The effects of time-delay and phase lags on symmetric circular formations of mobile agents

The mobile agents' circular formation problem is driven by applications that require a desired spatial and temporal separation, such as data collection or satellite constellations. There are numerous strategies to assign the cluster number and spacing, the radius of the trajectory, communication network, and other parameters. In this numerical study, we are particularly interested in the effects of time-delay and phase lag on circular formations with symmetric clusters in a first-order model with all-to-all interaction. When the information exchange time-delay exceeds some critical value, it leads to either incoherent or alternative symmetric arrangements. Besides, we show that formations can be effectively controlled using phase lags. As a paradigmatic example, we consider identical phase lags for half of the agents, while the other half receives an opposite phase lag. As a result, the system reaches more complicated stable arrangements. For instance, the agents can split into asymmetric clusters, each of them creating a symmetric formation.