Asymptotic methods in nonlinear dynamics are used to improve perturbation
theory results in the oscillations regime. However, for some problems of
nonlinear dynamics, particularly in the case of Higgs (Duffing) equation and
the Friedmann cosmological equations, not only small oscillations regime is of
interest but also the regime of rolling (climbing), more precisely the rolling
from a top (climbing to a top). In the Friedman cosmology, where the slow
rolling regime is often used, the rolling from a top (not necessary slow) is of
interest too.
In the present work a method for approximate solution to the Higgs equation
in the rolling regime is presented. It is shown that in order to improve
perturbation theory in the rolling regime turns out to be effective not to use
an expansion in trigonometric functions as it is done in case of small
oscillations but use expansions in hyperbolic functions instead. This regime is
investigated using the representation of the solution in terms of elliptic
functions. An accuracy of the corresponding approximation is estimated.Comment: Latex, 36 Pages, 8 figures, typos correcte