The Reachability Cones of Essentially Nonnegative Matrices

Let A be an n× n essentially nonnegative matrix and consider the linear differential system ẋ(t) = Ax(t), t ≥ 0. We show that there exists a constant h(A) > 0 such that the trajectory emanating from xo reaches R + at a finite time to = t(xo) ≥ 0 if and only if the sequence of points generated by a finite differences approximation from xo, with time–step 0 < h < h(A), reaches R + at a finite index ko = k(xo) ≥ 0. This generalizes and strengthens earlier results of two of the authors, where some additional spectral restrictions were imposed on A. Our proof makes use of the existence of a basis of nonnegative vectors to the Perron eigenspace.