BioND — Dynamics of Biological Networks

Global return maps for mixed-mode oscillations with one fast and two slow variables

Christian Kuehn
Chaos 21, 033107, 2011.


Phase-space diagram

Alternating patterns of small and large amplitude oscillations occur in a wide variety of physical, chemical, biological and engineering systems. These mixed-mode oscillations (MMOs) are often found in systems with multiple time scales. Previous differential equation modeling and analysis of MMOs has mainly focused on local mechanisms to explain the small oscillations. The goal of this paper is to provide a systematic starting point for the study of global return maps that generate the large oscillations. This paper contributes to the understanding of global return maps in the following ways: (a) We provide a detailed numerical study of the singular return maps for the Koper model which is a prototypical example for MMOs that also relates to local normal form theory, (b) we decompose the global return map to provide first steps towards towards analytical proofs of complicated MMO patterns, (c) we develop affine and quadratic return map models based on the previous analysis and (d) we demonstrate how to combine the global map models with small oscillations generated by two types of local normal forms. We find that the local-global decomposition is an efficient way to model, simulate and analyze MMOs.