A qualitative analysis of ẋ=Ax+b

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Abstract

For a dynamical system ẋ=Ax+b, where A is constant real n×n matrix and b is an n-vector there is special interest in the existence of a constant asymptotically stable attractor trajectory in positive orthant of Rn. For certain patterns of signs of entries (+, −, or 0) in A and b, such a trajectory always must exist, regardless of the magnitudes of those entries. This paper describes all such patterns in terms of digraphs. Applications to a nonlinear systems and ecosystem models are also mentioned.

Keywords

Lotka-Volterra
qualitative stability
sign solvability
sign stability
signed digraphs

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Research supported in part by a grant from the National Science Foundation.