Detecting and locating near-optimal almost-invariant sets and cycles

The behaviors of trajectories of nonlinear dynamical systems are notoriously hard to characterize and predict. Rather than characterizing dynamical behavior at the level of trajectories, we consider following the evolution of sets. There are often collections of sets that behave in a very predictable way, in spite of the fact that individual trajectories are entirely unpredictable. Such special collections of sets are invisible to studies of long trajectories. We describe a global set-oriented method to detect and locate these large dynamical structures. Our approach is a marriage of new ideas in modern dynamical systems theory and the novel application of graph dissection algorithms.