Symbiosis points for linear differential systems

Michael Neumann, Michael J. Tsatsomeros
1991 Linear and multilinear algebra  
Let A be an n × n essentially nonnegative matrix and consider the linear differential systemẋ = Ax. An initial point v lies in the reachability cone X A (R n + ) for the nonnegative orthant if the trajectory x(t) emanating from v reaches the nonnegative orthant at some time t 0 . Due to the essential nonnegativity of A, once x(t) enters the nonnegative orthant it remains in it thereafter.In this paper we introduce the notion of a symbiosis point for the system. This is a point in X A (R n + )
more » ... ch that also the velocity vector at the point is in the reachability cone. This means that not only does the trajectory become and remain nonnegative, but there comes a time such that from then on the components of the trajectory become and remain nondecreasing. We characterize all symbiosis points for the system. We also show that if 0 < h < h(A) = sup{h > 0|I + h A ≥ 0} and det(I + hA) = 0, then any sequence of finite differences approximation which initiates at a symbiosis point becomes nondecreasing in the ordering of the nonnegative orthant and vice versa. In the case that A is weakly stable, we use a result of Hans Schneider to show that symbiosis points can be described from a matrix-combinatorial point of view. For weakly stable systems we also characterize trajectories whose higher order derivatives are required to lie in the reachability cone.
doi:10.1080/03081089108818087 fatcat:gpd5fvblf5achifpfpzn3afynq