Hypergraphs, the Qualitative Solvability of κ · λ = 0, and Volterra Multipliers for Nonlinear Dynamical Systems
✍ Scribed by C. Jeffries; C. Lee; P. Vandendriessche
- Publisher
- Elsevier Science
- Year
- 1993
- Tongue
- English
- Weight
- 441 KB
- Volume
- 105
- Category
- Article
- ISSN
- 0022-0396
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✦ Synopsis
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 (k) of the same sign pattern, is there a real vector (\lambda), each component of which is positive, such that (k \cdot i=0) ? This paper presents graph theoretic sufficient conditions on a hypergraph generated from the sign pattern of (\kappa) which guarantee the existence of (\lambda). For (k) with more highly connected hypergraphs, this paper also presents sufficient quatitative conditions on the sign pattern of (\kappa) and certain quantitative conditions on sums of hypergraph path products which together guarantee the existence of (\lambda . \quad 1993) Academic Press. Inc.