Zero patterns and dominant modes of the state evolutions of autonomous continuous-time positive systems

In this paper, the zero pattern properties and the asymptotic evolution of the trajectories of an autonomous continuous-time positive system are investigated. To this end, a normal form for the exponential of a Metzler matrix is provided, and the concept of "echelon basis" is introduced. By making use of these two ingredients, the dominant mode of each single block appearing in the normal form of the exponential matrix is determined. As a result, the zero pattern as well as the dominant mode of every state evolution, depending on the zero pattern of the initial state, can be easily inferred.