Dynamics of locally coupled agents with next nearest neighbor interaction

TL;DR

This paper analyzes the stability and dynamic response of large systems of identical agents with next nearest neighbor interactions, revealing conditions for stability and discovering reflectionless wave solutions.

Contribution

It provides new stability conditions for agents with asymmetric next nearest neighbor coupling and identifies reflectionless waves as a novel dynamic feature.

Findings

Stability conditions depend on system parameters.

Reflectionless waves occur with next nearest neighbor interactions.

Numerical simulations confirm analytical predictions.

Abstract

We consider large but finite systems of identical agents on the line with up to next nearest neighbor asymmetric coupling. Each agent is modelled by a linear second order differential equation, linearly coupled to up to four of its neighbors. The only restriction we impose is that the equations are decentralized. In this generality we give the conditions for stability of these systems. For stable systems, we find the response to a change of course by the leader. This response is at least linear in the size of the flock. Depending on the system parameters, two types of solutions have been found: damped oscillations and reflectionless waves. The latter is a novel result and a feature of systems with at least next nearest neighbor interactions. Analytical predictions are tested in numerical simulations.