TITLE:
Using Automatic Differentiation to compute periodic orbits of Delay
Differential Equations

AUTHORS:
Joan Gimeno, Angel Jorba
Departament de Matematiques i Informatica
Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

E-mails: joan@maia.ub.es, angel@maia.ub.es

ABSTRACT:
In this paper we focus on the computation of periodic solutions of
Delay Differential Equations (DDEs) with constant delays. The method
is based on defining a Poincar\'e section in a suitable functional
space and looking for a fixed point of the flow in this section. This 
is done by applying a Newton method on a suitable discretization of
the section. To avoid computing and storing large matrices we use a
GMRES method to solve the linear system because in this case GMRES
converges very fast due to the compactness of the flow of the DDE. The
derivatives of the Poincar\'e map are obtained in a simple way, by
applying Automatic Differentiation to the numerical integration. The
stability of the periodic orbit is also obtained in a very efficient
way by means of Arnoldi methods. The examples considered include
temporal and spatial Poincar\'e sections.