TITLE:
Explicit Numerical Computation of Normal Forms for Poincare Maps

AUTHORS:
Joan Gimeno^(1), Angel Jorba^(1,3), Marc Jorba-Cusco^(2) and Maorong Zou^(4)

(1) Departament de Matematiques i Informatica
    Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain
(2) Departament de Matematiques, Universitat Politecnica de Catalunya
    Av. Diagonal 647, 08028, Barcelona, Spain
(3) Centre de Recerca Matematica (CRM), Barcelona
(4) Department of Mathematics, University of Texas at Austin,
    Austin, TX 78712, USA

E-mails: joan@maia.ub.es, angel@maia.ub.es, marc.jorba@upc.edu, mzou@math.utexas.edu

ABSTRACT:

We present a methodology for computing normal forms in discrete
systems, such as those described by Poincar\'e maps. Our approach
begins by calculating high-order derivatives of the flow with respect
to initial conditions and parameters, obtained via jet transport, and
then applying appropriate projections to the Poincar\'e section to
derive the power expansion of the map.  In the second step, we perform
coordinate transformations to simplify the local power expansion
around a dynamical object, retaining only the resonant terms. The
resulting normal form provides a local description of the dynamics
around the object, and shows its dependence on parameters. Notably,
this method does not assume any specific structure of the system
besides sufficient regularity.

To illustrate its effectiveness, we first examine the well-known
H\'enon-Heiles system. By fixing an energy level and using a spatial
Poincar\'e section, the system is represented by a 2D Poincar\'e map.
Focusing on an elliptic fixed point of this map, we compute a
high-order normal form, which is a twist map obtained explicitly. This
means that we have computed the invariant tori inside the energy level
of the Poincar\'e section. Furthermore, we explore how both the fixed
point and the normal form depend on the energy level of the Poincar\'e
section, deriving the coefficients of the twist map as a power series
of the energy level. This approach also enables us to obtain invariant
tori inside nearby energy levels. We also discuss how to obtain the
frequencies of the torus for the flow. We include a second example
involving an elliptic periodic orbit of the spatial Restricted
Three-Body Problem. In this case the map is 4D, and the normal form is
a multidimensional twist map.