I have read the paper by Lacarbonara [1]. The author did a good job. To clarify some of the issues, additional comments are given.
First of all, the non-linear operator notation used throughout the text [1] has been previously developed and used in a number of papers [2}7]. When I was working with Professor Nayfeh as a post-doc on the comparison of direct versus discretization} perturbation methods, we were treating special problems. I thought that it would be better to make the comparisons on a general equation having arbitrary quadratic and cubic non-linearities. For this reason, I developed an operator notation which is suitable for perturbative calculations. The "rst paper using that notation appeared in Mechanics Research Communications [2]. The paper was a single-mode comparison of direct versus discretization}perturbation methods for arbitrary quadratic and cubic non-linearities. The second paper [3], which is also referenced by the author, was an in"nite mode comparison between the methods in the absence of internal resonances. Although the author refers to this paper by saying &&However, in their analysis, one of the fundamental results was postulated instead of proved'', all conclusions presented in the paper were clearly shown and justi"ed by calculations. The general operator notation was further used for the subharmonic, superharmonic and combination resonance cases [4]. The case of arbitrary odd non-linearities were considered by using the same notation [5,6]. Finally, the notation has been developed to express and solve coupled systems of equations with arbitrary quadratic and cubic non-linearities [7].
In the comparison of direct perturbation and discretization}perturbation methods, two papers are worth mentioning. All papers dealing with this comparison issue treated non-linear problems. In fact, the problem arises also in linear equations and the direct-perturbation method yields more accurate results for "nite mode truncations [8]. The comparison of both methods in a gyroscopic system revealed another interesting result [9]. For such systems, researchers usually discretize the equations "rst and then apply perturbations. If travelling string eigenfunctions are used (better convergence properties than stationary string eigenfunctions), the equation of motion should be cast into a convenient "rst order form since the eigenfunctions do not have orthogonality properties. However, it is shown that, such transformations are unnecessary when using direct-perturbation methods, and the original equation of motion can be treated directly by perturbations.