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{{Distinguish|Multiresolution analysis}}

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'''.

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==

[[File:Duffing-equation-perturbative-approaches.svg|thumb|431x431px|Here the differences between <math display="inline">\mathcal{O}(\varepsilon)</math> approaches for both regular perturbation theory and multiple-scale analysis can be seen, and how they compare to the exact solution for <math display="inline">\varepsilon = \frac{1}{4}</math>]]

===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&nbsp;<&nbsp;''ε''&nbsp;≪&nbsp;1. The undamped Duffing equation is known to be a [[Hamiltonian system]]:

<math display="block">\frac{dp}{dt}=-\frac{\partial H}{\partial q}, \qquad \frac{dq}{dt}=+\frac{\partial H}{\partial p}, \quad \text{ with } \quad H = \tfrac12 p^2 + \tfrac12 q^2 + \tfrac14 \varepsilon q^4,</math>

with ''q''&nbsp;=&nbsp;''y''(''t'') and ''p''&nbsp;=&nbsp;''dy''/''dt''. Consequently, the Hamiltonian ''H''(''p'',&nbsp;''q'') is a conserved quantity, a constant, equal to ''H₀''&nbsp;=&nbsp;{{sfrac|1|2}}&nbsp;+&nbsp;{{sfrac|1|4}}&nbsp;''ε'' for the given [[initial conditions]]. This implies that both ''q'' and ''p'' have to be bounded:

<math display="block">\left| q \right| \le \sqrt{1 + \tfrac12 \varepsilon} \quad \text{ and } \quad \left| p \right| \le \sqrt{1 + \tfrac12 \varepsilon} \qquad \text{ for all } t.</math>The bound on q is found by equating H with p = 0 to H₀: <math>\tfrac12 q^2 + \tfrac14 \varepsilon q^4 = \tfrac12 + \tfrac14 \varepsilon</math>, and then dropping the q⁴ term. This is indeed an upper bound on |q|, though keeping the q⁴ term gives a smaller bound with a more complicated formula.

===Straightforward perturbation-series solution===

:<math>\frac{d^2 y}{d t^2} + y + \varepsilon y^3 = 0,</math> {{pad|3em}} <math>y(0)=1, \qquad \frac{dy}{dt}(0)=0,</math>

A regular [[perturbation theory|perturbation-series approach]] to the problem proceeds by writing <math display="inline">y(t) = y_0(t) + \varepsilon y_1(t) + \mathcal{O}(\varepsilon^2)</math> and substituting this into the undamped Duffing equation. Matching powers of <math display="inline">\varepsilon</math> gives the system of equations

<math display="block">\begin{align}

which is a second-order [[ordinary differential equation]] describing a [[nonlinear]] [[oscillator]]. A solution ''y''(''t'') is sought for small values of the (positive) nonlinearity parameter 0&nbsp;<&nbsp;''ε''&nbsp;≪&nbsp;1. The undamped Duffing equation is known to be a [[Hamiltonian system]]:

\frac{d^2 y_0}{dt^2} + y_0 &= 0,\\

\frac{d^2 y_1}{dt^2} + y_1 &= - y_0^3.

:<math>\frac{dp}{dt}=-\frac{\partial H}{\partial q}, \qquad \frac{dq}{dt}=+\frac{\partial H}{\partial p}, \quad \text{ with } \quad H = \tfrac12 p^2 + \tfrac12 q^2 + \tfrac14 \varepsilon q^4,</math>

\end{align}</math>

with ''q''&nbsp;=&nbsp;''y''(''t'') and ''p''&nbsp;=&nbsp;''dy''/''dt''. Consequently, the Hamiltonian ''H''(''p'',&nbsp;''q'') is a conserved quantity, a constant, equal to ''H''&nbsp;=&nbsp;½&nbsp;+&nbsp;¼&nbsp;''ε'' for the given [[initial conditions]]. This implies that both ''y'' and ''dy''/''dt'' have to be bounded:

:<math>\left| y(t) \right| \le \sqrt{1 + \tfrac12 \varepsilon} \quad \text{ and } \quad \left| \frac{dy}{dt} \right| \le \sqrt{1 + \tfrac12 \varepsilon} \qquad \text{ for all } t.</math>{{pad|3em}}

===Straightforward perturbation-series solution===

A regular [[perturbation theory|perturbation-series approach]] to the problem gives the result:

Solving these subject to the initial conditions yields

:<math>

<math display="block">

y(t) = \cos(t)

+ \varepsilon \left[ \tfrac{1}{32} \cos(3t) - \tfrac{1}{32} \cos(t) - \underbrace{\tfrac38tfrac 3 8\, t\, \sin(t)}_\text{secular} \right]

+ \mathcal{O}(\varepsilon^2).

