Emergent Dynamics of an Orientation Flocking Model With Distance-Dependent Delay

In this paper, we investigate the dynamical behaviors of an orientation flocking model that incorporates state-dependent delay. The research unfolds in three principal phases: First, we establish the global existence and uniqueness of solutions for our system. Subsequently, we develop a comprehensive theoretical framework that provides sufficient conditions for the emergence of orientation flocking phenomena. In particular, this framework remains valid regardless of the number of particles N$$ N $$. In the final phase of our analysis, we perform a systematic derivation of the mean-field limit for the discrete system, accompanied by a proof of global well-posedness for the resulting mean-field equations. Furthermore, we demonstrate that our theoretical results and analytical framework naturally extend to and encompass the case of orientation flocking models with constant time delays.