  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.
