TITLE: 
Automatic Differentiation Tools in Computational Dynamical Systems

AUTHOR: 
Alex Haro
Departament de Matematica Aplicada i Analisi
Universitat de Barcelona
E-mail: alex@maia.ub.es

ABSTRACT: 
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,
including some benchmarks of a software to manipulate (truncated) power series 
of arbitrary number of variables and degree, and tests of 
the computation of the 4D center
and the 5D center-(un)stable manifolds of a Lagrangian point of the 
Restricted Three Body Problem.