Robust outer synchronization between two complex networks with fractional order dynamics

Abstract

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.

Finite-time outer synchronization between two complex dynamical network with time delay and noise perturbation

“…The mixed outer synchronization of coupled complex networks with time-varying coupling delay was investigated in Ref. [2]. Most of above works primarily focus on the asymptotical or exponential synchronization of networks.…”

“…On the hand, there exists another type of network synchronization, i. e.. "outer synchronization" between two or more complex networks. Recently, there are some researches who have studied the outer synchronization between two complex DOI: 10.14736/kyb-2016-4-0607 networks [2,18,22,23,27,28]. In Ref.…”

Robust projective outer synchronization of coupled uncertain fractional-order complex networks

“…Fortunately, attempts have begun to treat the synchronization problem of FCNs [27][28][29][30][31][32][33][34]. Examples include the results presented in [27][28][29] which are limited to specific network topologies, and the work in [30] on the distributed coordination of networked fractional-order systems over a directed interaction graph.…”

“…By designing nonlinear controllers, Wu and Lu [32] first reported outer synchronization phenomena between two bidirectionally coupled FCNs. Also, Asheghan et al [33] considered outer synchronization between two FCNs coupled unidirectionally by an open-plus-closed-loop scheme. More recently, cluster synchronization in FCNs with community structure was discussed in [34] by linear pinning control.…”

“…More recently, cluster synchronization in FCNs with community structure was discussed in [34] by linear pinning control. In [33,34], due to the nonlinearity of node dynamics, only local stability conditions were given for verifying network synchronization. On the other hand, it is very common to encounter parametric uncertainties in real-world networks due to modeling inaccuracies or environmental perturbations, which may destroy the stability of the networks.…”