Multifrequency oscillations of singularly perturbed systems

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Multifrequency oscillations of singularly perturbed systems. / Bibikov, Yu N.; Bukaty, V. R.

In: Differential Equations, Vol. 48, No. 1, 01.01.2012, p. 19-25.

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

BibTeX

@article{700e47a8596a4beca31c420c4b0329f2,

title = "Multifrequency oscillations of singularly perturbed systems",

abstract = "We consider a system of differential equations that consists of two parts, a regularly perturbed and a singularly perturbed one. We assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic; i. e., all of its eigenvalues have nonzero real parts. We derive the so-called determining equation, to each of whose positive solutions there corresponds an invariant torus. We show that, in general position, there is an m-dimensional invariant torus bifurcating from the equilibrium as the small parameter passes through the critical zero point; here m is the number of pure imaginary eigenvalues. In addition, in the degenerate case, we find conditions for the coexistence of two- and three-dimensional invariant tori.",

author = "Bibikov, {Yu N.} and Bukaty, {V. R.}",

year = "2012",

month = jan,

day = "1",

doi = "10.1134/S001226611201003X",

language = "English",

volume = "48",

pages = "19--25",

journal = "Differential Equations",

issn = "0012-2661",

publisher = "Pleiades Publishing",

number = "1",

}

RIS

TY - JOUR

T1 - Multifrequency oscillations of singularly perturbed systems

AU - Bibikov, Yu N.

AU - Bukaty, V. R.

PY - 2012/1/1

Y1 - 2012/1/1

N2 - We consider a system of differential equations that consists of two parts, a regularly perturbed and a singularly perturbed one. We assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic; i. e., all of its eigenvalues have nonzero real parts. We derive the so-called determining equation, to each of whose positive solutions there corresponds an invariant torus. We show that, in general position, there is an m-dimensional invariant torus bifurcating from the equilibrium as the small parameter passes through the critical zero point; here m is the number of pure imaginary eigenvalues. In addition, in the degenerate case, we find conditions for the coexistence of two- and three-dimensional invariant tori.

AB - We consider a system of differential equations that consists of two parts, a regularly perturbed and a singularly perturbed one. We assume that the matrix of the linear part of the regularly perturbed system has pure imaginary eigenvalues, while the matrix of the singularly perturbed part is hyperbolic; i. e., all of its eigenvalues have nonzero real parts. We derive the so-called determining equation, to each of whose positive solutions there corresponds an invariant torus. We show that, in general position, there is an m-dimensional invariant torus bifurcating from the equilibrium as the small parameter passes through the critical zero point; here m is the number of pure imaginary eigenvalues. In addition, in the degenerate case, we find conditions for the coexistence of two- and three-dimensional invariant tori.

UR - http://www.scopus.com/inward/record.url?scp=84857804427&partnerID=8YFLogxK

U2 - 10.1134/S001226611201003X

DO - 10.1134/S001226611201003X

M3 - Article

AN - SCOPUS:84857804427

VL - 48

SP - 19

EP - 25

JO - Differential Equations

JF - Differential Equations

SN - 0012-2661

IS - 1

ER -

ID: 49227184