</math>

TheNote that the last term between the square braces is secular: it grows without bound for large |''t''|. In particular, makingfor the<math>t perturbation= solutionO(\varepsilon^{-1})</math> validthis forterm onlyis small''O''(1) valuesand has the same order of magnitude as the timeleading-order ''t''term. Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution.

===Method of multiple scales===

To construct a globallysolution that is valid solutionbeyond <math>t = O(\epsilon^{-1})</math>, the method of ''multiple-scale analysis'' is used. Introduce the slow scale ''t''₁:

<math display="block">t_1 = \varepsilon t</math>

:<math>t_1 = \varepsilon t\,</math>

and assume the solution ''y''(''t'') is a perturbation-series solution dependent both on ''t'' and ''t''₁, treated as:

<math display="block">y(t) = Y_0(t,t_1) + \varepsilon Y_1(t,t_1) + \cdots.</math>

:<math>y(t) = Y_0(t,t_1) + \varepsilon Y_1(t,t_1) + \cdots.</math>

So:

<math display="block"> \begin{align}

\frac{dy}{dt}

:<math>

&= \left( \frac{\partial Y_0}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_0}{\partial t_1} \right)

\begin{align}

+ \varepsilon \left( \frac{\partial Y_1}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_1}{\partial t_1} \right)

\frac{dy}{dt}

+ \cdots

&= \left( \frac{\partial Y_0}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_0}{\partial t_1} \right)

\\

+ \varepsilon \left( \frac{\partial Y_1}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_1}{\partial t_1} \right)

&= \frac{\partial Y_0}{\partial t}

+ \cdots

+ \varepsilon \left( \frac{\partial Y_0}{\partial t_1} + \frac{\partial Y_1}{\partial t} \right)

\\

&=+ \fracmathcal{\partial Y_0O}{(\partial t} varepsilon^2),

\end{align}</math>

+ \varepsilon \left( \frac{\partial Y_0}{\partial t_1} + \frac{\partial Y_1}{\partial t} \right)

+ \mathcal{O}(\varepsilon^2),

\end{align}

</math>

using ''dt''₁/''dt''&nbsp;=&nbsp;''ε''. Similarly:

<math display="block">\frac{d^2 y}{d t^2}

= \frac{\partial^2 Y_0}{\partial t^2}

:<math>

+ \varepsilon \left( 2 \frac{\partial^2 Y_0}{\partial t\, \partial t_1} + \frac{\partial^2 Y_1}{\partial t^2} \right)

\frac{d^2 y}{d t^2}

+ \mathcal{O}(\varepsilon^2).</math>

= \frac{\partial^2 Y_0}{\partial t^2}

+ \varepsilon \left( 2 \frac{\partial^2 Y_0}{\partial t\, \partial t_1} + \frac{\partial^2 Y_1}{\partial t^2} \right)

+ \mathcal{O}(\varepsilon^2).

</math>

Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become:

<math display="block">\begin{align}

\frac{\partial^2 Y_0}{\partial t^2} + Y_0 &= 0, \\

:<math>

\frac{\partial^2 Y_1}{\partial t^2} + Y_1 &= - Y_0^3 - 2\, \frac{\partial^2 Y_0}{\partial t\, \partial t_1}.

\begin{align}

\end{align}</math>

\frac{\partial^2 Y_0}{\partial t^2} + Y_0 &= 0,

\\

\frac{\partial^2 Y_1}{\partial t^2} + Y_1 &= - Y_0^3 - 2\, \frac{\partial^2 Y_0}{\partial t\, \partial t_1}.

\end{align}

</math>

===Solution===

The zeroth-order problem has the general solution:

<math display="block">Y_0(t,t_1) = A(t_1)\, e^{+it} + A^\ast(t_1)\, e^{-it},</math>

:<math>Y_0(t,t_1) = A(t_1)\, e^{+it} + A^\ast(t_1)\, e^{-it},</math>

with ''A''(''t''₁) a [[complex-valued amplitude]] to the zeroth-order solution ''Y''₀(''t'',&nbsp;''t''₁) and ''i''²&nbsp;=&nbsp;−1. Now, in the first-order problem the forcing in the [[right hand side]] of the differential equation is

<math display="block">\left[ -3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} \right]\, e^{+it} - A^3\, e^{+3it} + c.c.</math>

:<math>\left[ -3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} \right]\, e^{+it} - A^3\, e^{+3it} + c.c.</math>

where ''c.c.'' denotes the [[complex conjugate]] of the preceding terms. The occurrence of ''secular terms'' can be prevented by imposing on the – yet unknown – amplitude ''A''(''t''₁) the ''solvability condition''

<math display="block">-3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} = 0.</math>

