  Coupled oscillator networks often display transitions between qualitatively
different phase-locked solutions -- such as synchrony and rotating wave
solutions -- following perturbation or parameter variation. In the limit of
weak coupling, these transitions can be understood in terms of commonly studied
phase approximations. As the coupling strength increases, however, predicting
the location and criticality of transition, whether continuous or
discontinuous, from the phase dynamics may depend on the order of the phase
approximation -- or a phase description of the network dynamics that neglects
amplitudes may become impossible altogether. Here we analyze synchronization
transitions and their criticality systematically for varying coupling strength
in theory and experiments with coupled electrochemical oscillators. First, we
analyze bifurcations analysis of synchrony and splay states in an abstract
phase model and discuss conditions under which synchronization transitions with
different criticalities are possible. Second, we illustrate that transitions
with different criticality indeed occur in experimental systems. Third, we
highlight that the amplitude dynamics observed in the experiments can be
captured in a numerical bifurcation analysis of delay-coupled oscillators. Our
results showcase that reduced order phase models may miss important features
that one would expect in the dynamics of the full system.
