Profits and pitfalls of timescales in asymptotics (Q2808250)

Certain timescales, like \(t,\) \(\varepsilon t,\) \(\varepsilon^{2}t,\) \(\ldots,\) arise naturally in perturbation problems for ordinary differential equations involving a small positive parameter \(\varepsilon.\) Multiscale methods represent very useful tools of analysis and anticipation of time scales is helpful in many problems, but not always. This nice paper addresses both efficiency and inefficiency of this popular technique. After a careful analysis of several examples, two classes of problems, where multiple timing may not be efficients are indicated. It appears that averaging and renormalization perform better than multiscale methods and give correct results in bifurcation problems, where one has to use unexpected algebraic timescales and in problems with resonance manifolds where higher order algebraic timescales and asymptotically small domains are encountered.