Families of Kuramoto models and bounded symmetric domains

We define families of Kuramoto models related to bounded symmetric domains. The families include Lohe unitary and spherical models as special cases. Our approach is based on the construction proposed by Watanabe and Strogats. We replace the Poincare disc and its $$S^1$$ boundary with bounded symmetric domains and with its Bergman–Shilov boundaries. In Cartan’s classifications there are four classical domains of types I–IV. Here we consider the domains of types I, II, and III. For a fixed domain, there is a decreasing chain of components of Bergman–Shilov boundaries. This leads to the families of Kuramoto models that we describe here.