Institute of Mathematical Statistics Lecture Notes - Monograph Series

Nearly-integrable perturbations of the Lagrange top: applications of KAM-theory

H. W. Broer, H. Hanssmann, J. Hoo, and V. Naudot

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Abstract

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 non-linear dynamics. As a consequence, we find Cantor families of invariant isotropic tori of all dimensions suggested by the integrable approximation.

Chapter information

Source
Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy, eds., Dynamics & Stochastics: Festschrift in honor of M. S. Keane (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), >286-303

Dates
First available in Project Euclid: 28 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.lnms/1196285829

Digital Object Identifier
doi:10.1214/074921706000000301

Mathematical Reviews number (MathSciNet)
MR2306209

Zentralblatt MATH identifier
1125.70003

Subjects
Primary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol d diffusion
Secondary: 70H08: Nearly integrable Hamiltonian systems, KAM theory

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

Rights
Copyright © 2006, Institute of Mathematical Statistics

Citation

Broer, H. W.; Hanssmann, H.; Hoo, J.; Naudot, V. Nearly-integrable perturbations of the Lagrange top: applications of KAM-theory. Dynamics & Stochastics, >286--303, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000301. https://projecteuclid.org/euclid.lnms/1196285829


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