Explosive Synchronization and Multistability in Large Systems of Kuramoto Oscillators with Higher-Order Interactions

Abstract

Since its original formulation, the Kuramoto model and its many variants have served as critical tools for uncovering and understanding the emergence of nonlinear collective behavior. However, recent evidence suggests that in such phase-reduced systems, interactions beyond the typical pair-wise angle differences need to be considered to develop a full picture of the dynamics. In particular, higher-order interactions, namely non-additive, nonlinear interactions that take place between three or more oscillators are required. Here we explore these interactions and their effect on the macroscopic dynamics of coupled phase oscillator systems. The analysis for these systems begins with all-to-all coupled systems where a range of techniques including dimensionality reductions and self-consistency analyses may be employed. The effects of the various higher-order coupling terms on the macroscopic dynamics may then be explored, revealing a natural mechanism for nonlinear phenomena that includes abrupt (i.e., explosive) synchronization transitions and extensive multistability. These dynamics are qualitatively preserved under more heterogeneous network topologies. Moreover, the high degree of multistability in such networks allows for the system to store information and memory.