Robust outer synchronization between two complex networks with fractional order dynamics

Synchronization between two coupled complex networks with fractional-order dynamics, hereafter referred to as outer synchronization, is investigated in this work. In particular, we consider two systems consisting of interconnected nodes. The state variables of each node evolve with time according to a set of (possibly nonlinear and chaotic) fractional-order differential equations. One of the networks plays the role of a master system and drives the second network by way of an open-plus-closed-loop (OPCL) scheme. Starting from a simple analysis of the synchronization error and a basic lemma on the eigenvalues of matrices resulting from Kronecker products, we establish various sets of conditions for outer synchronization, i.e., for ensuring that the errors between the state variables of the master and response systems can asymptotically vanish with time. Then, we address the problem of robust outer synchronization, i.e., how to guarantee that the states of the nodes converge to common values when the parameters of the master and response networks are not identical, but present some perturbations. Assuming that these perturbations are bounded, we also find conditions for outer synchronization, this time given in terms of sets of linear matrix inequalities (LMIs). Most of the analytical results in this paper are valid both for fractional-order and integer-order dynamics. The assumptions on the inner (coupling) structure of the networks are mild, involving, at most, symmetry and diffusivity. The analytical results are complemented with numerical examples. In particular, we show examples of generalized and robust outer synchronization for networks whose nodes are governed by fractional-order Lorenz dynamics. After the seminal work in Ref. 1, complex network models have become ubiquitous in the analysis of many phenomena appearing in the physical, biological, and social sciences. In particular, the analysis of synchronization in this class of models has attracted the attention of many researchers, with the expectation of gaining new insights of the interactions taking place in real-world complex systems. Most of the work in the literature so far has been focused on the synchronization of a collection of interconnected nodes (forming one single neteork), where each node is a dynamical system governed by a set of nonlinear differential equations (possibly displaying chaotic dynamics). In this paper, we study an extended version of this problem. In particular, we consider a setup consisting of two complex networks which are coupled unidirectionally (such that a set of signals from the master network are injected into the response network) with the important peculiarity that the dynamics of the nodes, nonlinear and possibly chaotic, are governed by sets of fractional-order differential equations. Then, we study how the response network attains synchronization with the drive network. Our analysis is fairly general. We impose few conditions on the network structure, do not assume that the nodes in a single network are synchronized and provide an analytical characterization of the problem in which the master and response networks are non identical (due to the perturbation of their fixed parameters). Our analysis is based on simple definitions of the synchronization error and the stability of fractional-order systems of differential equations, but it is also valid for "ordinary" networks whose dynamics is described by integer-order differential equations. Although the main aim of the work is to provide analytical insights, some numerical illustrations (obtained by computer simulations) are also presented