Cucker–Smale type flocking models on a sphere

We present a Cucker–Smale type flocking model on a sphere including three terms: a centripetal force, multi-agent interactions on a sphere, and inter-particle bonding forces. We consider a rotation operator to compare velocity vectors on different tangent spaces. Due to the geometric restriction, the rotation operator is singular at antipodal points and the relative velocity between two agents located at these points is not well-defined. We assume that the communication rate between two antipodal points is zero to establish a well-defined flocking operator. We obtain the global-in-time existence and uniqueness of the solution to the flocking model. From the geometric property of the sphere, it is difficult to control the position difference between agents to avoid this singular position without bonding force. With a positive bonding force, we present a sufficient condition for the emergence of flocking.