Geometric insights from manifold computations

Global manifolds are the backbone of a dynamical system and key to the characterisation of its behaviour. They arise in the classical sense of invariant manifolds associated with saddle-type equilibria or periodic orbits and, more recently, in the form of finite-time invariant manifolds in systems that evolve on multiple time scales. Dynamical systems theory relies heavily on the knowledge of such manifolds, because of the geometric insight that they can offer into how observed behaviour arises. Global invariant manifolds need to be computed and visualised numerically, which is a serious challenge, but an effort that pays off. Our approach is based on the continuation of solutions to a two-point boundary value problem, and has successfully been used for vector fields, as well as invertible and noninvertible maps. We will explain how global invariant manifolds can be computed to study the geometric complexity arising from classical chaotic dynamics. We will also discuss recent research on a new type of chaos that arises robustly in higher-dimensional systems. Specifically, we will present results for a 3D Henon-like map that address the discrepancy between theoretical results and their manifestations in explicit examples.