Global Topological Dirac Synchronization

Abstract

Synchronization is a fundamental dynamical state of interacting oscillators, observed, e.g., in natural biological rhythms and in the brain. Global synchronization which occurs when non-linear or chaotic oscillators placed on the nodes of a network display the same dynamics has received great attention in network theory. Here we propose and investigate Global Topological Dirac Synchronization (GTDS) on higher-order networks such as cell and simplicial complexes. This is a state where oscillators associated to simplices and cells of arbitrary dimension, coupled by the Topological Dirac operator, operate at unison. By combining algebraic topology with non-linear dynamics and machine learning, we derive the topological conditions under which this state exists and the dynamical conditions under which it is stable. We provide evidence of 1-dimensional simplicial complexes (networks) and 2-dimensional simplicial and cell complexes where GTDS can be observed. Our results point out that GTDS is a possible dynamical state of cell complexes and simplicial complexes that occur only in some specific network topologies and geometries, the latter ones being determined by the weights of the higher-order networks.