i teorii kolebaniy, YarGU, Yaroslavl, 1978, 110–117, (in Russian).]
7. Кащенко С.А., “Асимптотика периодического решения обобщённого уравнения Хатчинсона”, Исследования по устойчивости и теории колебаний, ЯрГУ, Ярославль, 1981; [Kashchenko S. A., “Asimptotika periodicheskogo resheniya obobshchennogo uravneniya Khatchinsona”, Issledovaniya po ustoychivosti i teorii kolebaniy, YarGU, Yaroslavl, 1981, (in Russian).]
8. Kashchenko S., “Asymptotics of the Solutions of the Generalized Hutchinson Equation”, Automatic Control and Computer Sciences, 47:7 (2013), 470–494.
9. Hale J. K., Theory of functional differential equations, Springer Verlag, New York, 1977, 626 pp.
10. Hartman P., Ordinary Differential Equations, Wiley, New York, 1965, 626 pp.
11. Кащенко С.А., “Бифуркации в окрестности цикла при малых возмущениях с большим запаздыванием”, Журнал вычислительной математики и математической физики, 40:5 (2000), 693–702; English transl.: Kashchenko S. A., “Bifurcations in the neighborhood of a cycle under small perturbations with a large delay”, Comput. Math. Math. Phys., 40:5 (2000), 659–668.
12. Kashchenko S. A., “Bifurcational Features in Systems of Nonlinear Parabolic Equations with Weak Diffusion”, International Journal of Bifurcation and Chaos, 15:11 (2005), 3595– 3606.
13. Кащенко С.А., “Применение метода нормализации к изучению динамики дифференциально-разностных уравнений с малым множителем при производной”, Дифференциальные уравнения, 25:8 (1989), 1448–1451; English transl.: Kashchenko S. A., “Application of the normalization method to the study of the dynamics of a differential- difference equation with a small factor multiplying the derivative”, Differ. Uravn., 25:8 (1989), 1448–1451.
14. Кащенко И.С., “Асимптотический анализ поведения решений уравнения с большим запаздыванием”, Доклады РАН, 421:5 (2008), 586–589; [Kashchenko I. S., “Asymptotic analysis of the behavior of solutions to equations with large delay”, Doklady Mathematics, 78:1 (2008), 570–573, (in Russian).]
15. Кащенко И.С., “Локальная динамика уравнений с большим запаздыванием”, Журнал вычислительной математики и математической физики, 48:12 (2008), 2141–2150; English transl.: Kashchenko I. S., “Local dynamics of equations with large delay”, Comput. Math. Math. Phys., 48:12 (2008), 2172–2181.
16. Кащенко С.А., “Уравнение Гинзбурга 퍨 Ландау 퍨 нормальная форма для дифференциально-разностного уравнения второго порядка с большим запаздыванием”, Журнал вычислительной математики и математической физики, 38:3 (1998), 457–465; English transl.: Kashchenko S. A., “The Ginzburg–Landau equation as a normal form for a second-order difference-differential equation with a large delay”, Comput. Math. Math. Phys., 38:3 (1998), 443–451.
17. Ахромеева Т.С., Курдюмов С.П., Малинецкий Г.Г., Нестационарные структуры и диффузионный хаос, Наука, М., 1992, 544 с.; [Akhromeeva T. S., Kurdyumov S. P., Malinetskiy G. G., Nestatsionarnye struktury i diffuzionnyy khaos, Nauka, M., 1992, 544 pp., (in Russian).]
18. Aranson I. S., Kramer L., “The world of the complex Ginzburg–Landau equation”, Reviews of Modern Physics, 74:1 (2002), 99–143.
19. Кудряшов Н.А., Методы нелинейной математической физики, МИФИ, М., 2008, 352 с.; [Kudryashov N. A., Metody nelineynoy matematicheskoy fiziki, MIFI, M., 2008, 352 pp., (in Russian).]
20. Кащенко А.А., “Устойчивость бегущих волн в уравнении Гинзбурга ᰠ Ландау с малой диффузией”, Моделирование и анализ информационных систем, 18:3 (2011), 58–62; [Kashchenko A. A., “Analysis of running waves stability in the Ginzburg–Landau equation with small diffusion”, Model. Anal. Inform. Sist., 18:3 (2011), 58–62, (in Russian).]
21. Kashchenko A. A., “Analysis of Running Waves Stability in the Ginzburg–Landau Equation with Small Diffusion”, Automatic Control and Computer Sciences, 49:7 (2015), 514–517.
Review
For citations:
Kashchenko S.A. About Bifurcations at Small Perturbations in a Logistic Equation with Delay. Modeling and Analysis of Information Systems. 2017;24(2):168-185. (In Russ.) https://doi.org/10.18255/1818-1015-2017-2-168-185
Views: 9183
Email this article 