Combinatorics

Physics

Sociology

Simplicial set

Nonlinear system

Topology (electrical circuits)

Pure mathematics

Mathematics

Simplicial approximation theorem

Dynamical systems theory

Collective behavior

Statistical physics

Phase transition

Artificial intelligence

Simplicial complex

Anthropology

Multistability

Synchronization (alternating current)

Computer science

Complex system

Simplex

Quantum mechanics

Homotopy category

Homotopy

Physical Review Letters

Collective behavior in large ensembles of dynamical units with nonpairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.

Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes