In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. 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. 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 a sounder support for multiscale modelling (for example, see center manifold and slow manifold).

Example: undamped Duffing equation

Differential equation and energy conservation

As an example for the method of multiple-scale analysis, consider the undamped and unforced Duffing equation:^[1]

$${\displaystyle {\frac {d^{2}y}{dt^{2))}+y+\varepsilon y^{3}=0,}$$

$${\displaystyle y(0)=1,\qquad {\frac {dy}{dt))(0)=0,}$$

$${\displaystyle {\frac {dp}{dt))=-{\frac {\partial H}{\partial q)),\qquad {\frac {dq}{dt))=+{\frac {\partial H}{\partial p)),\quad {\text{ with ))\quad H={\tfrac {1}{2))p^{2}+{\tfrac {1}{2))q^{2}+{\tfrac {1}{4))\varepsilon q^{4},}$$

$${\displaystyle \left|y(t)\right|\leq {\sqrt {1+{\tfrac {1}{2))\varepsilon ))\quad {\text{ and ))\quad \left|{\frac {dy}{dt))\right|\leq {\sqrt {1+{\tfrac {1}{2))\varepsilon ))\qquad {\text{ for all ))t.}$$

Straightforward perturbation-series solution

A regular perturbation-series approach to the problem proceeds by writing ${\textstyle y(t)=y_{0}(t)+\varepsilon y_{1}(t)+{\mathcal {O))(\varepsilon ^{2})}$ and substituting this into the undamped Duffing equation. Matching powers of ${\textstyle \varepsilon }$ gives the system of equations

$${\displaystyle {\begin{aligned}{\frac {d^{2}y_{0)){dt^{2))}+y_{0}&=0,\\{\frac {d^{2}y_{1)){dt^{2))}+y_{1}&=-y_{0}^{3}.\end{aligned))}$$

Solving these subject to the initial conditions yields

$${\displaystyle y(t)=\cos(t)+\varepsilon \left[{\tfrac {1}{32))\cos(3t)-{\tfrac {1}{32))\cos(t)-\underbrace ((\tfrac {3}{8))\,t\,\sin(t)} _{\text{secular))\right]+{\mathcal {O))(\varepsilon ^{2}).}$$

Note that the last term between the square braces is secular: it grows without bound for large |t|. In particular, for ${\displaystyle t=O(\varepsilon ^{-1})}$ this term is O(1) and has the same order of magnitude as the leading-order 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 ${\displaystyle t=O(\epsilon ^{-1})}$, the method of multiple-scale analysis is used. Introduce the slow scale t₁:

$${\displaystyle t_{1}=\varepsilon t}$$

$${\displaystyle y(t)=Y_{0}(t,t_{1})+\varepsilon Y_{1}(t,t_{1})+\cdots .}$$

So:

$${\displaystyle {\begin{aligned}{\frac {dy}{dt))&=\left({\frac {\partial Y_{0)){\partial t))+{\frac {dt_{1)){dt)){\frac {\partial Y_{0)){\partial t_{1))}\right)+\varepsilon \left({\frac {\partial Y_{1)){\partial t))+{\frac {dt_{1)){dt)){\frac {\partial Y_{1)){\partial t_{1))}\right)+\cdots \\&={\frac {\partial Y_{0)){\partial t))+\varepsilon \left({\frac {\partial Y_{0)){\partial t_{1))}+{\frac {\partial Y_{1)){\partial t))\right)+{\mathcal {O))(\varepsilon ^{2}),\end{aligned))}$$

$${\displaystyle {\frac {d^{2}y}{dt^{2))}={\frac {\partial ^{2}Y_{0)){\partial t^{2))}+\varepsilon \left(2{\frac {\partial ^{2}Y_{0)){\partial t\,\partial t_{1))}+{\frac {\partial ^{2}Y_{1)){\partial t^{2))}\right)+{\mathcal {O))(\varepsilon ^{2}).}$$

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

$${\displaystyle {\begin{aligned}{\frac {\partial ^{2}Y_{0)){\partial t^{2))}+Y_{0}&=0,\\{\frac {\partial ^{2}Y_{1)){\partial t^{2))}+Y_{1}&=-Y_{0}^{3}-2\,{\frac {\partial ^{2}Y_{0)){\partial t\,\partial t_{1))}.\end{aligned))}$$

Solution

The zeroth-order problem has the general solution:

$${\displaystyle Y_{0}(t,t_{1})=A(t_{1})\,e^{+it}+A^{\ast }(t_{1})\,e^{-it},}$$

$${\displaystyle \left[-3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1))}\right]\,e^{+it}-A^{3}\,e^{+3it}+c.c.}$$

$${\displaystyle -3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1))}=0.}$$

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

$${\displaystyle A={\tfrac {1}{2))\,\exp \left({\tfrac {3}{8))\,i\,t_{1}\right).}$$

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

$${\displaystyle y(t)=\cos \left[\left(1+{\tfrac {3}{8))\,\varepsilon \right)t\right]+{\mathcal {O))(\varepsilon ),}$$

This new solution is valid until ${\displaystyle t=O(\epsilon ^{-2})}$. Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, i.e., t₂ = ε² t, t₃ = ε³ t, etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see Kevorkian & Cole 1996; Bender & Orszag 1999).^[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,^[3] as described next.

A solution ${\displaystyle y\approx r\cos \theta }$ is sought in new coordinates ${\displaystyle (r,\theta )}$ where the amplitude ${\displaystyle r(t)}$ varies slowly and the phase ${\displaystyle \theta (t)}$ varies at an almost constant rate, namely ${\displaystyle d\theta /dt\approx 1.}$ Straightforward algebra finds the coordinate transform^{[citation needed]}

$${\displaystyle y=r\cos \theta +{\frac {1}{32))\varepsilon r^{3}\cos 3\theta +{\frac {1}{1024))\varepsilon ^{2}r^{5}(-21\cos 3\theta +\cos 5\theta )+{\mathcal {O))(\varepsilon ^{3})}$$

${\displaystyle dr/dt=0}$

$${\displaystyle {\frac {d\theta }{dt))=1+{\frac {3}{8))\varepsilon r^{2}-{\frac {15}{256))\varepsilon ^{2}r^{4}+{\mathcal {O))(\varepsilon ^{3}).}$$

That is, Duffing's oscillations are of constant amplitude ${\displaystyle r}$ but have different frequencies ${\displaystyle d\theta /dt}$ 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.^[5]