### Vassili Gelfreich

## Numerics and exponential smallness

A single step integrator for an ordinary
differential equation may be considered as
a high frequency perturbation of the original
system. An averaging method may be applied
to check that there is a coordinate change, wich
eleminate this time-dependance up to
an exponentially small remainder. In general this
remainder is not zero.

Applying this idea to an autonomous Hamiltonian
system with 1 degree of freedom, we see that the
truncated system is also autonomos, and it is
$h^n$ close to the original system, where $h$
stends for the integrator step and $n$ is the
order of the integrator. In this case the
autonomous systems have a quite simple dynamics:
every trajectory is either periodic,
(be)asymptotic to periodic trajectories, or goes
to infinity. On the other hand, the remainder may be a
source of a chaotic behavior for numerical
solutions. Chaotic means here, that the dynamics
on some close invariant subset is semiconjucated to
the Bernulli schift on sequences of two symbols.
In other words, the symbolic dynamics may be
constructed. The creation of chaotic trajectories
may be modelled by the famous Smale horseshue. In
Hamiltonian systems horseshues appear near
transversal homoclinic or heteroclinic
points. It is easy to see that the original
autonomous system has no trajectories of that
type, and the exponentially small remainder can
only result in exponentially small transversality
of the separatrices. Thus the latter may not be
detected by the classical perturbation theory,
which is based on the expansion in powers of the
small parameter. We discuss some latest result on
the exponentially small splitting of separatrices, especially,
the methods for obtaining upper and lower bounds for the splitting.
Finally, some numerical methods, specially desined to study
exponentially small phenomena, are discussed.

*
Vassili Gelfreich *

gelf@maia.ub.es

gelf@math.fu-berlin.de

Jan 27 1998