Firstly, based on the Lyapunov stability theory, adaptive feedback controllers are designed for achieving cluster synchronization.
Secondly, periodically intermittent controllers are designed for achieving cluster synchronization and the synchronization conditions are derived by using mathematical induction method and the analysis technique.
By means of the proposed scheme, the sufficient conditions for achieving cluster synchronization are derived analytically by utilizing Lyapunov stability theory.
According to Lyapunov stability theory, several sufficient conditions for the network to achieve cluster mixed synchronization are derived.
However, in real community networks, different communities in networks usually synchronize with each other in a different manner, i.e., achieving cluster projective synchronization.
Based on Lyapunov stability theory, sufficient conditions for achieving cluster projective synchronization are derived through designing proper controllers.
Two different adaptive laws of time-varying coupling strength and a linear feedback control are designed to achieve the cluster synchronization of this network.
Based on the Lyapunov theorem and Lipschitz condition, the network controllers and the identification laws of uncertain parameters are designed, and they are efficiently used to achieve the cluster synchronization and the identification of uncertain parameters.
For the case with fixed topology, it is shown that a feasible feedback controller can be designed to achieve the given cluster synchronization pattern if the induced network topology of each cluster has a directed spanning tree and further compared to the couplings among different clusters, the couplings within the each cluster are sufficiently strong.
It is shown that an appropriate feedback controller can be designed to achieve the given cluster synchronization pattern if the induced network topology of each cluster has a directed spanning tree and further compared to the couplings among different clusters, the couplings within the each cluster are sufficiently strong.
For all the cases, the lower bounds for such strengths of couplings within each cluster that secure the synchronization as well as cluster synchronization rate are explicitly specified.
