This agrees with the nonlinear frequency changes found by employing the Lindstedt–Poincaré method. This new solution is valid until formula_18. Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, "i.e.": "t"2 = "ε"2 "t", "t"3 = "ε"3 "t", etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see ; ). Alternatively, modern sound approaches derive these sorts of models using coordinate transforms, as described next. A solution formula_19 is sought in new coordinates formula_20 where the amplitude formula_21 varies slowly and the phase formula_22 varies at an almost constant rate, namely formula_23 Straightforward algebra finds the coordinate transform  transforms Duffing's equation into the pair that the radius is constant formula_25 and the phase evolves according to  That is, Duffing's oscillations are of constant amplitude formula_27 but have different frequencies formula_28 depending upon the amplitude. More difficult examples are better treated using a time-dependent coordinate transform involving complex exponentials (as also invoked in the previous multiple time-scale approach). A web service will perform the analysis for a wide range of examples.