Small noise may diversify collective motion

Natural systems are undeniably affected by noises. How noise affects the collective behavior of self-organized systems has attracted wide interest from various fields in the past several decades. To describe the collective motion of multiple interacting particles, Vicsek et al. proposed a well-known self-propelled particle(SPP) system and conjectured it exhibited a second order phase transition from disordered to ordered motion by simulations. However, due to its nonequilibrium, randomness, and strong coupling nonlinear dynamics, the mathematical analysis of such system is still lack. To decouple the systems consisting of deterministic laws and randomness, this paper originally proposes a general method which transfers the analysis of these systems to the design of cooperative control algorithms. Using our method we rigorously analyze the origin Vicsek model under both open and periodic boundary conditions for the first time, and also make some extensions to the inhomogenous SPP systems including the leader-follower models. Theoretic results show that the SPP systems will switch infinite times between ordered and disordered states for arbitrary small noise and large population density, which implies the phase transition should have new form differing from traditional senses. Also, the robust consensus and connectivity of these systems will be investigated. Moreover, our research shows the noise can lead to the diversity of collective motion of flocks, such as the appearance of turn, vortex, bifurcation and merger.