Interacting slow manifolds in multiple-time-scale systems.
Bernd Krauskopf and Hinke Osinga
Department of Mathematics, The University of Auckland

Dynamics that evolves on at least two different time scales occurs in many applications; examples are pulsing dynamics, chemical reaction dynamics and spiking or bursting of neurons. From the mathematical point of view, this type of dynamics is organised in the phase space of relevant ODE models by special surfaces, called attracting and repelling slow manifolds. We show how slow manifolds can be computed and visualised via boundary value problem formulations. This allows us to determine how they interact and intersect in what are known as canard orbits, which determine the pattern of pulses or spikes that are observed.