Delay-dependent flocking dynamics of a two-group coupling system

A group coupling model for a system with large-scale nodes is investigated. The model is formulated as a system of functional differential equations. It incorporates two additional factors that exist in the evolution of flocking behavior, but are often ignored in modeling: (i) the diversity of interactions, including inter-group and intra-group interactions and (ii) the delayed response of particles to signals from the environment or neighbors, including transmission and processing delays. Theoretically, using the divide-and-conquer method and under different delay factors, sufficient conditions for self-organizing flocking are derived by constructing a dissipative differen-tial inequalities with continuous parameters respectively, which involve some analytical expressions of the upper bound of the delay that the system can tolerate. Results of systematic numerical simulations are presented. They not only validate the analytical results, but hint at a somehow surprising behavior of system, that is, weak flocking behavior occurs when two types of delays coexist.