Representation Formulae for Discrete 2D Autonomous Systems

In this paper, we provide solution formulae for higher order discrete two-dimensional (2D) autonomous systems. We first consider a special type of 2D autonomous systems. These systems are described by equations that satisfy a certain special property: the module of equations contains elements of the form (σn 2 + an−1(σ1)σ n−1 2 + · · · + a0(σ1))wj = 0, for each dependent variable wj , where ai(σ1) ∈ R[σ±1 1 ] and a0(σ1) is a unit in R[σ±1 1 ]. We show that this property is equivalent to the corresponding quotient module being finitely generated as a module over the 1-variable Laurent polynomial ring R[σ±1 1 ]. We then show that solutions to these special systems can be viewed as evolutions along the second coordinate direction of certain suitably chosen one-dimensional (1D) trajectories over the first coordinate direction. Consequently, we show that these solutions can be written in terms of various integer powers of a square 1-variable Laurent polynomial matrix A(σ1) acting on suitable 1D trajectories. Following the 1D terminology we call these 1D trajectories initial conditions. We call this form of expressing the solutions a representation formula. Then, in order to extend this result to general 2D autonomous systems, we obtain an analogue of a classical algebraic result, called Noether’s normalization lemma, for the Laurent polynomial ring in two variables. Using this result we show that every 2D autonomous system admits a representation formula through a suitable coordinate transformation in the domain Z2. Further, we analyze the set of initial conditions that appear in our representation formulae and resolve the issue of how freely these initial conditions can be chosen.