Emergent dynamics of the Cucker-Smale flocking model and its variants

TL;DR

This paper reviews the mathematical structures and flocking behaviors of Cucker-Smale models, highlighting conditions for emergent flocking based on interaction parameters and initial data, and discusses recent theoretical developments.

Contribution

It provides a comprehensive overview of the mathematical analysis and flocking theorems for Cucker-Smale models and their variants, emphasizing recent advances.

Findings

Conditions for emergent flocking depend on coupling strength and initial data

Cucker-Smale model exhibits flocking behavior under certain parameter regimes

Recent theoretical results extend understanding of flocking in complex interaction topologies

Abstract

In this chapter, we present the Cucker-Smale type flocking models, and discuss their mathematical structures and flocking theorems in terms of coupling strength, interaction topologies and initial data. In 2007, two mathematicians Felipe Cucker and Steve Smale introduced a second-order particle model which resembles Newton's equations in $N$-body system, and present how their simple model can exhibit emergent flocking behavior under sufficient conditions expressed only in terms of parameters and initial data. After Cucker-Smale's seminal works, their model has received lots of attention from applied math and control engineering communities. We discuss the state-of-art for the flocking theorems to Cucker-Smale type flocking models.