Pulse coupled oscillators is one of the most widely used models for describing collective phenomena such as synchronization. The most well studied such system is the Kuramoto model and it consists of first order oscillators. When there is no coupling, the phase of each oscillator increases at a constant speed, called the natural frequency. However, coupling the oscillators by letting them influence each other’s speed leads (for sufficiently strong coupling) to synchronization.
A generalization of the Kuramoto model is to consider second order oscillators, where both the speed and the acceleration become important. Such models have been used to describe several systems, for example, power distribution grids. In recent research with Jian Gao we have developed a self-consistent approach to the dynamics of such systems that is more accurate and simpler to obtain compared to self-consistent approaches that have appeared earlier in the literature. The more accurate self-consistent equations allow to make much better predictions on the dynamics of the system, especially, close to the incoherent (non-synchronized) state.
The preprint for this work is now available from the arXiv. More details and references to related work can be found there.