A note on higher order renormalization group method

The Renormalization Group Method (RGM) is a simple and powerful method to 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 to determine higher order approximations may give solutions that are at variance with those solutions obtained with the Multiple Scales Method (MSM) and the Generalized Method of Averaging (GMA). However, transforming a dierential equation to a complex-variable form and then applying RGM, one may obtain solutions in agreement with the MSM and GMA solutions. Furthermore, we consider a Hamiltonian 2DOF system and observe that the 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 can be overcomed by applying RGM to a complex-variable form of the equation, obtaining solutions that are derivable from a Lagrangian that are in agreement with the MSM and GMA solutions. Therefore, using RGM, correct results may be obtained by treating the equation as a complex-variable form.