Automatic Differentiation Tools in Computational Dynamical Systems

In this paper we describe a unified framework for the computation of power series expansions of invariant manifolds and normal forms of vector fields, and estimate the computational cost when applied to simple models. By simple we mean that the model can be written using a finite sequence of compositions of arithmetic operations and elementary functions. In this case, the tools of Automatic Differentiation are the key to produce efficient algorithms. By efficient we mean that the cost of computing the coefficients up to order k of the expansion of a d-dimensional invariant manifold attached to a fix point of a n-dimensional vector field (d = n for normal forms) is proportional to the cost of computing the truncated product of two d-variate power series up to order k. We present actual implementations of some of the algorithms, with special emphasis to the computation of the 4D center manifold of a Lagrangian point of the Restricted Three Body Problem. Mathematics Subject Classification: 34C20,34C30,34C30,34C30,65Pxx,68W30