Mixed Mode Oscillations due to the Generalized Canard Phenomenon

Mixed mode oscillations combine features of small oscillations and large oscillations of relaxation type. We describe a mechanism for mixed mode oscillations based on the presence of canard solutions, which are trajectories passing from a stable to an unstable slow manifold. An important ingredient of this mechanism are singularities known as folded nodes.

The main focus of this article is to show how the local dynamics near a folded node can combine with global features, leading to mixed mode oscillations. We review and extend the results of [26] on the dynamics near a folded node and state some results on mixed mode periodic orbits with Farey sequences of the form 1s.

We also show how to generalize the context of one fast variable to an arbitrary number of fast variables.