Dimensional-invariance principles in coupled dynamical systems

In this paper we study coupled dynamical systems and establish several invariance principles relating to the dimensions of the subspace spanned by solutions of each individual system. We consider two types of coupled systems, one with scalar couplings and the other with matrix couplings. Via the rank-preserving flow theory, we prove that scalar-coupled dynamical systems possess 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 are given to characterize dimensional-invariance principles. The established invariance principles provide additional characterizations and insights to analyze the transient behaviors and solution evolution for a large family of coupled systems, such as multi-agent consensus dynamics, distributed coordination systems, formation control systems, among others.