Bounding network spectra for network design

The identification of the limiting factors in the dynamical behavior of complex systems is an important interdisciplinary problem which often can be traced to the spectral properties of an underlying network. By deriving a general relation between the eigenvalues of weighted and unweighted networks, here I show that for a wide class of networks the dynamical behavior is tightly bounded by few network parameters. This result provides rigorous conditions for the design of networks with predefined dynamical properties and for the structural control of physical processes in complex systems. The results are illustrated using synchronization phenomena as a model process.Comment: 17 pages, 4 figure

Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical Networks

Synchronization is an important and prevalent phenomenon in natural and engineered systems. In many dynamical networks, the coupling is balanced or adjusted in order to admit global synchronization, a condition called Laplacian coupling. Many networks exhibit incomplete synchronization, where two or more clusters of synchronization persist, and computational group theory has recently proved to be valuable in discovering these cluster states based upon the topology of the network. In the important case of Laplacian coupling, additional synchronization patterns can exist that would not be predicted from the group theory analysis alone. The understanding of how and when clusters form, merge, and persist is essential for understanding collective dynamics, synchronization, and failure mechanisms of complex networks such as electric power grids, distributed control networks, and autonomous swarming vehicles. We describe here a method to find and analyze all of the possible cluster synchronization patterns in a Laplacian-coupled network, by applying methods of computational group theory to dynamically-equivalent networks. We present a general technique to evaluate the stability of each of the dynamically valid cluster synchronization patterns. Our results are validated in an electro-optic experiment on a 5 node network that confirms the synchronization patterns predicted by the theory.Comment: 6 figure

Sampled-data synchronization control of dynamical networks with stochastic sampling

Copyright @ 2012 IEEEThis technical note is concerned with the sampled-data synchronization control problem for a class of dynamical networks. The sampling period considered here is assumed to be time-varying that switches between two different values in a random way with given probability. The addressed synchronization control problem is first formulated as an exponentially mean-square stabilization problem for a new class of dynamical networks that involve both the multiple probabilistic interval delays (MPIDs) and the sector-bounded nonlinearities (SBNs). Then, a novel Lyapunov functional is constructed to obtain sufficient conditions under which the dynamical network is exponentially mean-square stable. Both Gronwall's inequality and Jenson integral inequality are utilized to substantially simplify the derivation of the main results. Subsequently, a set of sampled-data synchronization controllers is designed in terms of the solution to certain matrix inequalities that can be solved effectively by using available software. Finally, a numerical simulation example is employed to show the effectiveness of the proposed sampled-data synchronization control scheme.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Royal Society of the UK, the National Natural Science Foundation of China under Grants 61028008, 60974030, 61134009 and 61104125, the National 973 Program of China under Grant 2009CB320600, and the Alexander von Humboldt Foundation of Germany

Control of coupled oscillator networks with application to microgrid technologies

The control of complex systems and network-coupled dynamical systems is a topic of vital theoretical importance in mathematics and physics with a wide range of applications in engineering and various other sciences. Motivated by recent research into smart grid technologies we study here control of synchronization and consider the important case of networks of coupled phase oscillators with nonlinear interactions--a paradigmatic example that has guided our understanding of self-organization for decades. We develop a method for control based on identifying and stabilizing problematic oscillators, resulting in a stable spectrum of eigenvalues, and in turn a linearly stable synchronized state. Interestingly, the amount of control, i.e., number of oscillators, required to stabilize the network is primarily dictated by the coupling strength, dynamical heterogeneity, and mean degree of the network, and depends little on the structural heterogeneity of the network itself

Synchronization of dynamical networks with nonidentical nodes: Criteria and control

This paper presents a framework for global synchronization of dynamical networks with nonidentical nodes. Several criteria for synchronization are given using free matrices for both cases of synchronizing to a common equilibrium solution of all isolated nodes and synchronizing to the average state trajectory. These criteria can be viewed as generalizations of the master stability function method for local synchronization of networks with identical nodes to the case of nonidentical nodes. The controlled synchronization problem is also studied. The control action, which is subject to certain constraints, is viewed as reorganization of the connection topology of the network. Synchronizability conditions via control are put forward. The synchronizing controllers can be obtained by solving an optimization problem.published_or_final_versio