Optimizing synchronizability in networks of coupled systems
Of collective behaviors in networks of coupled systems, synchronization is of central importance and an extensively studied area.This is due to the fact that it is essential for the proper functioning of a wide variety of natural and engineered systems. Traditionally,uniform coupling strength has been the default choice and the synchronizability measure has been employed for analysis and enhancement of synchronizability. The main drawback of optimizing the synchronizability measure is that it can reach the Pareto frontier but not necessarily a unique point on the Pareto frontier. Additionally, the shortcoming of uniform coupling strength is that it can reach Pareto frontier in specific topologies including edge-transitive graphs. To achieve a unique optimal answer on the Pareto frontier, this paper
takes a different approach and addresses the synchronizability in networks of coupled dynamical systems with nonuniform coupling strength and optimizing the synchronizability via maximizing the minimum distance between the nonzero eigenvalues of the Laplacian and the acceptable boundaries for the stability of the system. Furthermore, two solution methods, namely the concave-convex fractional
programming and the Semidefinite Programming (SDP) formulations of the problem have been provided. The proposed solution methods have been compared over different topologies and branches of an arbitrary network, where the SDP based approach has shown to be less restricted and more suitable for a wider range of topologies.
Key words: Network of Coupled Dynamical Systems, Synchronizability, Semidefinite Programming, Pareto Frontier.