Discretization of Variable-Coefficient Dynamical Systems via Rota Algebras

Abstract: We introduce a novel integrability-preserving discretization for a broad class of differential equations with variable coefficients, encompassing both linear and nonlinear cases. The construction is achieved via a categorical approach that enables a unified treatment of continuous and discrete dynamical systems. Within our theoretical framework, based on the finite operator calculus proposed by G. C. Rota, all difference operators act as derivations in an algebra of power series with respect to a suitable functional product. The core of our theory relies on the existence of covariant functors between the newly proposed Rota category of Galois differential algebras and suitable categories of abstract dynamical systems. In this setting, under certain regularity assumptions, a differential equation and its discrete analogues are naturally interpreted as objects of the same category. This perspective enables the construction of a vast class of integrable maps that share with their continuous analogues a wide set of exact solutions and, in the linear case, the Picard-Vessiot group.