Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction

T.P. Valkering, Stephanus A. van Gils

Research output: Contribution to journalArticleAcademicpeer-review

3 Citations (Scopus)
33 Downloads (Pure)

Abstract

We investigate analytically the effect of perturbations on an integrable oscillator in one degree of freedom whose frequency shows a maximum as a function of the energy, i.e. a system with nonmonotone twist. The perturbation depends on three parameters: one parameter describes friction such that the Jacobian is constant and less than one. A second and a third describe the variation of the frequency and of the strength of the driving force respectively. The main result is the appearance of two chains of saddle node pairs in the phase portrait. This contrasts with the bifurcation of one chain of periodic orbits in the case of perturbations of monotone twist systems. This result is obtained for a mapping, but it is demonstrated that the same formalism and results apply for time continuous systems as well. In particular we derive an explicit expression for the stroboscopic mapping of a particle in a potential well, driven by a periodic force and under influence of friction, thus giving a clear physical interpretation to the bifurcation parameters in the mapping.
Original languageUndefined
Pages (from-to)103-130
Number of pages28
JournalZeitschrift für angewandte Mathematik und Physik
Volume44
Issue number1
DOIs
Publication statusPublished - 1993

Keywords

  • METIS-129608
  • IR-80008

Cite this

@article{df2a1953a2374688b46c72deb2552283,
title = "Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction",
abstract = "We investigate analytically the effect of perturbations on an integrable oscillator in one degree of freedom whose frequency shows a maximum as a function of the energy, i.e. a system with nonmonotone twist. The perturbation depends on three parameters: one parameter describes friction such that the Jacobian is constant and less than one. A second and a third describe the variation of the frequency and of the strength of the driving force respectively. The main result is the appearance of two chains of saddle node pairs in the phase portrait. This contrasts with the bifurcation of one chain of periodic orbits in the case of perturbations of monotone twist systems. This result is obtained for a mapping, but it is demonstrated that the same formalism and results apply for time continuous systems as well. In particular we derive an explicit expression for the stroboscopic mapping of a particle in a potential well, driven by a periodic force and under influence of friction, thus giving a clear physical interpretation to the bifurcation parameters in the mapping.",
keywords = "METIS-129608, IR-80008",
author = "T.P. Valkering and {van Gils}, {Stephanus A.}",
year = "1993",
doi = "10.1007/BF00914356",
language = "Undefined",
volume = "44",
pages = "103--130",
journal = "Zeitschrift f{\"u}r angewandte Mathematik und Physik",
issn = "0044-2275",
publisher = "Birkhauser Verlag Basel",
number = "1",

}

Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction. / Valkering, T.P.; van Gils, Stephanus A.

In: Zeitschrift für angewandte Mathematik und Physik, Vol. 44, No. 1, 1993, p. 103-130.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction

AU - Valkering, T.P.

AU - van Gils, Stephanus A.

PY - 1993

Y1 - 1993

N2 - We investigate analytically the effect of perturbations on an integrable oscillator in one degree of freedom whose frequency shows a maximum as a function of the energy, i.e. a system with nonmonotone twist. The perturbation depends on three parameters: one parameter describes friction such that the Jacobian is constant and less than one. A second and a third describe the variation of the frequency and of the strength of the driving force respectively. The main result is the appearance of two chains of saddle node pairs in the phase portrait. This contrasts with the bifurcation of one chain of periodic orbits in the case of perturbations of monotone twist systems. This result is obtained for a mapping, but it is demonstrated that the same formalism and results apply for time continuous systems as well. In particular we derive an explicit expression for the stroboscopic mapping of a particle in a potential well, driven by a periodic force and under influence of friction, thus giving a clear physical interpretation to the bifurcation parameters in the mapping.

AB - We investigate analytically the effect of perturbations on an integrable oscillator in one degree of freedom whose frequency shows a maximum as a function of the energy, i.e. a system with nonmonotone twist. The perturbation depends on three parameters: one parameter describes friction such that the Jacobian is constant and less than one. A second and a third describe the variation of the frequency and of the strength of the driving force respectively. The main result is the appearance of two chains of saddle node pairs in the phase portrait. This contrasts with the bifurcation of one chain of periodic orbits in the case of perturbations of monotone twist systems. This result is obtained for a mapping, but it is demonstrated that the same formalism and results apply for time continuous systems as well. In particular we derive an explicit expression for the stroboscopic mapping of a particle in a potential well, driven by a periodic force and under influence of friction, thus giving a clear physical interpretation to the bifurcation parameters in the mapping.

KW - METIS-129608

KW - IR-80008

U2 - 10.1007/BF00914356

DO - 10.1007/BF00914356

M3 - Article

VL - 44

SP - 103

EP - 130

JO - Zeitschrift für angewandte Mathematik und Physik

JF - Zeitschrift für angewandte Mathematik und Physik

SN - 0044-2275

IS - 1

ER -