Adaptive cluster synchronisation of coupled harmonic oscillators with multiple leaders

In this study, the authors investigate the cluster synchronisation of coupled harmonic oscillators with multiple leaders in an undirected fixed network. Unlike many existing algorithms for cluster synchronisation of complex dynamical networks or group consensus of multi-agent systems, which require global information of the underlying network such as eigenvalues of the coupling matrix or centralised control protocols, we propose a novel decentralised adaptive cluster synchronisation protocol for coupled harmonic oscillators. By using the decentralised adaptive cluster synchronisation protocol and without using any global information of the underlying network, all oscillators in the same group asymptotically synchronise with the corresponding leader even when only one oscillator in each group has access to the information of the corresponding leader. Numerical simulation results are presented to illustrate the theoretical results.