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 -