Mixed-mode oscillations in singularly perturbed three-timescale systems
In this work, we are concerned with the dynamics of three-dimensional, three-timescale systems of ordinary differential equations. Systems with two timescales have been extensively studied, while the theory of systems with three or more timescales is less developed, although in some cases they provide more realistic modelling of natural systems. Our focus is on mixed-mode oscillations (MMOs), i.e., on trajectories which consist of alternating small-amplitude oscillations (SAOs) and large excursions or large-amplitude oscillations (LAOs). We aim to understand the underlying mechanisms that are responsible for the qualitative properties of these trajectories, as well as to classify the various behaviours of these systems upon variation of their parameters, using geometric singular perturbation theory (GSPT). In the first part, we present a new prototypical system that captures a geometric mechanism which distinguishes between MMOs that feature SAOs only in one region of the phase space from those which feature SAOs in two distinct regions of the phase space; we refer to the former as MMOs with single SAO-epochs and to the latter as MMOs with double SAO-epochs. In essence, this distinction is based on the relative position of points where normal hyperbolicity with respect to the fast flow is lost. In the second part, we show that the Koper model, a well known system from chemical kinetics, is merely a particular realisation of the prototypical example that we introduced in the first part. This system has been extensively studied in the two-timescale context, but, to our knowledge, it has not been studied in the three-timescale context before. We hence classify its dynamics in dependence of its parameters in the three-timescale context, and we show that some phenomena that are delicate in the two-timescale setting, like MMOs with double SAO epochs, become robust in the three-timescale one. In the third part, we show that the four-dimensional Hodgkin-Huxley equations from mathematical neuroscience can be reduced to a three-dimensional, three-timescale system. We then illustrate that, in particular parameter regimes, this system features similar properties to the prototypical example that we introduced in the first part, and we show how the theory that we introduced in the previous chapters can be used to explain its behaviour. From another point of view, we show that the system can be written in the non-standard form of GSPT, and we show that its oscillatory dynamics can be explained by extending the notions that we introduced in the previous chapters. In the fourth part, we study a three-dimensional system that describes the El-Ni˜no Southern Oscillation (ENSO) phenomenon. This system has too been extensively studied in the two-timescale context but not in the three-timescale one. However, this system is more complicated in terms of its invariant manifold structure compared to the other two systems mentioned above. Extending some notions from the previous parts and based again on the relative position of sets where normal hyperbolicity with respect to the fast flow is lost, we distinguish between oscillations with different qualitative properties. Moreover, we perform a desingularisation analysis of the sets where normal hyperbolicity is lost and we give estimates on bifurcation-delay phenomena, which are points that had not been addressed in previous works. In summary, this work is concerned with local and global phenomena in three-timescale systems. The focus is (a) on systems that have previously been analysed only in the two-timescale context, like the Koper and ENSO models, thus exploring the dynamics in the three-timescale setting, and (b) on systems that have been studied in the multi-timescale context (more than two) in the past, but not in the GSPT framework, and where the mechanisms that encode transitions between qualitatively different behaviours were elusive, like the Hodgkin-Huxley equations. The various qualitative behaviours depend on the underlying geometry of these systems, and the extended prototypical example that we propose does not only provide a geometric mechanism that is directly applicable to other systems, like the Koper model and the Hodgkin-Huxley equations, but also, by simple extension of some notions, provides insight on the dynamics and possible behaviours in three-timescale systems with more complicated geometry, like the ENSO model. Finally, we remark on some phenomena that are not robust in the two-timescale setting, but become robust in the three timescale one.