TITLE:
Explicit Numerical Computation of Normal Forms for Poincare Maps

AUTHORS:
Joan Gimeno^(1), Rafael de la Llave^(2) and Jiaqi Yang^(3)

(1) Departament de Matematiques i Informatica
    Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain

(2) School of Mathematics (Emeritus) Georgia Institute of Technology, 686
    Cherry St., Atlanta GA. 30332-0160, USA
    
(3) School of Mathematical Sciences, Key Laboratory of Intelligent Computing
    and Applications (Ministry of Education), Tongji University, Shanghai
    200092, China

E-mails: joan@maia.ub.es, rafael.delallave@math.gatech.edu, jqyang@tongji.edu.cn

ABSTRACT:

We rigorously construct a variety of orbits for certain delay differential
equations, including the electrodynamic equations formulated by Wheeler and
Feynman in 1949. These equations involve de- lays and advances that depend on
the trajectory itself, making it unclear how to formulate them as evolution
equations in a conventional phase space. Despite their fundamental significance
in physics, their mathematical treatment remains limited.

Our method applies broadly to various functional differential equations that
have appeared in the literature, including advanced/delayed equations, neutral
or state-dependent delay equations, and nested delay equations, under
appropriate regularity assumptions.

Rather than addressing the notoriously difficult problem of proving the
existence of solutions for all the initial conditions in a set, we focus on the
direct construction of a diverse collection of solutions.  This approach is
often sufficient to describe physical phenomena. For instance, in certain
models, we establish the existence of families of solutions exhibiting symbolic
dynamics.  Our method is based on the assumption that the system is, in a weak
sense, close to an ordinary differential equation (ODE) with “hyperbolic”
solutions as defined in dynamical systems. We then derive functional equations
to obtain space-time corrections.  As a byproduct of the method, we obtain that
the solutions constructed depend very smoothly on parameters of the model.
Also, we show that many formal approximations currently used in physics are
valid with explicit error terms. Several of the relations between different
orbits of the ODE persist qualitatively in the full problem.