Emergent Behaviors of a Kuramoto Ensemble Under Fading Memory

We study the emergent dynamics of the Kuramoto model with memory effect. The Kuramoto model with memory effect belongs to the system of Volterra-type integro-differential equations on the unit circle. For the modeling of memory effect, we adopt nonlocal temporal interactions so that dynamic behaviors of oscillators are affected by memories of the past interactions. For the proposed model, we first study global well-posedness, and then, we provide a sufficient framework for the uniform boundedness of the phase diameter. We also show the emergence of complete frequency synchronization in two different ways. One is to use a bootstrap argument and the other is to use an energy functional. In both ways, boundedness of the phase diameter is needed. In particular, we show the emergence of complete phase synchronization for the case where natural frequencies are all identical. We provide several numerical examples and compare them with presented analytical results.