One Frequency Systems and First Result for Two Frequency Systems

Abstract

After the rather abstract result of the last chapter, we now come back to a more concrete viewpoint in a much more restricted setting. In this chapter we first examine the case of one frequency systems when no resonances are involved and then start our study of the resonance phenomena with a result due to Arnold [Ar7] which relies on a rather unrealistic hypothesis but provides a clear introduction to some of the techniques we shall meet again and again from Chapter 4 onwards. One frequency averaging was explored in [Bog], where the original papers of the Russian school are summarized; it also forms the main subject of [Sand3], these two books containing interesting examples, many of practical interest. Once and for all we thus refer to them for a more detailed exposition of the material in this chapter; the latter book also contains references to the recent original papers and a short history of the subject starting with Lagrange.