Phase Transitions in Swarmalators

Systems of oscillators called Swarmalators, whose phase and spatial dynamics are coupled, have been used to describe the dynamics of some living systems. Their collective behavior presents simultaneous aggregation in space and synchronization in phase which in turn leads in some cases to explosive synchronization in a finite population as a function of the coupling parameter between the phases of the internal dynamics. This phenomenon is described using the order parameter and the Hamiltonian formalism. Near the synchronization transition the internal phases of the particles are represented by the XY model, and their transition to synchronization, which will be discussed, can be of the first or second order. We shall also discuss a multilayer system of swarmalators, which presents some interesting phases on their way to synchronization.