On the reconstitution problem in the multiple time-scale method

Higher-order multiple-scale methods for general multiparameter mechanical s ystems are studied. The role played by the control and imperfection paramet ers in deriving the perturbative equations is highlighted. The definition o f the codimension of the problem, borrowed from the bifurcation theory, is extended to general systems, excited either externally or parametrically. T he concept of a reduced dynamical system is then invoked. Different approac hes followed in the literature to deal with reconstituted amplitude equatio ns are discussed, both in the search for steady-state solutions and in the analysis of stability. Four classes of methods are considered, based on the consistency or inconsistency of the approach, and on the completeness or i ncompleteness of the terms retained in the analysis. The four methods are c ritically compared and general conclusions drawn. Finally, three examples a re illustrated to corroborate the findings and to show the quantitative dif ferences between the various approaches.