We study a class of multi-parameter three-dimensional systems of ordinary differential equations that exhibit dynamics on three distinct timescales. We apply geometric singular perturbation theory to explore the dependence of the geometry of these systems on their parameters, with a focus on mixed-mode oscillations (MMOs) and their bifurcations. In particular, we uncover a novel geometric mechanism that encodes the transition from MMOs with single epochs of small-amplitude oscillations (SAOs) to those with double-epoch SAOs; the former feature SAOs or pseudo-plateau bursting either "below" or "above" in their time series, while in the latter, SAOs or pseudo-plateau bursting occur both "below" and "above." We identify a relatively simple prototypical three-timescale system that realizes our mechanism, featuring a onedimensional S-shaped 2-critical manifold that is embedded into a two-dimensional S-shaped critical manifold in a symmetric fashion. We show that the Koper model from chemical kinetics is merely a particular realization of that prototypical system for a specific choice of parameters; in particular, we explain the robust occurrence of mixed-mode dynamics with double epochs of SAOs therein. Finally, we argue that our geometric mechanism can elucidate the mixed-mode dynamics of more complicated systems with a similar underlying geometry, such as a three-dimensional, three-timescale reduction of the Hodgkin-Huxley equations from mathematical neuroscience.

In this work, we study the dynamics of piecewise smooth systems on a codimension-2 transverse intersection of two codimension-1 discontinuity sets. The Filippov convention can be extended to such intersections, but this approach does not provide a unique sliding vector and, as opposed to the classical sliding vector-field on codimension-1 discontinuity manifolds, there is no agreed notion of stability in the codimension-2 context. From a modelling perspective, one may interpret this lack of determinacy as a fact that additional modelling is required; knowing the four adjacent vector-fields is not enough to define a unique forward flow. In this paper, we provide additional information to the system by performing a regularization of the piecewise smooth system, introducing two regularization functions and a small perturbation parameter. Then, based on singular perturbation theory, we define sliding and stability of sliding through a critical manifold of the singularly perturbed, regularized system. We show that this notion of sliding vector-field coincides with the Filippov one. The regularized system gives a parameterized surface, the canopy [12], independent of the regularization functions. This surface serves as our natural basis to derive new and simple geometric criteria on the existence, multiplicity and stability of the sliding flow, depending only on the smooth vector fields around the intersection. Interestingly, we are able to show that if there exist two sliding vector-fields then one is a saddle and the other is of focus/node/center type. This means that there is at most one stable sliding vector-field. We then investigate the effect of the choice of the regularization functions, and, using a blowup approach, we demonstrate the mechanisms through which sliding behavior can appear or disappear on the intersection and describe what consequences this has on the dynamics on the adjacent codimension-1 discontinuity sets. This blowup method also shows that the PWS limit of the regularization may be well-defined, even in cases where the Filippov sliding vector-field is nonunique. Finally, we show the existence of canard explosions of regularizations of PWS systems in R 3 that depend on a single unfolding parameter. * P. Kaklamanos

We consider a non-dimensionalised version of the four-dimensional Hodgkin-Huxley equations [J. Rubin and M. Wechselberger, Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model, Biological Cybernetics, 97 (2007), pp. 5-32], and we present a novel and global three-dimensional reduction that is based on geometric singular perturbation theory (GSPT). We investigate the dynamics of the resulting reduced system in two parameter regimes in which the flow evolves on three distinct timescales. Specifically, we demonstrate that the system exhibits bifurcations of oscillatory dynamics and complex mixed-mode oscillations (MMOs), in accordance with the geometric mechanisms introduced in [P. Kaklamanos, N. Popović, and K. U. Kristiansen, Bifurcations of mixed-mode oscillations in three-timescale systems: An extended prototypical example, Chaos: An Interdisciplinary Journal of Nonlinear Science, 32 (2022), p. 013108], and we classify the various firing patterns in dependence of the external applied current. While such patterns have been documented in [S. Doi, S. Nabetani, and S. Kumagai, Complex nonlinear dynamics of the Hodgkin-Huxley equations induced by time scale changes, Biological Cybernetics, 85 (2001), pp. 51-64] for the multi-timescale Hodgkin-Huxley equations, we elucidate the geometry that underlies the transitions between them, which had not been previously emphasised.

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We study delayed loss of stability in a class of fast–slow systems with two fast variables and one slow one, where the linearisation of the fast vector field along a one-dimensional critical manifold has two real eigenvalues which intersect before the accumulated contraction and expansion are balanced along any individual eigendirection. That interplay between eigenvalues and eigendirections renders the use of known entry–exit relations unsuitable for calculating the point at which trajectories exit neighbourhoods of the given manifold. We illustrate the various qualitative scenarios that are possible in the class of systems considered here, and we propose novel formulae for the entry–exit functions that underlie the phenomenon of delayed loss of stability therein.

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