On the Notion of Neighborhood System

We are discussing the origin and meaning of the notion of neighborhood system. The transition from continuous to discrete erases the distinction between dynamic systems (i.e. systems of first order ordinary differential equations) and evolutionary partial differential equations: in both cases we get the same object, discrete dynamic system. The dynamic neighborhood systems can be considered as a subclass of dynamic discrete systems, corresponding substantially to evolutionary partial differential equations and characterized by the “sparsity” of the entering of variables in the equations. This “sparsity” is described by a digraph, which is called the neighborhood structure, while the neighborhood system itself can be considered as a system “over” this neighborhood structure.