Hypergraphs, the Qualitative Solvability of κ · λ = 0, and Volterra Multipliers for Nonlinear Dynamical Systems

Abstract

Certain sign equivalence classes of n-dimensional nonlinear dynamical systems correspond to n-vertex hypergraphs, The global stability of some such dynamical systems can be guaranteed if the associated hypergraphs have a simplicity of structure and meet certain quantitative path product conditions. A purely algebraic version of the same problem can be described as follows. Suppose we are given a rectangular matrix pattern of signs; each entry in the matrix is +, −, or 0. For every real matrix κ of the same sign pattern, is there a real vector λ, each component of which is positive, such that κ · λ = 0? This paper presents graph theoretic sufficient conditions on a hypergraph generated from the sign pattern of κ which guarantee the existence of λ. For κ with more highly connected hypergraphs, this paper also presents sufficient qualitative conditions on the sign pattern of κ and certain quantitative conditions on sums of hypergraph path products which together guarantee the existence of λ.