Slow Manifolds, Canard Orbits and the Organization of Mixed-Mode Oscillations

Abstract

A mixed-mode oscillation is a complex waveform with a pattern of alternating smallamplitude oscillations (SAOs) and large-amplitude oscillations (LAOs). MMOs have been observed in many applications, including semiconductor lasers, neuron models and chemical reactions. In this thesis, we aim to understand and elucidate the phenomenological behavior of MMOs inherited in slow-fast systems, which take the form of ordinary differential equations with a group of fast variables and a group of slow variables, with a time-scale separation parameter ". Geometric singular perturbation theory predicts the existence of locally invariant slow manifolds that organize the slow dynamics. The fast dynamics are typically organized by stable and unstable (fast) manifolds of slow manifolds. Interactions of slow manifolds give rise to so-called canard orbits, which create a mechanism for generating SAOs. On the other hand, LAOs require a global return mechanism. In this thesis, we present a framework for studying two-dimensional slow manifolds, canard orbits and their roles in organizing MMOs in parameter regimes where " is too large for applying classical results from the theory. This is achieved by employing advanced numerical methods based on a two-point boundary value problem setup. We first consider an autocatalytic chemical reaction model with one fast and two slow variables. In this system, we find that canard orbits, which occur along intersections of two-dimensional attracting and repelling slow manifolds, are organized in pairs that we call twin canard orbits. Consequently, the extended attracting slow manifold is divided into subsurfaces called ribbons. Ribbons and associated twin canard orbits organize MMOs when " is relatively large. A continuation analysis illustrates how twin canard orbits arise due to generic quadratic tangencies of slow manifolds. In systems with two fast and two slow variables, two-dimensional slow manifolds of saddle type play a key role in organizing MMOs. One goal of this thesis is to introduce a general approach for computing two-dimensional saddle slow manifolds and their threedimensional stable and unstable manifolds, as well as associated canard orbits in R4. We first test and demonstrate our methods for an extended normal form of a folded node. These methods are then reliably implemented for the full four-dimensional Hodgkin- Huxley neuron model, where " is again relatively large. Our results show that MMOs of this model are also organized by ribbons of the attracting slow manifold and bounding twin canard orbits. Overall, we conclude that our approach is suitable for computing two-dimensional slow manifolds in R3 as well as in R4, and in parameter regimes that are beyond what is known from established theory. In particular, we show that it is practical to study fourdimensional slow-fast vector fields without the need for applying any reduction technique.