# Multiple-scale analysis

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Approximations that apply at multiple scales

Not to be confused with [Multiresolution analysis](/source/Multiresolution_analysis) or [multiscale modeling](/source/Multiscale_modeling).

In [mathematics](/source/Mathematics) and [physics](/source/Physics), **multiple-scale analysis** (also called the **method of multiple scales**) comprises techniques used to construct uniformly valid [approximations](/source/Approximation) to the solutions of [perturbation problems](/source/Perturbation_theory), both for small as well as large values of the [independent variables](/source/Independent_variable). 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 terms](/source/Secular_variation). The latter puts constraints on the approximate solution, which are called **solvability conditions**.

Mathematics research from about the 1980s proposes[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling (for example, see [center manifold](/source/Center_manifold) and [slow manifold](/source/Slow_manifold)).

## Example: undamped Duffing equation

Here the differences between

            O

        (
        ε
        )

    {\textstyle {\mathcal {O}}(\varepsilon )}

 approaches for both regular perturbation theory and multiple-scale analysis can be seen, and how they compare to the exact solution for

        ε
        =

            1
            4

    {\textstyle \varepsilon ={\frac {1}{4}}}

### Differential equation and energy conservation

As an example for the method of multiple-scale analysis, consider the undamped and unforced [Duffing equation](/source/Duffing_equation):[1] d 2 y d t 2 + y + ε y 3 = 0 , {\displaystyle {\frac {d^{2}y}{dt^{2}}}+y+\varepsilon y^{3}=0,} y ( 0 ) = 1 , d y d t ( 0 ) = 0 , {\displaystyle y(0)=1,\qquad {\frac {dy}{dt}}(0)=0,} which is a second-order [ordinary differential equation](/source/Ordinary_differential_equation) describing a [nonlinear](/source/Nonlinear_system) [oscillator](/source/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](/source/Hamiltonian_system): d p d t = − ∂ H ∂ q , d q d t = + ∂ H ∂ p , with H = 1 2 p 2 + 1 2 q 2 + 1 4 ε q 4 , {\displaystyle {\frac {dp}{dt}}=-{\frac {\partial H}{\partial q}},\qquad {\frac {dq}{dt}}=+{\frac {\partial H}{\partial p}},\quad {\text{ with }}\quad H={\tfrac {1}{2}}p^{2}+{\tfrac {1}{2}}q^{2}+{\tfrac {1}{4}}\varepsilon q^{4},} with *q* = *y*(*t*) and *p* = *dy*/*dt*. Consequently, the Hamiltonian *H*(*p*, *q*) is a conserved quantity, a constant, equal to *H0* = ⁠1/2⁠ + ⁠1/4⁠ *ε* for the given [initial conditions](/source/Initial_conditions). This implies that both *q* and *p* have to be bounded: | q | ≤ 1 + 1 2 ε and | p | ≤ 1 + 1 2 ε for all t . {\displaystyle \left|q\right|\leq {\sqrt {1+{\tfrac {1}{2}}\varepsilon }}\quad {\text{ and }}\quad \left|p\right|\leq {\sqrt {1+{\tfrac {1}{2}}\varepsilon }}\qquad {\text{ for all }}t.} The bound on q is found by equating H with p = 0 to H0: 1 2 q 2 + 1 4 ε q 4 = 1 2 + 1 4 ε {\displaystyle {\tfrac {1}{2}}q^{2}+{\tfrac {1}{4}}\varepsilon q^{4}={\tfrac {1}{2}}+{\tfrac {1}{4}}\varepsilon } , 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

A regular [perturbation-series approach](/source/Perturbation_theory) to the problem proceeds by writing y ( t ) = y 0 ( t ) + ε y 1 ( t ) + O ( ε 2 ) {\textstyle y(t)=y_{0}(t)+\varepsilon y_{1}(t)+{\mathcal {O}}(\varepsilon ^{2})} and substituting this into the undamped Duffing equation. Matching powers of ε {\textstyle \varepsilon } gives the system of equations d 2 y 0 d t 2 + y 0 = 0 , d 2 y 1 d t 2 + y 1 = − y 0 3 . {\displaystyle {\begin{aligned}{\frac {d^{2}y_{0}}{dt^{2}}}+y_{0}&=0,\\{\frac {d^{2}y_{1}}{dt^{2}}}+y_{1}&=-y_{0}^{3}.\end{aligned}}}

Solving these subject to the initial conditions yields y ( t ) = cos ⁡ ( t ) + ε [ 1 32 cos ⁡ ( 3 t ) − 1 32 cos ⁡ ( t ) − 3 8 t sin ⁡ ( t ) ⏟ secular ] + O ( ε 2 ) . {\displaystyle y(t)=\cos(t)+\varepsilon \left[{\tfrac {1}{32}}\cos(3t)-{\tfrac {1}{32}}\cos(t)-\underbrace {{\tfrac {3}{8}}\,t\,\sin(t)} _{\text{secular}}\right]+{\mathcal {O}}(\varepsilon ^{2}).}

