  This papers studies the expressive and computational power of discrete
Ordinary Differential Equations (ODEs). It presents a new framework using
discrete ODEs as a central tool for computation and provides several implicit
characterizations of complexity and computability classes.
  The proposed framework presents an original point of view on complexity and
computability classes. It also unifies in an elegant settings various
constructions that have been proposed for characterizing these classes. This
includes Cobham's and, Bellantoni and Cook's definition of polynomial time and
later extensions on the approach, as well as recent characterizations of
computability and complexity by classes of ordinary differential equations. It
also helps understanding the relationships between analog computations and
classical discrete models of computation theory.
  At a more technical point of view, this paper points out the fundamental role
of linear (discrete) ordinary differential equations and classical ODE tools
such as changes of variables to capture computability and complexity measures,
or as a tool for programming various algorithms.
