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Zbl 1125.70003
Broer, H.W.; Han\ss mann, H.; Hoo, J.; Naudot, V.
Nearly-integrable perturbations of the Lagrange top: applications of KAM-theory. (English)
[A] Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference `Dynamical systems, probability theory, and statistical mechanics', Eindhoven, The Netherlands, January 3--7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics. Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 286-303 (2006). ISBN 0-940600-64-1/pbk

Summary: Motivated by the Lagrange top coupled to an oscillator, we consider the quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal linear stability theory of an invariant torus with a generic (i.e., non-semisimple) normal $1:-1$ resonance. This theory guarantees the persistence of the invariant torus in the Diophantine case and makes possible a further quasi-periodic normal form, necessary for investigation of the nonlinear dynamics. As a consequence, we find Cantor families of invariant isotropic tori of all dimensions suggested by the integrable approximation.

MSC 2000:: *70E20 Perturbation methods for Euler's equations
70E05 Motion of the gyroscope
37J40 Perturbations, etc.
70H08 Nearly integrable Hamiltonian systems, KAM theory

Keywords: KAM theory; quasi-periodic Hamiltonian Hopf bifurcation; singular foliation; Lagrange top; gyroscopic stabilization

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