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In [[mathematics]] and [[physics]], '''multiple-scale analysis''' (also called the '''method of multiple scales''') comprises techniques used to construct uniformly valid [[approximation]]s to the solutions of [[perturbation theory|perturbation problems]], both for small as well as large values of the [[independent variable]]s. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove (unwanted) [[secular variation|secular terms]]. The latter puts constraints on the approximate solution, which are called '''solvability conditions'''. Mathematics research from about the 1980s proposes that coordinate transforms and invariant manifolds provide much sounder support for multiscale modelling (for example, see [[center manifold]] and [[slow manifold]]).
==Example: undamped Duffing equation==
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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>
===Coordinate transform to amplitude/phase variables===
We seek a solution <math>y\approx r\cos\theta</math> 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
:<math>y=r\cos\theta +\frac1{32}\varepsilon r^3\cos3\theta +\frac1{1024}\varepsilon^2r^5(-21\cos3\theta+\cos5\theta)+\mathcal O(\varepsilon^3)</math>
transforms Duffing's equation into the pair that the radius is constant <math>dr/dt=0</math> and the phase evolves according to
:<math>\frac{d\theta}{dt}=1 +\frac38\varepsilon r^2 -\frac{15}{256}\varepsilon^2r^4 +\mathcal O(\varepsilon^3).</math>
That is, Duffing's oscillations are constant amplitude but a different frequencies depending upon the amplitude.
<ref>A. J. Roberts, <em>Modelling emergent dynamics in complex systems</em> [http://www.maths.adelaide.edu.au/anthony.roberts/modelling.php]</ref>
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. <ref>A. J. Roberts, <em>Construct centre manifolds of ordinary or delay differential equations (autonomous)</em> [http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php]</ref>
==See also==
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