Symbiosis points for linear differential systems

Abstract

Let A be an n × n essentially nonnegative matrix and consider the linear differential system . An initial point y lies in the reachability cone for the nonnegative orthant if the trajectory x(t) emanating from v reaches the nonnegative orthant at some time y ₀. Due to the essential nonnegativity of A, once t(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 such 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 ¹>0|I+h ¹ 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.

^*Research supported in part by US Air Force Research Grant No. AOSR-88-0047 and by NSF Grant No. DMS-8901860.

^†Present address: Department of Mathematics, University of Victoria, Victoria, BC, Canada V8W 2Y2.Present address: Department of Mathematics, University of Victoria, Victoria, BC, Canada V8w 2Y2.

^*Research supported in part by US Air Force Research Grant No. AOSR-88-0047 and by NSF Grant No. DMS-8901860.

^†Present address: Department of Mathematics, University of Victoria, Victoria, BC, Canada V8W 2Y2.Present address: Department of Mathematics, University of Victoria, Victoria, BC, Canada V8w 2Y2.

Notes

^*Research supported in part by US Air Force Research Grant No. AOSR-88-0047 and by NSF Grant No. DMS-8901860.

^†Present address: Department of Mathematics, University of Victoria, Victoria, BC, Canada V8W 2Y2.Present address: Department of Mathematics, University of Victoria, Victoria, BC, Canada V8w 2Y2.

Additional information

Notes on contributors

Michael J. Tsatsomeros

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