Algorithms for verification of characteristic sets of discrete autonomous nD systems with n ≥ 2

Abstract

An important question for multidimensional systems is to characterize the minimum information required to uniquely specify a ‘trajectory’. For a discrete nD system described by partial difference equations with real constant coefficients, the notion of a characteristic set formalizes this idea. The issue of characteristic sets for n ≥ 3 has largely remained open since Valcher’s seminal work for n = 2 (Valcher (2000)). In this paper, we provide a necessary and sufficient condition for a given subset (either a cone or a sub-lattice) in Zn to be a characteristic set for a discrete scalar autonomous nD system. This necessary and sufficient condition enables us to formulate an algebraic test for verifying whether a given cone or a sub-lattice is a characteristic set for a given discrete scalar autonomous nD system. We then provide algorithms - that are implementable using standard computational algebra packages - for doing this check. This is achieved by first converting the above-mentioned necessary and sufficient condition to another equivalent algebraic condition that is more suited for applying Grobner bases theory. We further pursue the question of ‘minimality’ of characteristic sets. We show how the idea of minimal characteristic sub-lattices is related with the notion of autonomy degree and Krull dimension. Thus, we provide a complete solution to the problem of determining if a given cone or a sub-lattice is a characteristic set for a scalar autonomous discrete nD system.