A new model for self-organized dynamics and its flocking behavior

TL;DR

This paper introduces a novel self-organized dynamics model that addresses limitations of the Cucker-Smale model by using relative distances, leading to a new framework for analyzing flocking behavior in asymmetric systems.

Contribution

The paper proposes a new influence model based on relative distances, removing explicit dependence on agent count and introducing a framework for asymmetric flocking analysis.

Findings

Unconditional flocking proven for slowly decaying influence functions.

Model accommodates non-symmetric influence matrices.

Framework extends from particle to hydrodynamic descriptions.

Abstract

We introduce a model for self-organized dynamics which, we argue, addresses several drawbacks of the celebrated Cucker-Smale (C-S) model. The proposed model does not only take into account the distance between agents, but instead, the influence between agents is scaled in term of their relative distance. Consequently, our model does not involve any explicit dependence on the number of agents; only their geometry in phase space is taken into account. The use of relative distances destroys the symmetry property of the original C-S model, which was the key for the various recent studies of C-S flocking behavior. To this end, we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems, which covers systems with non-symmetric influence matrices. In particular, we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with "leaders". The methodology presented in this paper, based on the notion of active sets, carries over from the particle to kinetic and hydrodynamic descriptions. In particular, we discuss the hydrodynamic formulation of our proposed model, and prove its unconditional flocking for slowly decaying influence functions.