On the reconstitution problem in the multiple time-scale method
The authors consider general systems of the form \(\ddot q+F(q, \dot q,t;\mu)=0\), where \(\mu\) is a vector containing small physical quantities, or small deviations from the critical values of other parameters. A perturbation parameter \(\varepsilon\) is introduced by the equation \(\mu= \varepsilon \widehat\mu\), \(\widehat\mu= O(1)\), then the Lagrangian coordinates are expanded in series in \(\varepsilon\): \(q=\varepsilon q_1+ \varepsilon^2 q_2+\dots\), and temporal scales \(t_k=\varepsilon^kt\) \((k=0,1, \dots)\) are introduced. The authors derive perturbative equations for \(q_1,q_2,\dots\), and finally obtain a reconstructed equation for the amplitudes by using phases \(a\). This reconstructed equation is an asymptotic representation of a reduced dynamical system \(\dot a=G(a;\mu)\) able to capture the main aspects of the dynamics of the original system. The different methods used in literature to study this equations can be subdivided into two main classes of consistent or inconsistent methods, and in two sub-classes of complete or incomplete methods. In the consistent approach the asymptotic nature of the reduced dynamical system is taken into account, whereas in the inconsistent approach this feature is ignored. Moreover, in the complete method all terms derived from the solvability conditions are retained, while in the incomplete methods some of them are neglected. By combining the alternatives, the authors identify and discuss the four approaches. Three interesting illustrative examples are studied in details.
scientific article; zbMATH DE number 1396389