Computational problems in multiple scale analysis

This paper discusses the computational problems that arise in the application of higher-order multiple scale methods to general multiresonant multiparameter systems. Complex amplitude equations are first analytically derived and their structure analyzed solely in the light of existing resonance conditions. A qualitative study of the complex amplitude equations is performed, using both an analytical and a geometrical approach able to furnish a complete description of the classes of motion admitted by the system and of the structure of the variational matrix governing their stability. The problem of representing the amplitude equations using real quantities is then addressed. It is shown that the use of a mixed polar-Cartesian form leads to a standard form of equation if suitable hypotheses are satisfied. This form makes it possible to overcome problems arising in the stability analysis of (incomplete) classes of motion when the most popular polar form is used. The final section discusses the reconstitution method, consisting in alternative techniques that can be used to combine the amplitude equations on different scales into a unique equation on a single scale. Four classes of method are considered, based on the consistency or inconsistency of the approach and on the completeness or incompleteness of the terms retained in the analysis. The four methods are critically compared and general conclusions drawn. Examples are given to clarify all the procedures discussed.