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arXiv · 1411.7976

Quasi-periodicity in relative quasi-periodic tori

Francesco Fassò·Luis C. García-Naranjo·Andrea Giacobbe
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Abstract

At variance from the cases of relative equilibria and relative periodic orbits of dynamical systems with symmetry, the dynamics in relative quasi-periodic tori (namely, subsets of the phase space that project to an invariant torus of the reduced system on which the flow is quasi-periodic) is not yet completely understood. Even in the simplest situation of a free action of a compact and abelian connected group, the dynamics in a relative quasi-periodic torus is not necessarily quasi-periodic. It is known that quasi-periodicity of the unreduced dynamics is related to the reducibility of the reconstruction equation, and sufficient conditions for it are virtually known only in a perturbation context. We provide a different, though equivalent, approach to this subject, based on the hypothesis of the existence of commuting, group-invariant lifts of a set of generators of the reduced torus. Under this hypothesis, which is shown to be equivalent to the reducibility of the reconstruction equation, we give a complete description of the structure of the relative quasi-periodic torus, which is a principal torus bundle whose fibers are tori of a dimension which exceeds that of the reduced torus by at most the rank of the group. The construction can always be done in such a way that these tori have minimal dimension and carry ergodic flow.

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Francesco Fassò, Luis C. García-Naranjo, Andrea Giacobbe. 2015-10-20. Quasi-periodicity in relative quasi-periodic tori. https://doi.org/10.1088/0951-7715%2F28%2F11%2F4281

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