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.
1
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