  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.
