The hidden sensitivity of non-smooth dynamics

Switches in dynamical systems are known to exhibit wildly different behaviours depending on how they are modelled, for instance whether they occur as smooth transitions or abrupt jumps, and whether they involve delays or discrete perturbations. These differences arise because switches are sensitive to perturbation, but there is limited knowledge about where this sensitivity comes from. Here we take a switch in a simple one-dimensional system, then discretise time and introduce a small delay. The resulting system is described by a piecewise-linear map with incredibly complex dynamics, and is sensitive to parameter changes even if the time-steps and delays are made infinitesimally small. We show that this sensitivity reveals itself in the more versatile numerical tool of the transition count, which captures the likelihood of switching occurring at any instant in a simulation. We use this to show that sensitivity to parameters persists in a system with two switches, where it then has large-scale dynamical effects. The use of transition counts in this way may prove a versatile tool for studying more complex switching processes in general.