The solution to the solvability condition, also satisfying the initial conditions {{math|1=''y''(0) = 1}} and {{math|1=''dy''/''dt''(0) = 0}}, is:

:<math>-3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} = 0.</math>

<math display="block">A = \tfrac 1 2\, \exp \left(\tfrac 3 8\, i \, t_1 \right).</math>

The solution to the solvability condition, also satisfying the initial conditions ''y''(0)&nbsp;=&nbsp;1 and ''dy''/''dt''(0)&nbsp;=&nbsp;0, is:

:<math>A = \tfrac12\, \exp \left(\tfrac38\, i \, t_1 \right).</math>

As a result, the approximate solution by the multiple-scales analysis is

<math display="block">y(t) = \cos \left[ \left( 1 + \tfrac38\, \varepsilon \right) t \right] + \mathcal{O}(\varepsilon),</math>

using {{math|1=''t''₁ = ''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''₂ = ''ε''² ''t''}}, {{math|1=''t''₃ = ''ε''³ ''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>

:<math>y(t) = \cos \left[ \left( 1 + \tfrac38\, \varepsilon \right) t \right] + \mathcal{O}(\varepsilon),</math>

using ''t''₁&nbsp;=&nbsp;''εt'' and valid for ''εt''&nbsp;=&nbsp;O(1). This agrees with the nonlinear [[frequency]] changes found by employing the [[Lindstedt–Poincaré method]].

Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, ''i.e.'': ''t''₂&nbsp;=&nbsp;''ε''²&nbsp;''t'', ''t''₃&nbsp;=&nbsp;''ε''³&nbsp;''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 approaches derive these sorts of models using coordinate transforms, like in the [[Normal form (dynamical systems)|method of normal forms]],<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 |bibcode=2012NonDy..70.1931L |hdl=10985/7473 |s2cid=254862552 |url=https://hal.archives-ouvertes.fr/hal-00880968/file/LSIS-INSM_nonli_dyn_2012_thomas.pdf }}</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}}

:<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>

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}}

<math display="block">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 display="block">\frac{d\theta}{dt} = 1 + \frac 3 8 \varepsilon r^2 -\frac{15}{256}\varepsilon^2r^4 +\mathcal O(\varepsilon^3).</math>

That is, Duffing's oscillations are of constant amplitude <math>r</math> but have different frequencies <math>d\theta/dt</math> depending upon the amplitude.<ref>{{citation |first=A.J. |last=Roberts |title=Modelling emergent dynamics in complex systems |url=http://www.maths.adelaide.edu.au/anthony.roberts/modelling.php |accessdate=2013-10-03 }}</ref>

:<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>{{citation |first=A.J. |last=Roberts |title=Modelling emergent dynamics in complex systems |url=http://www.maths.adelaide.edu.au/anthony.roberts/modelling.php |accessdate=2013-10-03 }}</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.{{When|date=July 2024}}<ref>{{citation |first=A.J. |last=Roberts |title=Construct centre manifolds of ordinary or delay differential equations (autonomous) |url=http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php |accessdate=2013-10-03 }}</ref>

==See also==

* [[Method of matched asymptotic expansions]]

* [[WKB approximation]]

*[[Method of averaging]]

*[[Krylov–Bogoliubov averaging method]]

==Notes==

{{reflist}}

A good reference is Nayfeh's perturbation method book

==References==

{{refbegin}}

*{{citation

| last1=Kevorkian | first1=J.

Line 128 ⟶ 107:

| year=1996

| publisher=Springer

| isbn=978-0-387-94202-5

}}

*{{citation

Line 136 ⟶ 115:

| publisher=Springer

| year=1999

| isbn=978-0-387-98931-50

| pages=544–568

}}

* {{citation

| title = Perturbation methods

| first = A.H. | last = Nayfeh | author-link = Ali H. Nayfeh

| year = 2004

| publisher = Wiley–VCH Verlag | isbn = 978-0-471-39917-9

}}

* {{citation

| title = Physics of Wave Turbulence

| first = S. | last = Galtier

| year = 2023

| publisher = Cambridge University Press | isbn = 978-1-009-27588-0

}}

{{refend}}

==External links==

*{{scholarpedia | title=Multiple scale analysis | urlname=Multiple_scale_analysis | curator=Carson C. Chow}}

*{{citation

| url=http://www.scholarpedia.org/article/Multiple_scale_analysis

| publisher=[[Scholarpedia]]

| title=Multiple scale analysis

| first=Carson C. | last=Chow

| accessdate=2009-08-09

| doi=10.4249/scholarpedia.1617

| year=2007

| journal=Scholarpedia

| volume=2

| issue=10

| pages=1617

|bibcode = 2007SchpJ...2.1617C }}

[[Category:Mathematical physics]]