Flocking of Cucker-Smale model with intrinsic dynamics

In this paper, we consider the flocking problem of modified continuous-time and discrete-time Cucker-Smale models where every agent has its own intrinsic dynamics with Lipschitz property. The dynamics of the models are governed by the interplay between agents’ own intrinsic dynamics and Cucker-Smale coupling dynamics. Based on the explicit construction of Lyapunov functionals, we show that conditional flocking would occur. And then we study the relationship between the Lipschitz constant L of the intrinsic dynamics and exponent β measuring the strength of the interaction between agents when flocking occurs. We also give two examples to show flocking might not occur for enough large L or unconditionally for β > 0. At last, we provide several numerical simulations to illustrate our theoretical results.