An algebraic approach for the weak coupling of multiple Lohe tensor models

TL;DR

This paper introduces an algebraic framework using characteristic symbols and a fusion operation to systematically analyze weak coupling in multiple Lohe tensor models, enabling new insights into their emergent dynamics.

Contribution

It develops a novel algebraic approach with a fusion operation that models weak coupling of Lohe tensor models, forming a monoid structure for systematic analysis.

Findings

Fusion operation is associative and has an identity element.

Weak coupling of models leads to emergent collective behavior.

Framework applied to swarm sphere and Lohe matrix models.

Abstract

We present a systematic algebraic approach for the weak coupling of Cauchy problems to multiple Lohe tensor models. For this, we identify an admissible Cauchy problem to the Lohe tensor (LT) model with a characteristic symbol consisting of four tuples in terms of a size vector, a natural frequency tensor, a coupling strength tensor and admissible initial configuration. In this way, the collection of all admissible Cauchy problems to the LT models is equivalent to the space of characteristic symbols. On the other hand, we introduce a binary operation, namely "{\it fusion operation}" as a binary operation between the characteristic symbols. It turns out that the fusion operation satisfies the associativity and admits the identity element in the space of characteristic symbols which naturally forms a monoid. By virtue of the fusion operation, the weakly coupled system of multiple LT models can be obtained by applying the fusion operation of multiple characteristic symbols corresponding to the LT models. As a concrete example, we consider a weak coupling of the swarm sphere model and the Lohe matrix model, and provide a sufficient framework leading to emergent dynamics to the proposed weakly coupled model.