Localization of an Unstable Solution of a System of Three Nonlinear Ordinary Differential Equations with a Small Parameter | Sciact - Система учета научной деятельности ИМ СО РАН

Localization of an Unstable Solution of a System of Three Nonlinear Ordinary Differential Equations with a Small Parameter Научная публикация

Журнал

Journal of Applied and Industrial Mathematics
ISSN: 1990-4789 , E-ISSN: 1990-4797

Вых. Данные

Год: 2022, Том: 16, Номер: 4, Страницы: 606–620 Страниц : 15 DOI: 10.1134/S1990478922040032

Ключевые слова

Andronov–Hopf bifurcation, nonlinear ordinary differential equation (ODE), ODE with small parameter, asymptotic expansion, Lyapunov function

Авторы

Chumakov G.A. ^1,3 , Chumakova N.A. ^2,3

Организации

1	Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
2	Boreskov Institute of Catalysis, Novosibirsk, 630090 Russia
3	Novosibirsk State University, Novosibirsk, 630090 Russia

In the present paper, we study some nonlinear autonomous systems of three nonlinear ordinary differential equations (ODE) with small parameter μ such that two variables (x, y) are fast and the remaining variable z is slow. In the limit as μ → 0, from this “complete dynamical system” we obtain the “degenerate system,” which is included in a one-parameter family of two-dimensional subsystems of fast motions with parameter z in some interval. It is assumed that there exists a monotone function ρ(z) that, in the three-dimensional phase space of a complete dynamical system, defines a parametrization of some arc L of a slow curve consisting of the family of fixed points of the degenerate subsystems. Let L have two points of the Andronov–Hopf bifurcation in which some stable limit cycles arise and disappear in the two-dimensional subsystems. These bifurcation points divide L into the three arcs; two arcs are stable, and the third arc between them is unstable. For the complete dynamical system, we prove the existence of a trajectory that is located as close as possible to both the stable and unstable branches of the slow curve L as μ tends to zero for values of z within a given interval.

Chumakov G.A. , Chumakova N.A.
Localization of an Unstable Solution of a System of Three Nonlinear Ordinary Differential Equations with a Small Parameter
Journal of Applied and Industrial Mathematics. 2022. V.16. N4. P.606–620. DOI: 10.1134/S1990478922040032 Scopus РИНЦ OpenAlex

Чумаков Г.А. , Чумакова Н.А.
О локализации неустойчивого решения одной системы трех нелинейных обыкновенных дифференциальных уравнений с малым параметром
Сибирский журнал индустриальной математики. 2022. Т.25. №4. С.221–238. DOI: 10.33048/SIBJIM.2022.25.417 РИНЦ

Поступила в редакцию:	15 июл. 2022 г.
Принята к публикации:	29 сент. 2022 г.
Опубликована online:	6 мар. 2023 г.
Опубликована в печати:	19 апр. 2023 г.

Scopus:	2-s2.0-85150187806
РИНЦ:	50731353
OpenAlex:	W4323344661

БД	Цитирований
РИНЦ	2