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Localization of an Unstable Solution of a System of Three Nonlinear Ordinary Differential Equations with a Small Parameter. / Chumakov, G. A.; Chumakova, N. A.
в: Journal of Applied and Industrial Mathematics, Том 16, № 4, 2022, стр. 606-620.Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование
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TY - JOUR
T1 - Localization of an Unstable Solution of a System of Three Nonlinear Ordinary Differential Equations with a Small Parameter
AU - Chumakov, G. A.
AU - Chumakova, N. A.
N1 - Публикация для корректировки.
PY - 2022
Y1 - 2022
N2 - In the present paper, we study some nonlinear autonomous systems of three nonlinearordinary differential equations (ODE) with small parameter (Formula presented.) such that two variables (x, y) are fast and the remaining variable z is slow. In the limit as (Formula presented.), 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 withparameter z in some interval. It is assumed that there exists a monotone function (Formula presented.) 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 thefamily of fixed points of the degenerate subsystems. Let L have two points of the Andronov–Hopf bifurcation in which some stable limitcycles 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 isunstable. For the complete dynamical system, we prove the existence of a trajectory that islocated as close as possible to both the stable and unstable branches of the slow curve L as (Formula presented.) tends to zero for values of z within a given interval.
AB - In the present paper, we study some nonlinear autonomous systems of three nonlinearordinary differential equations (ODE) with small parameter (Formula presented.) such that two variables (x, y) are fast and the remaining variable z is slow. In the limit as (Formula presented.), 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 withparameter z in some interval. It is assumed that there exists a monotone function (Formula presented.) 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 thefamily of fixed points of the degenerate subsystems. Let L have two points of the Andronov–Hopf bifurcation in which some stable limitcycles 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 isunstable. For the complete dynamical system, we prove the existence of a trajectory that islocated as close as possible to both the stable and unstable branches of the slow curve L as (Formula presented.) tends to zero for values of z within a given interval.
KW - Andronov–Hopf bifurcation
KW - Lyapunov function
KW - ODE with small parameter
KW - asymptotic expansion
KW - nonlinear ordinary differential equation (ODE)
UR - https://www.mendeley.com/catalogue/d7f98d6b-b9ab-391e-b476-75f3e18bc14a/
U2 - 10.1134/S1990478922040032
DO - 10.1134/S1990478922040032
M3 - Article
VL - 16
SP - 606
EP - 620
JO - Journal of Applied and Industrial Mathematics
JF - Journal of Applied and Industrial Mathematics
SN - 1990-4789
IS - 4
ER -
ID: 55697546