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.
Similar content being viewed by others
References
A. J. Lichtenberg and M. A. Lieberman,Regular and Stochastic Motion, Springer, New York 1983.
J. P. van der Weele, T. P. Valkering, H. W. Capel and T. Post, Physica A153, 283–294 (1988).
J. E. Howard and S. M. Hohs, Phys. Rev. A29, 418–421 (1984).
J. P. van der Weele and T. P. Valkering, Physica A169, 42–72 (1990).
G. M. Zaslavsky, M. Y. Zakharov, R. Z. Sagdeev, D. A. Usikov and A. A. Chemikov, Sov. Phys. JETP64, 294 (1986).
A. A. Chernikov, R. Z. Sagdeev and G. M. Zaslavsky, Physica D33, 65–76 (1988).
T. Zeegers, J. Phys. A: Math. Gen.24, 2287–2314 (1991).
J. Hale and P. Z. Táboas, Applic. Math.11, 21–37 (1980).
R. S. MacKay,Introduction to the Dynamics of Area Preserving Maps, in: Proc. US Particle Accelerator School 1985, M. Months ed., SLAC 1986.
H. Goldstein,Classical Mechanics, Addison Wesley, Reading, Massachusetts 1980.
J. Dieudonné,Foundations of Modern Analysis, Academic Press, New York, London 1960.
F. Rothe, Can. Math. Soc. Conf. Proc.8, 621–635 (1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Valkering, T.P., van Gils, S.A. Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction. Z. angew. Math. Phys. 44, 103–130 (1993). https://doi.org/10.1007/BF00914356
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00914356