Asymptotic methods in nonlinear dynamics are used usually to improveView original: http://arxiv.org/abs/1202.4395
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 (claiming), more precisely
the rolling from a top (claiming to the 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 it 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.