From cluster splitting to cluster synchronization: adaptive control in delayed complex dynamical networks with dynamic links

To address the synchronization challenges caused by dynamic links, inter-cluster delays, and topological uncertainties in complex dynamical networks (CDNs), this paper proposes a control framework integrating clustering and adaptive coordination. Unlike traditional methods that require predefined cluster numbers and rely on constant coupling matrices, we employ the spectral clustering algorithm to split networks into user-defined clusters and introduce dynamic links to characterize node coupling interactions, overcoming the inflexibility of conventional models in scenarios such as unmanned aerial vehicle (UAV) formations and communication networks. A unified state equation is adopted for node dynamics, reducing the complexity of predefined independent equations for each cluster in traditional multi-cluster modeling. In the control strategy, dynamic links act both as node coupling media and as control targets. Specifically, intra-cluster dynamic links track predefined topologies via adaptive controllers, while inter-cluster uncertain links are stabilized within bounded ranges through adaptive update laws. The asymptotic stability of CDNs is rigorously proven by using a composite Lyapunov function that integrates tracking errors of node states and coupling links. A simulation of UAV formation demonstrates that the proposed framework achieves on-demand clustering, with UAV flight parameters (node states) and intra-cluster communication strengths (intra-cluster links) asymptotically converging to desired trajectories, while inter-cluster interactions (inter-cluster link) remain controllable. This framework provides a flexible, stable, and universal theoretical solution for cluster synchronization in CDNs.