Numerical methods for control-based continuation of relaxation oscillations

TL;DR

This paper introduces an adaptive B-spline discretisation and a phase-locking method to extend control-based continuation to relaxation oscillations, enabling efficient analysis of slow-fast systems in experiments.

Contribution

The paper presents a novel adaptive B-spline discretisation and phase constraint, allowing CBC to effectively study relaxation oscillations in systems previously difficult to analyze.

Findings

Successfully applied to synthetic gene network simulations.

Demonstrated on Oregonator model simulations.

Extended CBC applicability to slow-fast systems.

Abstract

Control-based continuation (CBC) is an experimental method that can reveal stable and unstable dynamics of physical systems. It extends the path-following principles of numerical continuation to experiments, and provides systematic dynamical analyses without the need for mathematical modelling. CBC has seen considerable success in studying the bifurcation structure of mechanical systems. Nevertheless, the method is not practical for studying relaxation oscillations. Large numbers of Fourier modes are required to describe them, and the length of the experiment significantly increases when many Fourier modes are used, as the system must be run to convergence many times. Furthermore, relaxation oscillations often arise in autonomous systems, for which an appropriate phase constraint is required. To overcome these challenges, we introduce an adaptive B-spline discretisation, that can produce a parsimonious description of responses that would otherwise require many Fourier modes. We couple this to a novel phase constraint, that phase-locks control target and solution phase. Results are demonstrated on simulations of a slow-fast synthetic gene network and an Oregonator model. Our methods extend CBC to a much broader range of systems than have been studied so far, opening up a range of novel experimental opportunities on slow-fast systems.