  Natural systems are inextricably affected by noise. Within recent decades,
the manner in which noise affects the collective behavior of self-organized
systems, specifically, has garnered considerable interest from researchers and
developers in various fields. To describe the collective motion of multiple
interacting particles, Vicsek et al. proposed a well-known self-propelled
particle (SPP) system, which exhibits a second-order phase transition from
disordered to ordered motion in simulation; due to its non-equilibrium,
randomness, and strong coupling nonlinear dynamics, however, there has been no
rigorous analysis of such a system to date. To decouple systems consisting of
deterministic laws and randomness, we propose a general method which transfers
the analysis of these systems to the design of cooperative control algorithms.
In this study, we rigorously analyzed the original Vicsek model under both open
and periodic boundary conditions for the first time, and developed extensions
to heterogeneous SPP systems (including leaderfollower models) using the
proposed method. Theoretical results show that SPP systems switch an infinite
number of times between ordered and disordered states for any noise intensity
and population density, which implies that the phase transition indeed takes a
nontraditional form. We also investigated the robust consensus and connectivity
of these systems. Moreover, the findings presented in this paper suggest that
our method can be used to predict possible configurations during the evolution
of complex systems, including turn, vortex, bifurcation and flock merger
phenomena as they appear in SPP systems.
