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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{{Citation needed|date=December 2024}} that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling (for example, see [[center manifold]] and [[slow manifold]]).
==Example: undamped Duffing equation==
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===Differential equation and energy conservation===
As an example for the method of multiple-scale analysis, consider the undamped and unforced [[Duffing equation]]:<ref>This example is treated in: Bender & Orszag (1999) pp. 545–551.</ref>
<math display="block">\frac{d^2 y}{d t^2} + y + \varepsilon y^3 = 0,</math> <math display="block">y(0)=1, \qquad \frac{dy}{dt}(0)=0,</math>
which is a second-order [[ordinary differential equation]] describing a [[nonlinear system|nonlinear]] [[oscillator]]. A solution ''y''(''t'') is sought for small values of the (positive) nonlinearity parameter 0 < ''ε'' ≪ 1. The undamped Duffing equation is known to be a [[Hamiltonian system]]:
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using {{math|1=''t''<sub>1</sub> = ''εt''}} and valid for {{math|1=''εt'' = O(1)}}. This agrees with the nonlinear [[frequency]] changes found by employing the [[Lindstedt–Poincaré method]].
This new solution is valid until <math>t = O(\epsilon^{-2})</math>. Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, i.e., {{math|1=''t''<sub>2</sub> = ''ε''<sup>2</sup> ''t''}}, {{math|1=''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|
===Coordinate transform to amplitude/phase variables===
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==References==
{{refbegin}}
*{{
| last1
| last2=Cole | first2=J. D.
| title=Multiple scale and singular perturbation methods
| publisher
▲| year = 2013
| isbn
}}
*{{citation
| first1=C.M. | last1=Bender | authorlink1=Carl M. Bender
| |||