N Frequency Systems; Neistadt’s Result Based on Anosov’s Method

Abstract

This chapter and the next are devoted to n frequency systems in which the fast variables φ belong to Tⁿ, the n dimensional torus, and the unperturbed motion is quasiperiodic. The various resonant surfaces are no longer disjoint (cf. Appendix 3) and a detailed study of passage through resonance leading to global estimates is no longer possible. At least two approaches are possible: The first, which is the subject of this chapter, is based on Neistadt’s application [Nei3] of Anosov’s work to n frequency systems (see Chapter 2). The second approach, discussed in the following chapter and also due to Neistadt [Nei4], relies on Kasuga’s idea of an average (L¹) solution to the linearized equation; the original version of this idea which arised in the context of ergodic systems may be found in Chapter 9. We should point out that the second approach furnishes the optimal result, and for this reason the present chapter may be viewed as an illustrative application of Anosov’s general theorem presented in Chapter 2.