  The improved formulations of the Lindstedt Poincare perturbation method, the
harmonic balance method and its modifications using the rational functions, the
energy balance method and the two terms Fourier series expansion have been
employed to find approximate solution to the equation of motion for the cubic
quartic potential AHO and its special cases (the cubic, the quartic, and the
purely cubic quartic potential AHOs), to assess the relative merits and
demerits of these techniques. Two new rational functions as approximate
solution for the harmonic balance method have been proposed and the rational
functions have been used as trial functions in the energy balance method for
the first time. The time periods T+ (for q(t) > 0), T- (for q(t) < 0) and the
total period T so determined have been compared with the corresponding exact
values for an extensive range of the AHO parameter values. The results have
been thoroughly analysed in terms of their percentage difference from the
relevant exact time period. It is concluded that the methods based on the
Lindstedt Poincare perturbation approach do not lead to reliable results for T+
and T-, which makes their applicability for the asymmetric potential AHOs
doubtful. The harmonic balance technique together with its different variants
is the most successful, while the Fourier series approach also yields
commendably accurate results. The outcome of the third-order energy balance
method is reasonably good though the usage of the rational functions for this
has not been encouraging.
