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Generic bifurcations of forced oscillations of integrable mechanical systems

Published online by Cambridge University Press:  24 October 2008

J. P. Cleave
J. P. Cleave
Affiliation:
University of Bristol
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Extract

Although integrable Hamiltonian systems are non-generic (Robinson (13)) they have some importance in classical mechanics, e.g. the two-body problem, a free rigid body not subject to a gravitational field, the Toda lattice (Moser(12)). Arnol'd (cf. (2) and (1), appendix 26) proved that under quite general conditions action-angle coordinates can be introduced. Accordingly we consider a fixed system with n degrees of freedom in the standard form

where H0 is a smooth function of the action variables Li only and I ∈ I and I is an open set of n-tuples of positive reals. now subjected to a periodic impressed force, the resulting system, , being determined by a small, periodic perturbation of the energy function

where e is a small positive real and K(c, I; α0, α1, …, an) is a smooth function of parameters c ranging over a smooth r-manifold E, I ∈ I, and K has period 2π in each of the angles αi, i.e. (α0, …, αn) ∈ Tn (n-torus). The purpose of this paper is to define the forms of bifurcation of oscillations of the perturbed system (K):

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 1 , January 1979 , pp. 125 - 142
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

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