Stability analysis of a double integrator swarm model related to position and velocity

In this paper, we propose a biologically inspired, continuous double integrator model based on individual Newton's law in an n-dimensional space. The double integrator model is related to positions and velocities of individuals. We discuss three cases of the attraction/repulsion functions, which are odd for the attractive force and repulsion force taking effect in opposite directions, and we obtain the swarm size and ultimate motions for cohesion in every case. Stability analysis and numerical simulations are also presented to demonstrate the effectiveness of our model.