Network structure changes local stability of universal equilibria for swarm sphere model

We investigate the asymptotic dynamics of the swarm sphere model with second-order coupling when the network structure is introduced. Specifically, we explore various interactions between agents, allowing them to be influenced by others, and discover the essential roles of the network by studying the stability of specific equilibria. In particular, we are interested in universal equilibria that do not depend on the network. To this end, we verify that the prescribed network structure changes the local stability of universal equilibria. Our results mainly rely on linear stability analysis supplemented by Lyapunov functional analysis to determine the size of basin of attraction.