PART l: UNFOLDINGS OF QUASI-PERIODIC TORI

H.W. Broer, G.B. Huitema and F.Takens

§1. INTRODUCTION

This part is concerned with dynamical systems having invariant tori with

quasi-periodic dynamics. These systems depend on (external) parameters and may

preserve a given structure like a volume or a symplectic form, a certain kind

of symmetry, etc.. The main problem is the persistence of such invariant tori,

under small perturbations of these systems. We shall present an unfolding

theory in which suitable (generic) dependence on the parameters implies this

persistence. Here we follow Moser [55,56], where our unfolding parameters

replace the so-called modifying terms. For more details compare §7b, below.

The unfolding approach, which also uses ArnoPd [3], enables us to apply

KAM-perturbation-theory as developed by Poschel [61]. Therefore our

persistence results are in a sense (structural) stability results, where we

work with (Whitney-) smooth conjugacies defined on foliations of

quasi-periodic tori which are parametrized over "Cantor sets" of large

measure. In this respect we introduce the term quasi - periodic stability. The

stability result will be explicitly formulated for the general (dissipative)

case, the volume preserving and the symplectic case. The proof, however, is

expressed in terms of Lie algebra and it implies these results as special

cases, compare [56]. In this paper we restrict to vector fields, but a

completely analogous theory exists for diffeomorphisms.

To be more explicit we briefly sketch our set-up. Let M be the phase

space and P the parameter space. Also let T =R /(ZKIL) be the standard

n-torus. We assume that M admits a free T -action, which is compatible with

the structure that has to be preserved. A vector field on M is said to be

integrable if it is equivariant (or symmetric) with respect to this action.

1

Received by the editor June 20, 1988.

2

The third author was partially supported by the Netherlands organization for

the Advancement of Pure Scientific Research (Z.W.O.), grant 10-70-10.

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