Asymptotical stability for complex dynamical networks via link dynamics

Abstract

From the perspective of large‐scale system, a complex dynamical network can be regarded as the interconnected system with the node subsystem and link subsystem, which implies that the node subsystem and the link subsystem are the two main bodies of dynamical behaviors of network. Therefore, the whole stability of network is influenced by not only the dynamics of node subsystem but also the dynamics of link subsystem. According to the above view, the wholly asymptotical stability (WAS) is defined in this paper for the complex dynamical network with the model of differential equations. The WAS is used to describe the node subsystem achieves asymptotical stability when link subsystem achieves also the asymptotic stability in Lyapunov sense. For the WAS of the complex dynamical network, the corresponding criteria are derived by checking whether certain matrices are Hurwitz. The analysis results show that even if the isolated nodes are not asymptotically stable in Lyapunov sense, employing the dynamics of link can also force the node subsystem to achieve the asymptotical stability. Finally, the simulation examples show the validity of methods in this paper.

Tracking control for the nonlinear complex dynamical network assisted with outgoing links dynamics

“…It is remarkable that these studies as reviewed in References 9‐15 only focus on the collective behavior of nodes such as synchronization, nevertheless, the all the links are regarded only as playing the auxiliary and secondary role for all the nodes tending synchronization. In order words, the dynamics of links in the complex dynamical network should be transformed into an independent dynamical subsystem and their dynamics should be considered as reviewed in References 16‐22.…”

“…In other words, the topology and function of the complex dynamical network can be influenced by NS and LS, respectively. For instance, the synchronization [9][10][11][12][13][14][15][16] and stabilization 17 are regarded as the dynamical features of NS, and the structural balance [18][19][20][21][22] is regarded as the dynamical features of LS. If the weight-values of links are treated as the state variables for LS, the dynamical evolution of LS can also help to emerge certain dynamical features for NS.…”

Asymptotical outer synchronization control of the complex dynamical networks via unknown interaction and links dynamics

“…From the perspective of large-scale system, a CDN can be considered as the interconnected system with the nodes subsystem (NS) and the links subsystem (LS), which implies that NS and LS are the two main parts to emerge the dynamical behavior of CDN. 24,25 Therefore, the whole synchronization of network is influenced by not only the dynamics of NS but also the dynamics of LS. According to the above view, the achievements in previous studies [12][13][14][15][16][17][18] could be explained that the links assisted the nodes to emerge the synchronization of NS, but the whole dynamics of LS was ignored.…”

“…From the perspective of large‐scale system, a CDN can be considered as the interconnected system with the nodes subsystem (NS) and the links subsystem (LS), which implies that NS and LS are the two main parts to emerge the dynamical behavior of CDN 24,25 . Therefore, the whole synchronization of network is influenced by not only the dynamics of NS but also the dynamics of LS.…”

Double Tracking Control for the Directed Complex Dynamic Network via the Observer of Outgoing Links

“…By employing the above perspectives, the dynamic behaviors of the complex dynamic network can be reflected by the NS and the LS respectively. For example, the synchronization [11][12][13][14] and stability [15] are regarded as the dynamic behavior of NS, and the asymptotic structural balance [16][17][18] is regarded as the dynamic behavior of LS.…”