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

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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.

In: Journal of Applied and Industrial Mathematics, Vol. 16, No. 4, 2022, p. 606-620.

Research output: Contribution to journal › Article › peer-review

BibTeX

@article{4d0d5744958848c29221d369530cab48,

title = "Localization of an Unstable Solution of a System of Three Nonlinear Ordinary Differential Equations with a Small Parameter",

abstract = "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.",

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

author = "Chumakov, {G. A.} and Chumakova, {N. A.}",

note = "Публикация для корректировки.",

year = "2022",

doi = "10.1134/S1990478922040032",

language = "English",

volume = "16",

pages = "606--620",

journal = "Journal of Applied and Industrial Mathematics",

issn = "1990-4789",

publisher = "Maik Nauka-Interperiodica Publishing",

number = "4",

}

RIS

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