25 research outputs found
Spectral Network Principle for Frequency Synchronization in Repulsive Laser Networks
Network synchronization of lasers is critical for reaching high-power levels
and for effective optical computing. Yet, the role of network topology for the
frequency synchronization of lasers is not well understood. Here, we report our
significant progress toward solving this critical problem for networks of
heterogeneous laser model oscillators with repulsive coupling. We discover a
general approximate principle for predicting the onset of frequency
synchronization from the spectral knowledge of a complex matrix representing a
combination of the signless Laplacian induced by repulsive coupling and a
matrix associated with intrinsic frequency detuning. We show that the gap
between the two smallest eigenvalues of the complex matrix generally controls
the coupling threshold for frequency synchronization. In stark contrast with
Laplacian networks, we demonstrate that local rings and all-to-all networks
prevent frequency synchronization, whereas full bipartite networks have optimal
synchronization properties. Beyond laser models, we show that, with a few
exceptions, the spectral principle can be applied to repulsive Kuramoto
networks. Our results may provide guidelines for optimal designs of scalable
laser networks capable of achieving reliable synchronization
The continuum limit of the Kuramoto model on sparse random graphs
In this paper, we study convergence of coupled dynamical systems on
convergent sequences of graphs to a continuum limit. We show that the solutions
of the initial value problem for the dynamical system on a convergent graph
sequence tend to that for the nonlocal diffusion equation on a unit interval,
as the graph size tends to infinity. We improve our earlier results in [Arch.
Ration. Mech. Anal., 21 (2014), pp. 781--803] and extend them to a larger class
of graphs, which includes directed and undirected, sparse and dense, random and
deterministic graphs.
There are three main ingredients of our approach. First, we employ a flexible
framework for incorporating random graphs into the models of interacting
dynamical systems, which fits seamlessly with the derivation of the continuum
limit. Next, we prove the averaging principle for approximating a dynamical
system on a random graph by its deterministic (averaged) counterpart. The proof
covers systems on sparse graphs and yields almost sure convergence on time
intervals of order where is the number of vertices. Finally, a
Galerkin scheme is developed to show convergence of the averaged model to the
continuum limit.
The analysis of this paper covers the Kuramoto model of coupled phase
oscillators on a variety of graphs including sparse Erd\H{o}s-R{\' e}nyi,
small-world, and power law graphs.Comment: To appear in Communications in Mathematical Science