Algorithms for specifying initial data for discrete autonomous nD systems

Abstract

For nD systems, initial data is obtained by specifying trajectories on special subsets of the domain, known as characteristic sets. In this paper, we consider a special class of systems that admits a union of a coordinate sublattice and finitely many parallel translates of it as a characteristic set; we call such systems as strongly relevant systems. Using the discrete Noether's normalization lemma, every discrete autonomous nD system can be transformed to a strongly relevant system. For a strongly relevant system, the set of allowable initial conditions, obtained by restricting trajectories on the characteristic set, is characterized. We then provide an implementable algorithm, based on Gröbner basis, for obtaining a representation of the set of allowable initial conditions. Once such a representation is obtained, important deductions, such as, arbitrary assignability of initial data can be easily made.