Note that the last term between the square braces is secular: it grows without bound for large |*t*|. In particular, for t = O ( ε − 1 ) {\displaystyle t=O(\varepsilon ^{-1})} this term is *O*(1) and has the same [order of magnitude](/source/Order_of_magnitude) as the [leading-order](/source/Leading-order_term) 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 solution that is valid beyond t = O ( ϵ − 1 ) {\displaystyle t=O(\epsilon ^{-1})} , the method of *multiple-scale analysis* is used. Introduce the slow scale *t*1: t 1 = ε t {\displaystyle t_{1}=\varepsilon t} and assume the solution *y*(*t*) is a perturbation-series solution dependent both on *t* and *t*1, treated as: y ( t ) = Y 0 ( t , t 1 ) + ε Y 1 ( t , t 1 ) + ⋯ . {\displaystyle y(t)=Y_{0}(t,t_{1})+\varepsilon Y_{1}(t,t_{1})+\cdots .}

So: d y d t = ( ∂ Y 0 ∂ t + d t 1 d t ∂ Y 0 ∂ t 1 ) + ε ( ∂ Y 1 ∂ t + d t 1 d t ∂ Y 1 ∂ t 1 ) + ⋯ = ∂ Y 0 ∂ t + ε ( ∂ Y 0 ∂ t 1 + ∂ Y 1 ∂ t ) + O ( ε 2 ) , {\displaystyle {\begin{aligned}{\frac {dy}{dt}}&=\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)+\cdots \\&={\frac {\partial Y_{0}}{\partial t}}+\varepsilon \left({\frac {\partial Y_{0}}{\partial t_{1}}}+{\frac {\partial Y_{1}}{\partial t}}\right)+{\mathcal {O}}(\varepsilon ^{2}),\end{aligned}}} using *dt*1/*dt* = *ε*. Similarly: d 2 y d t 2 = ∂ 2 Y 0 ∂ t 2 + ε ( 2 ∂ 2 Y 0 ∂ t ∂ t 1 + ∂ 2 Y 1 ∂ t 2 ) + O ( ε 2 ) . {\displaystyle {\frac {d^{2}y}{dt^{2}}}={\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}).}

Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become: ∂ 2 Y 0 ∂ t 2 + Y 0 = 0 , ∂ 2 Y 1 ∂ t 2 + Y 1 = − Y 0 3 − 2 ∂ 2 Y 0 ∂ t ∂ t 1 . {\displaystyle {\begin{aligned}{\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{aligned}}}

### Solution

The zeroth-order problem has the general solution: Y 0 ( t , t 1 ) = A ( t 1 ) e + i t + A ∗ ( t 1 ) e − i t , {\displaystyle Y_{0}(t,t_{1})=A(t_{1})\,e^{+it}+A^{\ast }(t_{1})\,e^{-it},} with *A*(*t*1) a [complex-valued amplitude](/source/Complex-valued_amplitude) to the zeroth-order solution *Y*0(*t*, *t*1) and *i*2 = −1. Now, in the first-order problem the forcing in the [right hand side](/source/Right_hand_side) of the differential equation is [ − 3 A 2 A ∗ − 2 i d A d t 1 ] e + i t − A 3 e + 3 i t + c . c . {\displaystyle \left[-3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1}}}\right]\,e^{+it}-A^{3}\,e^{+3it}+c.c.} where *c.c.* denotes the [complex conjugate](/source/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* − 3 A 2 A ∗ − 2 i d A d t 1 = 0. {\displaystyle -3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1}}}=0.}

The solution to the solvability condition, also satisfying the initial conditions *y*(0) = 1 and *dy*/*dt*(0) = 0, is: A = 1 2 exp ⁡ ( 3 8 i t 1 ) . {\displaystyle A={\tfrac {1}{2}}\,\exp \left({\tfrac {3}{8}}\,i\,t_{1}\right).}

As a result, the approximate solution by the multiple-scales analysis is y ( t ) = cos ⁡ [ ( 1 + 3 8 ε ) t ] + O ( ε ) , {\displaystyle y(t)=\cos \left[\left(1+{\tfrac {3}{8}}\,\varepsilon \right)t\right]+{\mathcal {O}}(\varepsilon ),} using *t*1 = *εt* and valid for *εt* = O(1). This agrees with the nonlinear [frequency](/source/Frequency) changes found by employing the [Lindstedt–Poincaré method](/source/Lindstedt%E2%80%93Poincar%C3%A9_method).

This new solution is valid until t = O ( ϵ − 2 ) {\displaystyle t=O(\epsilon ^{-2})} . 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 [Kevorkian & Cole 1996](#CITEREFKevorkianCole1996); [Bender & Orszag 1999](#CITEREFBenderOrszag1999)).[2]

### Coordinate transform to amplitude/phase variables

Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the [method of normal forms](/source/Normal_form_(dynamical_systems)),[3] as described next.

