arXiv subjects

Panagiotis Kaklamanos

Publications and source records attributed to Panagiotis Kaklamanos.

5 recordsLinked to original sources

arXiv2023

A geometric analysis of the SIRS model with secondary infections

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 $\mathcal{R}_0 < 1$. 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.

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arXiv2023

Geometric singular perturbation analysis of the multiple-timescale Hodgkin-Huxley equations

We present a novel and global three-dimensional reduction of 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] that is based on geometric singular perturbation theory (GSPT). 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 (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 terms 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 multiple-timescale Hodgkin-Huxley equations, we elucidate the geometry that underlies the transitions between them, which had not been previously emphasised.

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arXiv2022

Entry-exit functions in fast-slow systems with intersecting eigenvalues

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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arXiv2022

Complex oscillatory dynamics in a three-timescale El Niño Southern Oscillation model

We study the three-timescale dynamics of a model that describes the El Niño Southern Oscillation (ENSO) phenomenon, which was proposed in [A. Roberts, J. Guckenheimer, E. Widiasih, A. Timmermann, and C. K. Jones, Mixed-mode oscillations of El Niño--Southern Oscillation, Journal of the Atmospheric Sciences, 73 (2016), pp. 1755--1766]. 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 system exhibits a variety of qualitatively different behaviours, 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. 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.

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arXiv2021

Bifurcations of mixed-mode oscillations in three-timescale systems: an extended prototypical example

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.

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