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

Panagiotis Kaklamanos, Nikola Popović, Kristian Uldall Kristiansen

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.

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

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 -shaped supercritical manifold that is embedded into a two-dimensional -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.

Paper Structure

This paper contains 19 sections, 15 theorems, 74 equations, 16 figures.

Table of Contents

  1. Introduction
  2. The double singular limit: geometry and singular cycles
  3. The critical manifold $\mathcal{M}_1$
  4. The supercritical manifold $\mathcal{M}_2$
  5. Relative geometry
  6. Singular cycles
  7. Singular perturbation
  8. Aligned or connected singularities
  9. Remote singularities
  10. Summary
  11. The Koper model revisited
  12. Singular geometry
  13. Classification of three-timescale dynamics
  14. Numerical verification
  15. Conclusions
  16. ...and 4 more sections

Key Result

Lemma 1

Let where $F$ is defined as in (eq:Fx). Then, for $\delta$ sufficiently small, the strong stable manifold of the origin for Equation (eq:classy-des) can be written as the graph where $I$ is an appropriately defined, fixed interval about $x=0$.

Figures (16)

  • Figure 1: Oscillatory dynamics in the Koper model, Equation (\ref{['eq:koper1']}), for different values of the parameters $k$ and $\lambda$. (a) MMO trajectory with single epochs of SAOs and Farey sequence $2^{s_1}2^{s_2}2^{s_3}\cdots$; (b) MMO trajectory with single epochs of SAOs and Farey sequence $2_{s_1}2_{s_2}2_{s_3}\cdots$; (c) MMO trajectory with double epochs of SAOs and Farey sequence $1^{s_1}1_{s_2}1^{s_3}1_{s_4}\cdots$; (d) relaxation oscillation.
  • Figure 2: Two-parameter bifurcation diagram of the three-timescale Koper model, Equation (\ref{['eq:koper1']}), to leading order in $\varepsilon$ and $\delta$; see Section \ref{['sec:koper']} for details.
  • Figure 3: (a) The critical manifold $\mathcal{M}_1$ as the set of equilibria for the fast flow of (\ref{['eq:norm12lay']}); the fast fibres are parallel to the $x$-direction. (b) The supercritical manifold $\mathcal{M}_2$ as the set of equilibria for the intermediate flow of (\ref{['eq:intermediate']}); the intermediate fibres are confined to $\mathcal{M}_2$ and evolve on planes with $z$ constant.
  • Figure 4: Projection of the supercritical manifold $\mathcal{M}_2$ and of the fold lines $\mathcal{L}^\mp$ of the critical manifold $\mathcal{M}_1$ onto the $(x,z)$-plane: in dependence on the parameters $\alpha$ and $\beta$, the pair of fold points $p^{\mp}$ of $\mathcal{M}_2$ lies either on $\mathcal{S}^r$ (panels (c) and (f)), on $\mathcal{S}^{a^\mp}$ (panels (a) and (d)), or on $\mathcal{L}^\mp$ (panels (b) and (e)).
  • Figure 5: Relative geometry of the folded singularities $q^\mp$ of $\mathcal{M}_1$ according to Definition \ref{['definition:singalign']} (top row); bifurcation of the resulting singular cycles, as described in Proposition \ref{['proposition:double']} (bottom row).
  • ...and 11 more figures

Theorems & Definitions (38)

  • Lemma 1
  • proof
  • Proposition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 2
  • proof
  • Definition 1
  • Definition 2
  • ...and 28 more