Numerical methods for control-based continuation of relaxation oscillations

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.