The Watanabe-Strogatz transform and constant of motion functionals for kinetic vector models

TL;DR

This paper extends the Watanabe-Strogatz transform to kinetic vector models, deriving constants of motion and analyzing stability, thereby advancing the understanding of collective dynamics in these systems.

Contribution

It introduces a kinetic Watanabe-Strogatz transform for vector models, identifies conditions for constants of motion, and applies these to analyze stability in kinetic swarm models.

Findings

Derived a kinetic WS-transform for vector models.

Identified conditions for existence of constants of motion.

Proved instability of bipolar states in kinetic swarm models.

Abstract

We present a kinetic version of the Watanabe-Strogatz(WS) transform for vector models in this paper. From the generalized WS-transform, we obtain the cross-ratio type constant of motion functionals for kinetic vector models under suitable conditions. We present the sufficient and necessary conditions for the existence of the suggested constant of motion functional. As an application of the constant of motion functional, we provide the instability of bipolar states of the kinetic swarm sphere model. We also provide the WS-transform and constant of motion functional for non-identical kinetic vector models.