ABSTRACT
A good theory is one where the intuitive concepts of the linear theory can be carried over to the nonlinear regime at least formally. Otherwise a totally new apparatus must be developed—an apparatus where the linear tools look ugly and complicated for the sake of compatibility with the nonlinear treatment. The map approach to rings succeeds in avoiding this problem. In this chapter, the authors describe intuitive tools which they use in the context of leading order perturbation theory: it is a map theory in disguise. The usage of maps will permit the authors to develop well-defined algorithms which are recursive and conceptually trivial to extend to high orders. In the case of k-jets produced by automatic differentiation libraries, they can truly perform calculations to arbitrary orders. In analytical calculation, the Dragt-Finn factorization could be in terms of a smallness parameter and the Lie transforms need not be polynomials.
