Novel representation formulae for discrete 2D autonomous systems

In this paper, we provide explicit solution formulae for higher order discrete 2D autonomous systems. We first consider a special type of 2D autonomous systems, namely, systems whose quotient modules are finitely generated as modules over the one variable Laurent polynomial ring ℝ[σ1±1].We then 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. 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.