Profits and Pitfalls of Timescales in Asymptotics

Abstract

Abstract. The method of multiple timescales is widely used in engineering and mathematical physics.In this article we draw attention to the literature on the comparison of various perturbation methods. We indicate where we can obtain an advantage from the concept of timescales, and we present examples where the anticipation of timescales makes sense and cases where, because of resonances or bifurcations, the analysis is less straightforward. In a number of problems second order approximations are essential to understand the phenomena. We will conclude from these examples that the anticipation of timescales as in the multiple timescales method may give misleading results; methods that do not anticipate timescales a priori such as averaging and renormalization should be used in research problems.

Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction

“…which lie precisely on the W 1 (τ (t 0 )) slice (or fiber) of the SSM, W 1 (τ (t)), whose Taylor expansion we computed in (50). We show the unique analytic SSM for this example in Fig.…”

“…These perturbed NNMs are routinely sought via formal asymptotic expansions with various a priori postulated time scales (see, e.g., Nayfeh [32] for a survey of such intuitive methods). There is generally limited concern for the the validity of these formal approximations (see Verhulst [50] for a discussion). Formal computability of the first few terms of the assumed asymptotic expansion for NNMs, however, does not imply that the targeted structure actually exists, as we discussed in the Introduction.…”

“…(50) The requirement of 2π-periodicity on a i (t) given in ( 47) defines a boundary-value problem for the ODEs in (50), whose unique solutions are…”

Resonances and vibrations in an elevator cable system due to boundary sway

“…This example might add extra resonance manifolds and might produce new asymptotic timescales in the dynamics of the cable. One should be also aware of intersection of resonance zones [17], what makes the analysis of the problem even more interesting, but complex. In conclusion, the analytic methodology developed in this paper can be implemented in other types of gyroscopic systems which are governed by differential equations with in time slowly varying coefficients.…”

Nonlinear dynamics in mechanics and engineering: 40 years of developments and Ali H. Nayfeh’s legacy

“…Notwithstanding some mathematical warnings about possible pitfalls occurring in the asymptotics with the method of multiple time scales, Nayfeh's methodological and operational contributions on multiple scales in the first outstanding part of his scientific career, yet continued all over his activity, are certainly a fundamental part of his legacy and are worldwide utilized for solving nonlinear dynamics problems. Pitfalls are mostly associated with the anticipated choice of timescales, as contrasted with methods of averaging and renormalization which could be preferred because of timescales emerging by the non-secularity conditions with no a priori assumptions [101]. This is particularly true in the presence of resonances or bifurcations, where naturally ensuing algebraic timescales (including some with fractional powers of the small parameter ) can play an important role in obtaining reliable asymptotic estimates on long intervals of time.…”