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A complex dynamical network (CDN) can be considered as the composition system with the nodes subsystem (NS) and the links subsystem (LS), and both subsystems are coupled with each other. In this paper, two vector differential equations (VDE) are used to describe the dynamical behaviours of NS and LS, respectively, in which the dynamical behaviour of NS is considered as the VDE with the second derivative term (SDT). This paper mainly focuses on the dynamics of LS, which is represented as VDE with the intuitive topologic feature of outgoing links, and investigates the design of the tracking controller for NS and the auxiliary tracking objectives (ATO) for LS. Firstly, the dynamical models of NS and LS in CDN are proposed, and the corresponding assumptions are given. Secondly, based on Lyapunov stability theory, the controller of NS and the ATO of LS are designed so that the state of NS can asymptotically track the given reference signal. Finally, the effectiveness of the proposed control strategy in this paper is verified by the numerical simulation example with N two-links robots. Abbreviations: ATO: auxiliary tracking objectives; CDN: complex dynamical network; LS: links subsystem; MDE: matrix differential equation; NS: nodes subsystem; SDT:second derivative term; VDE: vector differential equation; [ABSTRACT FROM AUTHOR]

For the complex dynamical network (CDN) with uncertainties, this paper defines the links synchronization and synthesizes the adaptive control scheme to realize it. Generally speaking, a CDN can be considered as the composition system with the two coupled dynamic subsystems: the nodes subsystem (NS) and the links subsystem (LS). We observed that there are many results in the existing literature on the NS synchronization, which implies that the state at each node tends to be the same when the NS synchronization happens. However, the LS synchronization is rarely discussed in the existing literature due to its unknown meaning and engineering practice background. Inspired by the ego-networks, this paper employs the time-varying outgoing links at each node to geometrically describe the changing of network topologic structure and regards the values-weighted of outgoing links as the state variables of LS, by which the LS synchronization is introduced, the corresponding control scheme for its implementation is synthesized under the condition that the state of NS is available and the state of LS is unavailable. The control scheme includes the adaptive controller for NS and the coupling strategy for LS. Finally, the effectiveness of proposed control scheme in this paper is verified by a numerical example. [ABSTRACT FROM AUTHOR]