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 $2$D autonomous systems. These systems are described by equations that satisfy a certain special property: the module of equations contains elements of the form (σ₂ⁿ+aₙ₋₁(σ₁)σ₂ⁿ⁻¹+⋯+a₀(σ₁))wⱼ=0, for each dependent variable wⱼ, where aᵢ(σ₁)∈R[σ₁± 1] and a₀(σ₁) is a unit in R[σ₁± 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[σ₁± 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 $1$D 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 $2$D 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.