Smooth Invariant Foliations in Infinite Dimensional Spaces

One of the most useful properties of dynamical systems is the existence of invariant manifolds and their invariant foliations near an equilibrium or a periodic orbit. These manifolds and foliations serve as a convenient setting to describe the qualitative behavior of the local flows, and in many cases they are useful tools for technical estimates which facilitate the study of the local bifurcation diagram (see [6]). Many other important concepts in dynamical systems are closely related to the invariant manifolds and foliations. In finite dimensional space, the relations among invariant manifolds, invariant foliations, l-lemma, linearization, and homoclinic bifurcation have been studied in [ll]. It is well known that if each leaf is used as a coordinate, the original system is completely decoupled and the linearization follows easily (for example, see [27, 221). As a motivation, let us consider a linear system in Rmfn