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Abstract: Center manifold theory forms one of the cornerstones of the theory of dynamical systems. This is already true for finite-dimensional systems, but it holds a fortiori in the infinite-dimensional case. In its simplest form center manifold theory reduces the study of a system near a (non-hyperbolic) equilibrium point to that of an ordinary differential equation on a low-dimensional invariant center manifold. For finite-dimensional systems this means a (sometimes considerable) reduction of the dimension, leading to simpler calculations and a better geometric insight. When the starting point is an infinite-dimensional problem, such as a partial, a functional or an integro differential equation, then the reduction forms also a qualitative simplification. Indeed, most infinite-dimensional systems lack some of the nice properties which we use almost automatically in the case of finite-dimensional flows. For example, the initial value problem may not be well posed, or backward solutions may not exist; and one has to worry about the domains of operators or the regularity of solutions. Therefore the reduction to a finite-dimensional center manifold, when it is possible, forms a most welcome tool, since it allows us to recover the familiar and easy setting of an ordinary differential equation.

Abstract: The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.

Abstract: Introduction Notation and preliminaries Statements of theorems Local coordinate systems Cone lemmas Center-unstable manifold Center-stable manifold Smoothness of center-stable manifold Smoothness of center-unstable manifold Persistence of invariant manifold Persistence of normal hyperbolicity Invariant manifolds for perturbed semiflow References.

Abstract: In recent years quaternionic Kahler and hyperkähler manifolds have received a great deal of attention. They appear in many different areas of mathematics and mathematical physics. It has been argued that these recent advances in quaternionic geometry vindicate Hamilton's conviction that the algebra of quaternions should play an important role in mathematical physics [At], [Hil]. The purpose of this paper is to describe the geometry and topology of a class of Riemannian Einstein manifolds that is closely related to both hyperkähler and quaternionic Kahler manifolds. These manifolds, known äs manifolds with a Sasakian 3-structure, first appeared in a paper by Kuo in 1970 [Ku] which was published a few years before Ishihara and Calabi introduced the now commonly accepted terms \"quaternionic Kahler\" and \"hyperkähler\", respectively. We shall refer to manifolds with a Sasakian 3-structure äs 3-Sasakian manifolds.