Asymptotic expansions beyond all orders: towards asymptotic description of chaotic dynamics
Heteroclinic or homoclinic structures are the well-known mechanism for a
genesis of chaos in conservative dynamical systems. While numerical
iterations of low-dimensional maps easily provide these complicated
structures, it is very difficult to derive them analytically, as addressed
by Poincare (French Mathematician) more than a century ago.
However, the difficulty can now be overcome by using a scheme of
the asymptotic expansions beyond all orders. Nakamura and Hamada(1996) ,
Tovbis et al (1998), and Nakamura and Kushibe(1998) attempted to
describe analytically the chaotic dynamics for the Henon and cubic maps.
However, they failed to obtain qualitatively essential corrections to
the Stokes multiplier. In order to overcome this problem, some new
systematic approach should be invented.
In this talk, dealing with the standard and cubic maps., i.e.,
time-discrete dynamical systems in the presence of cosine and double-well
potentials, respectively, we shall provide an asymptotic analytical
description of the complicated heteroclinic and homoclinic structures
on exteremely fine scales and compare the issue with that of an exact
numerical iteration of the maps. Technical problems around the Borel
summability and Stokes phenomena are also discussed. The talk is based
on the latest work by Nakamura and Kushibe (1999).