Emergent behaviors of relativistic flocks with adaptive coupling laws

In this paper, we investigate the emergent behaviors of the relativistic Cucker–Smale (RCS) model equipped with adaptive couplings. To do this, we first divide adaptive couplings into two types, Hebbian or anti-Hebbian. For the Hebbian case, we demonstrate the asymptotic flocking of the RCS model in two ways based on the Lyapunov functional approach and continuous argument. Meanwhile, for the anti-Hebbian case, depending on the regularity of the adaptive law at the origin, we prove the various emergent behaviors such as the slow velocity alignment or the group formation.