Abstract: We propose a compartmental model for a disease with temporary immunity and secondary infections. From our assumptions on the parameters involved in the model, the system naturally evolves in three time scales. We characterize the equilibria of the system and analyze their stability. We find conditions for the existence of two endemic equilibria, for some cases in which $R_0<1$. Then, we unravel the interplay of the three time scales, providing conditions to foresee whether the system evolves in all three scales, or only in the fast and the intermediate ones. We conclude with numerical simulations and bifurcation analysis, to complement our analytical results.
Abstract: We present a novel and global three-dimensional reduction of a nondimensionalized version of the four-dimensional Hodgkin–Huxley equations [J. Rubin and M. Wechselberger, Biol. Cybernet., 97 (2007), pp. 5–32] that is based on geometric singular perturbation theory. We investigate the dynamics of the resulting three-dimensional 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, in accordance with the geometric mechanisms introduced in [P. Kaklamanos, N. Popović, and K. U. Kristiansen, Chaos, 32 (2022), 013108], and we classify the various firing patterns in terms of the external applied current. While such patterns have been documented in [S. Doi, S. Nabetani, and S. Kumagai, Biol. Cybernet., 85 (2001), pp. 51–64] for the multiple-timescale Hodgkin–Huxley equations, we elucidate the geometry that underlies the transitions between them, which had not been previously emphasized.
Abstract: 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 contrac- tion 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 con- sidered here, and we propose novel formulae for the entry–exit functions that underlie the phenomenon of delayed loss of stability therein.
Abstract: We study the three-timescale dynamics of a model that describes the El Niño Southern Oscillation (ENSO) phenomenon, which was proposed in Roberts et al. (2016). While ENSO phenomena are inherently characterised by the presence of multiple distinct timescales, the above model has previously been studied in a two-timescale context only. Here, we uncover the geometric mechanisms that are responsible for complex oscillatory dynamics in a three-timescale regime, and we demonstrate that the model exhibits a variety of qualitatively different behaviours in that regime, such as mixed-mode oscillation (MMO) with “plateaus” – trajectories where epochs of quiescence alternate with dramatic excursions – and relaxation oscillation. The latter, although emergent also in the two-timescale context in appropriate parameter regimes, had not been documented previously for this particular model. Moreover, we show that these mechanisms are relevant to models from other fields of ecological and population dynamics, as the underlying geometry is similar to the unfolding of Rosenzweig–MacArthur-type models in three dimensions.
Abstract: 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. We identify a relatively simple prototypical three-timescale system that realises our mechanism, featuring a one-dimensional S-shaped supercritical 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 realisation 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 of a three-dimensional, three-timescale reduction of the Hodgkin-Huxley equations from mathematical neuroscience.
Abstract: 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. In this paper, we perform 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) 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. Finally, we show the existence of canard explosions of regularizations of PWS systems in ℝ3 that depend on a single unfolding parameter.