A note on higher order renormalization group method

The renormalization group method (rgm) is a simple and powerful methodto obtain analytical solutions for dierential equations. in this paper, with some examples,we show that the application of rgm to the second order form of dierential equations todetermine higher order approximations may give solutions that are at variance with thosesolutions obtained with the multiple scales method (msm) and the generalized methodof averaging (gma). however, transforming a dierential equation to a complex-variableform and then applying rgm, one may obtain solutions in agreement with the msm andgma solutions. furthermore, we consider a hamiltonian 2dof system and observe thatthe application of rgm to the second order form results in non-hamiltonian rg equations,and the result is at variance with the msm and gma solutions. again, this problem canbe overcomed by applying rgm to a complex-variable form of the equation, obtainingsolutions that are derivable from a lagrangian that are in agreement with the msm andgma solutions. therefore, using rgm, correct results may be obtained by treating theequation as a complex-variable form.