Coordinated collective motion in a motile particle group with a leader

We study the collective dynamics of a group of motile particles with a leader. The leader is unaffected by the particle members whereas each member is influenced by the leader and the other members. By using coupled oscillators theory and Lyapunov function method, we show how the group dynamics depends on the motion of the leader and the coupling weights among all particles. Generally speaking, two types of collective motions will occur, depending on different ranges of the coupling weights among the member particles. One is that all the member particles will move in the same direction and the other is that all the member particles move in such a way that the weighted centroid of the group approaches a fixed position. In each case, all the member particles eventually move in the same manner except for the directions of the motion in certain cases. Numerical simulations are worked out to demonstrate the theoretical results. The study suggests potential approaches to control a group motion by steering the motion of the leader and adjusting coupling patterns. This is of practical interest in applications of multiagent systems.