Guest lecturer: Bernd Krauskopf

Computing global invariant manifolds as a sequence of geodesic level sets

Bernd Krauskopf, University of Auckland

joint work with

Hinke Osinga, University of Auckland

May 16, 2018

Abstract: Steady-states and periodic orbits of saddle type of dynamical systems defined by ordinary differential equations come with stable and unstable manifolds. These invariant, global objects play a crucial role in organising the dynamics, for example, as basin boundaries and in the creation of chaotic attractors. We consider here global invariant manifolds of dimension two in three-dimensional systems. These are implicitely defined surfaces that can generally only be found by numerical means.

We present arguably the most geometric approach to finding and visualising these surfaces: we grow a manifold step by step from the steady-state of periodic orbit by computing a sequence of rings or bands at suitable geodesic distances. Our method is illustrated with several examples, including the Lorenz system.