# Dimensional-invariance principles in coupled dynamical systems-- A unified analysis and applications

## TL;DR

This paper establishes unified dimensional-invariance principles for coupled dynamical systems with scalar and matrix couplings, simplifying analysis and extending applicability to time-varying and state-dependent cases.

## Contribution

It provides a unified framework and necessary conditions for dimensional invariance in coupled systems, extending previous results and simplifying proofs using rank-preserving flow theory.

## Key findings

- Dimensional invariance holds for scalar-coupled systems with various couplings.
- Necessary and sufficient conditions for matrix-coupled systems are derived.
- The analysis simplifies previous proofs and extends to time-varying and state-dependent couplings.

## Abstract

In this paper we study coupled dynamical systems and investigate dimension properties of the subspace spanned by solutions of each individual system. Relevant problems on \textit{collinear dynamical systems} and their variations are discussed recently by Montenbruck et. al. in \cite{collinear2017SCL}, while in this paper we aim to provide a unified analysis to derive the dimensional-invariance principles for networked coupled systems, and to generalize the invariance principles for networked systems with more general forms of coupling terms. To be specific, we consider two types of coupled systems, one with scalar couplings and the other with matrix couplings. Via the \textit{rank-preserving flow theory}, we show that any scalar-coupled dynamical system (with constant, time-varying or state-dependent couplings) possesses the dimensional-invariance principles, in that the dimension of the subspace spanned by the individual systems' solutions remains invariant. For coupled dynamical systems with matrix coefficients/couplings, necessary and sufficient conditions (for constant, time-varying and state-dependent couplings) are given to characterize dimensional-invariance principles. The proofs via a rank-preserving matrix flow theory in this paper simplify the analysis in \cite{collinear2017SCL}, and we also extend the invariance principles to the cases of time-varying couplings and state-dependent couplings. Furthermore, subspace-preserving property and signature-preserving flows are also developed for coupled networked systems with particular coupling terms. These invariance principles provide insightful characterizations to analyze transient behaviors and solution evolutions for a large family of coupled systems, such as multi-agent consensus dynamics, distributed coordination systems, formation control systems, among others.