Martin Wechselberger

Mixed-Mode Oscillations with Multiple Time Scales

Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

Existence and Bifurcation of Canards in

We give a geometric analysis of canards of folded node type in singularly perturbed systems with two-dimensional (2D) folded critical manifold using the blow-up technique. The existence of two primary canards is known provided a nonresonance condition

\mathbbR^3

\mu\notin\mathbb{N}

in the Case of a Folded Node

is satisfied, where

Canard Induced Mixed-Mode Oscillations in a Medial Entorhinal Cortex Layer II Stellate Cell Model ∗

\mu=\lambda_1/\lambda_2

The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales.

denotes the ratio of the eigenvalues of the associated folded singularity of the reduced flow. We show that, due to resonances, bifurcation of secondary canards occurs. We give a detailed geometric explanation of this phenomenon using an extension of Melnikov theory to prove a transcritical bifurcation of canards for odd

Chaotic attractors of relaxation oscillators

\mu

Canards, Clusters, and Synchronization in a Weakly Coupled Interneuron Model

. Furthermore, we show numerically the existence of a pitchfork bifurcation for even

The Dynamic Range of Bursting in a Model Respiratory Pacemaker Network

\mu

Mixed Mode Oscillations as a Mechanism for Pseudo-Plateau Bursting

and a novel turning point bifurcation close to