Directed acyclic decomposition of Kuramoto equations

Abstract

The Kuramoto model is one of the most widely studied models for describing synchronization behaviors in a network of coupled oscillators, and it has found a wide range of applications. Finding all possible frequency synchronization configurations in a general non-uniform, heterogeneous, and sparse network is important yet challenging due to complicated nonlinear interactions. From the view point of homotopy deformation, we develop a general framework for decomposing a Kuramoto network into smaller directed acyclic subnetworks, which lays the foundation for a divideand-conquer approach to studying the configurations of frequency synchronization of large Kuramoto networks.The spontaneous synchronization of a network of oscillators is an emergent phenomenon that naturally appears in many seemingly independent complex systems including mechanical, chemical, biological, and even social systems. The Kuramoto model is one of the most widely studied and successful mathematical models for analyzing synchronization behaviors. While much is known about the macro-scale question of whether or not a Kuramoto network can be synchronized, detailed analysis of the possible configurations of the oscillator once it has reached synchronization remains difficult for large networks partly due to the nonlinear interactions involved. In this work, we demonstrate that by dividing the link between two oscillators into two one-way interactions, complex networks can indeed be decomposed into much simpler subnetwork. This is a crucial step toward fully understanding synchronization configurations in large networks. arXiv:1903.04492v2 [math.CO]

Facets and facet subgraphs of symmetric edge polytopes

“…These concepts are defined and studied closely in recent works [3,10]. The notations are the mirror images of the map G → ∇ G , and the two interact in an expected way -for any subset X of ∇ G , we have ∇ G X = X, and for any directed subgraph H of G, we have G ∇ H = H. The same holds for their directed counterparts.…”

“…facet) F of ∇ G such that H = G F . Such facet subgraphs have been studied in connection to homotopy methods for solving algebraic Kuramoto equations [3]. In the broader context, facet subgraphs and face subgraphs were also classified algebraically [10,12].…”

“…Recent results on facets and facet subgraphs. The role of facets of ∇ G as building blocks in the study of algebraic Kuramoto systems has been explored recently from the viewpoint of homotopy continuation methods [3]. Facets of ∇ G associated with even cycles are described from the viewpoint of Ehrhart theory by Ohsugi and Shibata [18].…”

“…Applications to algebraic Kuramoto equations. Facets of ∇ G play important roles in the study of algebraic Kuramoto equations [3,4,7], which has attracted interests from electric engineering, biology, and chemistry. The original Kuramoto equations models the synchronization behaviors of a network of coupled oscillators [15], which can be represented by a weighted graph G with the nodes V(G) = {0, .…”

“…Clearly, H G (x, 1) = F R G (x). As t varies between 0 and 1 within the interval (0, 1), H G (x, t) represents a smooth deformation of the system F R G , and the corresponding complex roots also vary smoothly, forming smooth paths reaching all complex zeros of F R G [3,14]. The starting points of the smooth paths, however, are not well defined, as the limit points of these paths as t → 0 are not contained in (C * ) n .…”