A Note on Higher Order Renormalization Group ‎Method

Renormalization group method (RGM) is a simple and powerful method to obtain analytical solution for differential equation. In this paper, with some examples, we show that the application of RGM to second order form of differential equation to determine the higher order approximation may give solution that is at variance with those solutions obtained with multiple scales method (MSM) and generalized method of averaging (GMA). However, transforming the differential equation to 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 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 overcame by applying the RGM to the complex-variable form of equations and obtaining solutions, which are derivable from a Lagrangian and are in agreement with the MSM and GMA solutions. Therefore, in use of RGM correct results may be obtained by treating the equations in complex-variable form.