Lyapunov analysis of collective behaviors in self-propelled particle systems

We study the detailed mechanism of collective behaviors in self-propelled particle (SPP) systems by using Lyapunov exponents spectrum. We treat a simple 2-dimensional continuous time dynamical model, which includes the self-controlling effect as well as the interaction effect in the swarm. It is shown that the swarm reveals three types of cohesive clustering modes corresponding to marching, rotating, and swarming. Those modes appear alternatively and intermittently in the course of time in general. We elucidate the stability/instability of these cohesive states from the Lyapunov exponents related to the stretching or contracting manifolds. Furthermore, it is shown that the clustering ratio approaches to unity when the number of particles goes to large enough, and that the size dependence of the clustering ratio is clearly shown by the Lyapunov dimension. These numerical results seem to suggest that the tangent space plays an essential role in self-organization of collective swarm dynamics.