Polyhedral Dynamics and the Geometry of Systems
Ronald H. Atkin, John L. Casti · IIASA PURE (International Institute of Applied Systems Analysis) · 1977
Using elementary concepts from algebraic topology, a program is outlined for the application of topological notions to the problem of characterizing the connective structure of large-scale systems. The basic approach is to generate appropriate mappings which associate a given system with a simplicial complex. Topological tools are then used in a non-standard manner to investigate the connectivity pattern of the system. Auxiliary notions such as eccentricity of a simplex, patterns on a complex, and the homological structure of a complex are also shown to have system- theoretic relevance. The general ideas of the paper are illustrated on a number of simple examples, including the standard linear system set-up. The paper concludes with a discussion of several research topics to be carried out in an attempt to connect the algebraic-topological ideas with other branches of mathematical system theory.