A nonlinear functional approach for emergent behaviour of discrete infinite Motsch–Tadmor model

In this paper, we investigate the emergent dynamics of the time-discrete infinite Motsch–Tadmor (IMT) model, focusing on flocking behaviours, uniform-in-time stability, and uniform-in-time continuous transition from discrete to continuous dynamics. First, we develop a framework to analyse flocking dynamics in the time-discrete IMT model under the sender network. To overcome the loss of momentum conservation, we employ the projection method and a convex envelope argument to construct a system of dissipative differential inequalities (SDDI), which enables us to obtain an exponential decay in the velocity diameter. Second, we show uniform-in-time stability and continuous transitions within the sender network framework. Our analysis of uniform-in-time stability and continuous transitions both focus on shape discrepancies without requiring the same asymptotic velocity limit for different initial data. By leveraging flocking estimates and a convex envelope argument, we establish the corresponding SDDI to demonstrate uniform-in-time stability. Finally, we derive uniform-in-time continuous transitions using flocking estimates and classical finite-in-time transition results.