Invariant Fiber Bundles

Abstract

In a broad range of qualitative studies on nonlinear dynamical systems, invariant manifolds are omnipresent and play a crucial role for local as well as global questions: For instance, local stable and unstable manifolds dictate the saddle-point behavior in the vicinity of hyperbolic solutions (or surfaces) of a system. As illustrated by the celebrated reduction principle of Pliss, center manifolds are a paramount tool to simplify given dynamical systems in terms of a reduction of their state space dimension. Concerning amore global perspective, stable manifolds serve as separatrix between different domains of attractions and allow a classification of solutions with a specific asymptotic behavior. Systems with a gradient structure possess global attractors consisting of unstable manifolds (and equilibria). Finally, so-called inertial manifolds are global versions of the classical center-unstable manifolds and yield a global reduction principle for typically infinite-dimensional dissipative equations.