Narrow Results

We present the mathematical analysis of bi-cluster flocking phenomenon for the short-ranged Cucker–Smale model with some well-prepared initial data. For this, we derive a system of differential inequalities for the functionals measuring the local spatial and velocity fluctuations and differences of local velocity averages, and then estimate the upper bound of spatial fluctuations and the lower bound of the difference between local velocity averages. We explicitly present an admissible class of initial configurations leading to the asymptotic emergence of bi-cluster flocking phenomenon. Unlike global flocking (a mono-cluster flocking configuration in velocity), where the convergence rate is always exponential, the asymptotic convergence to bi-cluster flocking configurations is affected by the far-field decay rate of communication weights so that it can be algebraic.

We study the emergence of bi-cluster flocking for a generalized Justh–Krishnaprasad (J–K) model and the Cucker–Smale (C–S) model with the unit speed constraint. We provide a sufficient framework for the emergence of bi-cluster flocking in terms of initial configurations and communication weights between agents. Our analytical results on bi-cluster flocking illustrate that the spatial-decay rate of communication weights is transferred to the temporal decay rate toward the bi-cluster flocking configuration. We also provide several numerical simulations and compare them to our analytical results.