Linearization of a quasi-periodically forced flow on 𝕋^m under Brjuno–Rüssmann non-resonant condition

Abstract

In this paper, we prove that a class of quasi-periodically forced flow with external parameters on a m-torus is linearizable, i.e. conjugable to the quasi-periodic rotation, under some appropriate conditions. More concretely, utilizing Pöschel–Rüssmann KAM method we can find a quasi-periodic transformation such that this flow becomes a quasi-periodic linear flow in a Cantor subset of parameters when forcing frequency and unperturbed rotation vector to satisfy the Brjuno–Rüssmann’s non-resonant condition and a weaker non-degeneracy condition, respectively. While in the case where the unperturbed rotation vector does not satisfy the weaker non-degeneracy condition, Herman’s method is applied to overcome the degeneracy so that we show that some perturbed systems can conjugate to the original system with prescribed rotation vector Finally, as an application, our results are used to investigate the spectrum and eigenfunctions of a quasi-periodic Schrödinger operators.

Keywords:

• Linearization of a quasi-periodically forced flow
• Pöschel–Rüssmann KAM method
• weaker non-degeneracy condition
• the Brjuno–Rüssmann’s non-resonant condition
• Herman’s method

AMS Subject Classifications:

• 34J40
• 70H08
• 70K43

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China [grant number 11171185], [grant number 11571201].