Emergent dynamics of various Cucker-Smale type models with a fractional derivative

In this paper, we demonstrate emergent dynamics of various Cucker-Smale type models, especially standard Cucker-Smale (CS), thermodynamic Cucker-Smale (TCS), and relativistic Cucker-Smale (RCS) with a fractional derivative in time variable. For this, we adopt the Caputo fractional derivative as a widely used standard fractional derivative. We first introduce basic concepts and previous properties based on fractional calculus to explain its unusual aspects compared to standard calculus. Thereafter, for each proposed fractional model, we provide several sufficient frameworks for the asymptotic flocking of the proposed systems. Unlike the flocking dynamics which occurs exponentially fast in the original models, we focus on the flocking dynamics that occur slowly at an algebraic rate in the fractional systems.