The Kate Edger Department of Mathematics

Manifolds of systems with multiple time scales


Solutions of a system with multiple time scales frequently exhibit slow and fast epochs characterised by the speed at which the solution advances. During a slow epoch, we say that the solution lies on a slow manifold, which is followed by a fast transition to another slow manifold. Slow manifolds are not invariant, because solutions do not stay on them for infinite time. How do slow manifolds interact with other dynamical objects, such as equilibria, periodic orbits, and their invariant manifolds, to organise the overall observed dynamics?

We work with John Guckenheimer from Cornell University to develop a theory that explains the geometry behind such complex interactions. Our focus is on a class of models that are representative for the types of behaviours that can be observed in systems with multiple time scales, but we are also interested in high-dimensional systems that are much harder to visualise and for which it is more challenging to extract the different time scales.

The image above is taken from SIAM Journal on Applied Dynamical Systems 7: 1131-1162 (2008)

Researchers at The University of Auckland