Cluster Projective Synchronization of Fractional-Order Complex Network via Pinning Control

Abstract

Synchronization is the strongest form of collective phenomena in complex systems of interacting components. In this paper, the problem of cluster projective synchronization of complex networks with fractional-order nodes based on the fractional-order differential equation stability theory is investigated. Only the nodes in one community which have direct connections to the nodes in other communities are controlled. Some sufficient synchronization conditions are derived via pinning control. Numerical simulations are provided to show the effectiveness of the theoretical results.

Cluster Synchronization of Fractional-Order Nonlinearly-Coupling Community Networks With Time-Varying Disturbances and Multiple Delays

“…Remark 3. Compared with the integer-order adaptive controller in [35]- [36], our controller adds a positive control parameter d * i , which ensures the condition (12) in Theorem existing works [24]- [27]. The first part [−(d i (t)+d * i )Γe i (t)] of our controller is to synchronize all the nodes of each community to the same state.…”

“…Definition 5. [27] For each continuous function f l (.) : R n → R n (l = 1, 2, ..., m) in fractional-order system (1), there exist positive definite matrices P = diag{p 1 , p 2 , ..., p n }, ∆ l = diag{δ l1 , δ l2 , ..., δ ln } and constants…”

“…Liu et al [26] considered cluster synchronization in the fractionalorder dynamical network by pinning control inter-act nodes and discussed the influence of control gain and coupling strength on the clustering synchronization process. Yang et al [27] focused on cluster project synchronization for the fractional-order dynamical network with identical nodes and non-delayed linear coupling. In [28], based on the fractional stability theory, the authors investigated local and global cluster synchronization in fractional-order dynamical systems with asymmetric coupling matrix via using fractionalorder adaptive pinning control.…”