I study a simple model of synchronisation proposed by Jensen (2008). The relevant degrees of freedom are expected to be strictly increasing functions of time, such as the total angle swept out by an oscillator. The model is rooted in Winfree’s mean-field model for spontaneous synchronisation; some of Winfree’s basic assumptions, such as identical or nearly identical dynamics and identical couplings, are therefore retained. I investigated the behaviour of the present model with respect to synchronisation without and in the presence of time delay.

The mathematical treatment focuses on characterising the synchronised state as either at- tractive or repulsive, producing a theory (which ultimately leads to a phase diagram) that compares well with numerics. I employed a perturbative approach, linearising in small time delays and small phase differences. The interaction between individual oscillators is captured by an interaction matrix, which does not require further approximation, i.e. lattice structure enters exactly. To link with established results in the literature, a mean field theory, however, is also studied.

The main result is that these typically systems synchronise due to a time delay.