Emergence of anomalous flocking in the fractional Cucker-Smale model

Abstract

In this paper, we study the emergent behaviors of the Cucker-Smale (C-S) ensemble under the interplay of memory effect and flocking dynamics. As a mathematical model incorporating aforementioned interplay, we introduce the fractional C-S model which can be obtained by replacing the usual time derivative by the Caputo fractional time derivative. For the proposed fractional C-S model, we provide a sufficient framework which admits the emergence of anomalous flocking with the algebraic decay and an \begin{document}$\ell^2$\end{document} -stability estimate with respect to initial data. We also provide several numerical examples and compare them with our theoretical results.