  All interesting and fascinating collective properties of a complex system
arise from the intricate way in which its components interact. Various systems
in physics, biology, social sciences and engineering have been successfully
modelled as networks of coupled dynamical systems, where the graph links
describe pairwise interactions. This is, however, too strong a limitation, as
recent studies have revealed that higher-order many-body interactions are
present in social groups, ecosystems and in the human brain, and they actually
affect the emergent dynamics of all these systems. Here, we introduce a general
framework that allows to study coupled dynamical systems accounting for the
precise microscopic structure of their interactions at any possible order. We
consider the most general ensemble of identical dynamical systems, organized on
the nodes of a simplicial complex, and interacting through
synchronization-non-invasive coupling function. The simplicial complex can be
of any dimension, meaning that it can account, at the same time, for pairwise
interactions, three-body interactions and so on. In such a broad context, we
show that complete synchronization exists as an invariant solution, and we give
the necessary condition for it to be observed as a stable state in terms of a
Master Stability Function. This generalizes the existing results valid for
pairwise interactions (i.e. graphs) to the case of complex systems with the
most general possible architecture. Moreover, we show how the approach can be
simplified for specific, yet frequently occurring, instances, and we verify all
our theoretical predictions in synthetic and real-world systems. Given the
completely general character of the method proposed, our results contribute to
the theory of dynamical systems with many-body interactions and can find
applications in an extremely wide range of practical cases.
