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Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, ''i.e.'': ''t''<sub>2</sub> = ''ε''<sup>2</sup> ''t'', ''t''<sub>3</sub> = ''ε''<sup>3</sup> ''t'', etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see {{harvnb|Kevorkian|Cole|1996}}; {{harvnb|Bender|Orszag|1999}}).<ref>Bender & Orszag (1999) p. 551.</ref>
Alternatively, modern sound approaches derive these sorts of models using coordinate transforms<ref>{{citation| first1=C.-H. |last1=Lamarque |first2=C. |last2=Touze |first3=O. |last3=Thomas |title=An upper bound for validity limits of asymptotic analytical approaches based on normal form theory |journal=[[Nonlinear Dynamics (journal)|Nonlinear Dynamics]] |pages=1931–1919 |year=2012 |volume=70 |issue=3 |doi=10.1007/s11071-012-0584-y }}</ref> as also described next.▼
===Coordinate transform to amplitude/phase variables===
▲Alternatively, modern sound approaches derive these sorts of models using coordinate transforms,<ref>{{citation| first1=C.-H. |last1=Lamarque |first2=C. |last2=Touze |first3=O. |last3=Thomas |title=An upper bound for validity limits of asymptotic analytical approaches based on normal form theory |journal=[[Nonlinear Dynamics (journal)|Nonlinear Dynamics]] |pages=1931–1919 |year=2012 |volume=70 |issue=3 |doi=10.1007/s11071-012-0584-y }}</ref> as
A solution <math>y\approx r\cos\theta</math> is sought in new coordinates <math>(r,\theta)</math> where the amplitude <math>r(t)</math> varies slowly and the phase <math>\theta(t)</math> varies at an almost constant rate, namely <math>d\theta/dt\approx 1</math>.
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