Forced linear oscillators and the dynamics of Euclidean group extensions

Abstract

We study the generic dynamical behaviour of skew-product extensions generated by cocycles arising from equations of forced linear oscillators of special form. This work extends our earlier work on cocycles into compact Lie groups arising from differential equations of special form, (cf. [21]), to the case of non-compact fiber groups of Euclidean type. The earlier techniques do not work in the non-compact case. In the non-compact case one of the main obstacle is the lack of `recurrence'. Thus, our approach to studying Euclidean group extensions is : (i) first, to use a `twisted version' of the so called `conjugation approximation method' and then (ii) to use `geometric-control theoretic methods' developed in our earlier work (cf. [20] and [21]). Even then, our arguments only work for base flows that admit a global Poincae section, (e.g. for the irrational rotation flows on tori and for certain nil flows). We apply these results to study generic spectral behaviour of the forced quantum harmonic oscillator with time dependent stationary force restricted to satisfy given constraints.