A solution y ≈ r cos ⁡ θ {\displaystyle y\approx r\cos \theta } is sought in new coordinates ( r , θ ) {\displaystyle (r,\theta )} where the amplitude r ( t ) {\displaystyle r(t)} varies slowly and the phase θ ( t ) {\displaystyle \theta (t)} varies at an almost constant rate, namely d θ / d t ≈ 1. {\displaystyle d\theta /dt\approx 1.} Straightforward algebra finds the coordinate transform[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] y = r cos ⁡ θ + 1 32 ε r 3 cos ⁡ 3 θ + 1 1024 ε 2 r 5 ( − 21 cos ⁡ 3 θ + cos ⁡ 5 θ ) + O ( ε 3 ) {\displaystyle y=r\cos \theta +{\frac {1}{32}}\varepsilon r^{3}\cos 3\theta +{\frac {1}{1024}}\varepsilon ^{2}r^{5}(-21\cos 3\theta +\cos 5\theta )+{\mathcal {O}}(\varepsilon ^{3})} transforms Duffing's equation into the pair that the radius is constant d r / d t = 0 {\displaystyle dr/dt=0} and the phase evolves according to d θ d t = 1 + 3 8 ε r 2 − 15 256 ε 2 r 4 + O ( ε 3 ) . {\displaystyle {\frac {d\theta }{dt}}=1+{\frac {3}{8}}\varepsilon r^{2}-{\frac {15}{256}}\varepsilon ^{2}r^{4}+{\mathcal {O}}(\varepsilon ^{3}).}

That is, Duffing's oscillations are of constant amplitude r {\displaystyle r} but have different frequencies d θ / d t {\displaystyle d\theta /dt} depending upon the amplitude.[4]

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?](https://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/Dates_and_numbers#Chronological_items)*][5]

## See also

- [Method of matched asymptotic expansions](/source/Method_of_matched_asymptotic_expansions)

- [WKB approximation](/source/WKB_approximation)

- [Method of averaging](/source/Method_of_averaging)

- [Krylov–Bogoliubov averaging method](/source/Krylov%E2%80%93Bogoliubov_averaging_method)

## Notes

1. **[^](#cite_ref-1)** This example is treated in: Bender & Orszag (1999) pp. 545–551.

1. **[^](#cite_ref-2)** Bender & Orszag (1999) p. 551.

1. **[^](#cite_ref-3)** Lamarque, C.-H.; Touze, C.; Thomas, O. (2012), ["An upper bound for validity limits of asymptotic analytical approaches based on normal form theory"](https://hal.archives-ouvertes.fr/hal-00880968/file/LSIS-INSM_nonli_dyn_2012_thomas.pdf) (PDF), *[Nonlinear Dynamics](/source/Nonlinear_Dynamics_(journal))*, **70** (3): 1931–1949, [Bibcode](/source/Bibcode_(identifier)):[2012NonDy..70.1931L](https://ui.adsabs.harvard.edu/abs/2012NonDy..70.1931L), [doi](/source/Doi_(identifier)):[10.1007/s11071-012-0584-y](https://doi.org/10.1007%2Fs11071-012-0584-y), [hdl](/source/Hdl_(identifier)):[10985/7473](https://hdl.handle.net/10985%2F7473), [S2CID](/source/S2CID_(identifier)) [254862552](https://api.semanticscholar.org/CorpusID:254862552)

1. **[^](#cite_ref-4)** Roberts, A.J., [*Modelling emergent dynamics in complex systems*](http://www.maths.adelaide.edu.au/anthony.roberts/modelling.php), retrieved 2013-10-03

1. **[^](#cite_ref-5)** Roberts, A.J., [*Construct centre manifolds of ordinary or delay differential equations (autonomous)*](http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php), retrieved 2013-10-03

## References

- Kevorkian, J.; Cole, J. D. (1996), *Multiple scale and singular perturbation methods*, Springer, [ISBN](/source/ISBN_(identifier)) [978-0-387-94202-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-94202-5)

- [Bender, C.M.](/source/Carl_M._Bender); [Orszag, S.A.](/source/Steven_A._Orszag) (1999), *Advanced mathematical methods for scientists and engineers*, Springer, pp. 544–568, [ISBN](/source/ISBN_(identifier)) [978-0-387-98931-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98931-0)

- [Nayfeh, A.H.](/source/Ali_H._Nayfeh) (2004), *Perturbation methods*, Wiley–VCH Verlag, [ISBN](/source/ISBN_(identifier)) [978-0-471-39917-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-39917-9)

- Galtier, S. (2023), *Physics of Wave Turbulence*, Cambridge University Press, [ISBN](/source/ISBN_(identifier)) [978-1-009-27588-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-009-27588-0)

## External links

- Carson C. Chow (ed.). ["Multiple scale analysis"](http://www.scholarpedia.org/article/Multiple_scale_analysis). *[Scholarpedia](/source/Scholarpedia)*.

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Adapted from the Wikipedia article [Multiple-scale analysis](https://en.wikipedia.org/wiki/Multiple-scale_analysis) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Multiple-scale_analysis?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.