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===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=1931–1949|year=2012 |volume=70 |issue=3 |doi=10.1007/s11071-012-0584-y |hdl=10985/7473 }}</ref> as described next.
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> Straightforward algebra finds the coordinate transform{{citation needed|date=June 2015}}
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| year=1996
| publisher=Springer
| isbn=978-0-387-94202-5
}}
*{{citation
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| publisher=Springer
| year=1999
| isbn=978-0-387-98931-
| pages=544–568
}}
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| first = A.H. | last = Nayfeh | author-link = Ali H. Nayfeh
| year = 2004
| publisher = Wiley–VCH Verlag | isbn = 978-0-471-39917-
}}
{{ref end}}
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