# Bifurcation of an Oscillatory Mode under a Periodic Perturbation of a Special Oscillator

Результат исследований: Научные публикации в периодических изданияхстатья

1 цитирование (Scopus)

### Выдержка

We study a bifurcation from the zero solution of the differential equation ẍ + xp/q = 0, where p > q > 1 are odd coprime numbers, under periodic (in particular, time-invariant) perturbations depending on a small positive parameter ε. The motion separation method is used to derive the bifurcation equation. To each positive root of this equation, there corresponds an invariant two-dimensional torus (a closed trajectory in the time-invariant case) shrinking to the equilibrium position x = 0 as ε → 0. The proofs use methods of the Krylov-Bogolyubov theory to study time-periodic perturbations and the implicit function theorem in the case of time-invari ant perturbations.

Язык оригинала английский 753-757 Differential Equations 55 6 15 июл 2019 https://doi.org/10.1134/S001226611906003X Опубликовано - 2019

### Отпечаток

Bifurcation
Perturbation
Invariant
Implicit Function Theorem
Coprime
Shrinking
Torus
Odd
Roots
Trajectory
Differential equation
Closed
Motion
Zero

### Предметные области Scopus

• Анализ
• Математика (все)

### Цитировать

title = "Bifurcation of an Oscillatory Mode under a Periodic Perturbation of a Special Oscillator",
abstract = "We study a bifurcation from the zero solution of the differential equation ẍ + xp/q = 0, where p > q > 1 are odd coprime numbers, under periodic (in particular, time-invariant) perturbations depending on a small positive parameter ε. The motion separation method is used to derive the bifurcation equation. To each positive root of this equation, there corresponds an invariant two-dimensional torus (a closed trajectory in the time-invariant case) shrinking to the equilibrium position x = 0 as ε → 0. The proofs use methods of the Krylov-Bogolyubov theory to study time-periodic perturbations and the implicit function theorem in the case of time-invari ant perturbations.",
author = "Bibikov, {Yu. N.} and Bukaty, {V. R.}",
note = "Bibikov, Y.N., Bukaty, V.R. Bifurcation of an Oscillatory Mode under a Periodic Perturbation of a Special Oscillator. Diff Equat 55, 753–757 (2019) doi:10.1134/S001226611906003X",
year = "2019",
doi = "10.1134/S001226611906003X",
language = "English",
volume = "55",
pages = "753--757",
journal = "Differential Equations",
issn = "0012-2661",
number = "6",

}

В: Differential Equations, Том 55, № 6, 2019, стр. 753-757.

Результат исследований: Научные публикации в периодических изданияхстатья

TY - JOUR

T1 - Bifurcation of an Oscillatory Mode under a Periodic Perturbation of a Special Oscillator

AU - Bibikov, Yu. N.

AU - Bukaty, V. R.

N1 - Bibikov, Y.N., Bukaty, V.R. Bifurcation of an Oscillatory Mode under a Periodic Perturbation of a Special Oscillator. Diff Equat 55, 753–757 (2019) doi:10.1134/S001226611906003X

PY - 2019

Y1 - 2019

N2 - We study a bifurcation from the zero solution of the differential equation ẍ + xp/q = 0, where p > q > 1 are odd coprime numbers, under periodic (in particular, time-invariant) perturbations depending on a small positive parameter ε. The motion separation method is used to derive the bifurcation equation. To each positive root of this equation, there corresponds an invariant two-dimensional torus (a closed trajectory in the time-invariant case) shrinking to the equilibrium position x = 0 as ε → 0. The proofs use methods of the Krylov-Bogolyubov theory to study time-periodic perturbations and the implicit function theorem in the case of time-invari ant perturbations.

AB - We study a bifurcation from the zero solution of the differential equation ẍ + xp/q = 0, where p > q > 1 are odd coprime numbers, under periodic (in particular, time-invariant) perturbations depending on a small positive parameter ε. The motion separation method is used to derive the bifurcation equation. To each positive root of this equation, there corresponds an invariant two-dimensional torus (a closed trajectory in the time-invariant case) shrinking to the equilibrium position x = 0 as ε → 0. The proofs use methods of the Krylov-Bogolyubov theory to study time-periodic perturbations and the implicit function theorem in the case of time-invari ant perturbations.

UR - http://www.scopus.com/inward/record.url?scp=85069203853&partnerID=8YFLogxK

U2 - 10.1134/S001226611906003X

DO - 10.1134/S001226611906003X

M3 - Article

AN - SCOPUS:85069203853

VL - 55

SP - 753

EP - 757

JO - Differential Equations

JF - Differential Equations

SN - 0012-2661

IS - 6

ER -