Stability and bifurcation for periodic perturbations of the equilibrium of an oscillator with infinite or infinitesimal oscillation frequency

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Stability and bifurcation for periodic perturbations of the equilibrium of an oscillator with infinite or infinitesimal oscillation frequency. / Bibikov, Yu N.

In: Mathematical Notes, Vol. 65, No. 3-4, 01.01.1999, p. 269-279.

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

BibTeX

@article{c1ac456e8ed04da993c27d019df019fe,

title = "Stability and bifurcation for periodic perturbations of the equilibrium of an oscillator with infinite or infinitesimal oscillation frequency",

abstract = "We consider small perturbations periodic in time of an oscillator whose restoring force has a leading term with exponent 3 or 1/3. The first case corresponds to oscillations with infinitesimal frequency and the second case to oscillations with infinite frequency. The smallness of the perturbation is determined both by the smallness of the considered neighborhood of the equilibrium point and by a small nonnegative parameter ε. For ε = 0, the stability of the equilibrium point is studied. For ε > 0, we find conditions for an invariant two-dimensional torus to branch off with {"}soft{"} or {"}rigid{"} loss of stability with loss index 1/2.",

keywords = "Bifurcation, Loss of stability, Oscillator, Stability of equilibrium",

author = "Bibikov, {Yu N.}",

year = "1999",

month = jan,

day = "1",

doi = "10.1007/BF02675068",

language = "English",

volume = "65",

pages = "269--279",

journal = "Mathematical Notes",

issn = "0001-4346",

publisher = "Pleiades Publishing",

number = "3-4",

}

RIS

TY - JOUR

T1 - Stability and bifurcation for periodic perturbations of the equilibrium of an oscillator with infinite or infinitesimal oscillation frequency

AU - Bibikov, Yu N.

PY - 1999/1/1

Y1 - 1999/1/1

N2 - We consider small perturbations periodic in time of an oscillator whose restoring force has a leading term with exponent 3 or 1/3. The first case corresponds to oscillations with infinitesimal frequency and the second case to oscillations with infinite frequency. The smallness of the perturbation is determined both by the smallness of the considered neighborhood of the equilibrium point and by a small nonnegative parameter ε. For ε = 0, the stability of the equilibrium point is studied. For ε > 0, we find conditions for an invariant two-dimensional torus to branch off with "soft" or "rigid" loss of stability with loss index 1/2.

AB - We consider small perturbations periodic in time of an oscillator whose restoring force has a leading term with exponent 3 or 1/3. The first case corresponds to oscillations with infinitesimal frequency and the second case to oscillations with infinite frequency. The smallness of the perturbation is determined both by the smallness of the considered neighborhood of the equilibrium point and by a small nonnegative parameter ε. For ε = 0, the stability of the equilibrium point is studied. For ε > 0, we find conditions for an invariant two-dimensional torus to branch off with "soft" or "rigid" loss of stability with loss index 1/2.

KW - Bifurcation

KW - Loss of stability

KW - Oscillator

KW - Stability of equilibrium

UR - http://www.scopus.com/inward/record.url?scp=29244481230&partnerID=8YFLogxK

U2 - 10.1007/BF02675068

DO - 10.1007/BF02675068

M3 - Article

VL - 65

SP - 269

EP - 279

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 3-4

ER -

ID: 49227519