Combinatorics

Discrete mathematics

Physics

Exponential function

Topology (electrical circuits)

Digraph

Mathematics

Exponential growth

Synchronization networks

Laplace operator

Mathematical analysis

Eigenvalues and eigenvectors

Synchronization (alternating current)

Quantum mechanics

Strongly connected component

Spanning tree

Infimum and supremum

IEEE Transactions on Automatic Control

For a high-dimensional Kuramoto model with identical oscillators under a general digraph that has a directed spanning tree, although the exponential synchronization has been proved under some initial state constraints, exponential synchronization rates have not been described exactly until now. In this article, the supremum of exponential synchronization rates is precisely determined as the smallest real part of the nonzero Laplacian eigenvalues of the digraph. Our obtained result extends the existing results from the special case of strongly connected balanced digraphs to the condition of general digraphs owning directed spanning trees, which is the weakest condition for synchronization from the aspect of network structure. Moreover, our adopted method is completely different from and much more elementary than the previous differential geometry method.

On Exponential Synchronization Rates for High-Dimensional Kuramoto Models With Identical Oscillators and Digraphs