Comparing the direct normal form method with harmonic balance and the method of multiple scales

Approximate analytical methods have been used extensively for finding approximate solutions to nonlinear ordinary differential equations. In this paper we compare the recently developed direct normal form transformation with two other very well known and long standing methods, harmonic balance and the method of multiple scales. We will show that the direct normal form method combines some of the key advantages of harmonic balance and multiple scales whilst reducing some of the limitations.