# Multiple scale analysis

Curator and Contributors

1.00 - Carson C. Chow

Multiple-scale analysis is a global perturbation scheme that is useful in systems characterized by disparate time scales, such as weak dissipation in an oscillator. These effects could be insignificant on short time scales but become important on long time scales. Classical perturbation methods generally break down because of resonances that lead to what are called secular terms.

The first scheme to address this problem is what Van Dyke (1975) refers to as the method of strained coordinates. The method is sometimes attributed to Poincare, although Poincare credits the basic idea to the astronomer Lindstedt (Kevorkian and Cole, 1996). Lighthill introduced a more general version in 1949. Later Krylov and Bogoliubov and Kevorkian and Cole introduced the two-scale expansion, which is now the more standard approach.

## Secular growth

The classic oft-used example (Bender and Orszag, 1999; Kevorkian and Cole, 1996) for the use of multiple-scale analysis is the weakly nonlinear Duffing oscillator given by $\tag{1} \frac{d^2y}{dt^2}+y+\epsilon y^3=0$

with boundary conditions $$y(0)=1$$ and $$y'(0)=0$$ and the positive parameter $$\epsilon\ll 1\ .$$ Although the Duffing oscillator can be solved exactly in terms of elliptical functions, it serves as a paradigmatic example for demonstrating multiple-scale analysis.

First consider standard perturbation theory where it is assumed that an expansion in a power series $\tag{2} y=y_0+\epsilon y_1 +\epsilon^2 y_2 + \cdots = \sum_{n=0}^\infty \epsilon^n y_n$

exists where $$y_0(0)=1\ ,$$ $$y'_0(0)=0\ ,$$ $$y_n(0)=y'_n(0)=0\ ,$$ $$n>0\ .$$ The implicit assumption is that (2) is an asymptotic expansion so that the $$n+1$$th term becomes arbitrarily small compared to the $$n$$th term as $$\epsilon \rightarrow 0\ .$$ Substituting the expansion (2) into (1) and equating coefficients of like powers of $$\epsilon\ ,$$ gives for the first two orders of $$\epsilon\ :$$ $\tag{3} y''_0+y_0=0$

$\tag{4} y''_1+y_1=-y_0^3$

The solution to (3) that satisfies the boundary conditions is $$y_0(t)=\cos(t)\ .$$ Hence, (4) becomes $\tag{5} y''_1+y_1= -\left[ \frac{1}{4} \cos 3t+\frac{3}{4}\cos t \right]$

where the identity $$\cos^3 t= \frac{1}{4} \cos 3t+\frac{3}{4}\cos t$$ is used. The general solution to (5) is then given by $\tag{6} y_1(t)=\frac{1}{32}\cos 3t -\frac{1}{32} \cos t - \frac{3}{8} t \sin t$

Hence, the first order perturbative solution to the Duffing oscillator is given by $\tag{7} y(t)\simeq \cos t + \epsilon\left[\frac{1}{32}\cos 3t -\frac{1}{32} \cos t - \frac{3}{8} t \sin t \right]$

An examination of (7) shows that the perturbation theory will break down when $$t\sim 1/\epsilon$$ since $$\epsilon y_1(t)$$ will be of the same order as $$y_0$$ and violate the uniformity of the convergence of the asymptotic expansion. This $$t$$ dependence in $$y_1(t)$$ is known as secular growth and arises whenever there is a resonance between $$y_0$$ and $$y_1\ .$$

At this point it should be noted that the secular growth is entirely an artifact of the perturbation scheme. The Duffing oscillator (1) is well behaved and always remains bounded. This can be seen from the fact that it contains a conserved quantity. Multiplying (1) by $$\frac{dy}{dt}$$ converts the equation into a total derivative and integrating gives $\tag{8} E=\frac{1}{2}\left(\frac{dy}{dt}\right)^2+\frac{1}{2}y^2+\frac{1}{4}\epsilon y^4$

The use of the symbol $$E$$ is deliberate since the Duffing oscillator is a Hamiltonian system with total energy $$E$$ given by (8).

## Motivation for multiple-scale analysis

So why does the perturbation expansion breakdown? The resolution is that the individual terms in the sum may be secular but the sum remains bounded. For the Duffing oscillator, the lowest order effect of the cubic term is to distort the frequency so a more appropriate approximation would be of the form $$y_0\simeq \cos(t + a\epsilon t)$$ where $$a$$ is a constant. Substituting this ansatz into (1) yields $-(1+a\epsilon)^2\cos(1+a\epsilon)t +\cos(1+a\epsilon)t + \epsilon\left(\frac{1}{4}\cos 3(1+a\epsilon)t +\frac{3}{4}\cos(1+a\epsilon)t\right)=0$ Equating terms order by order in $$\epsilon$$ gives $$a=\frac{3}{8}$$ and thus $$y= \cos(1+\frac{3}{8}\epsilon) t+O(\epsilon)$$ would be the expected lowest order approximation to the solution of the Duffing oscillator.

## Two-scale expansion

The ad hoc procedure of obtaining an ansatz to use can be made systematic using a two-scale expansion. The trick is to introduce a new variable $$\tau=\epsilon t\ .$$ This variable is called the slow time because it does not become significant until $$t\sim 1/\epsilon\ .$$ Then take an expansion of the form $\tag{9} y(t)=Y_0(t,\tau)+\epsilon Y_1(t,\tau)+\cdots$

Using the chain rule this implies $\frac{dy}{dt}=\frac{\partial Y_0}{\partial t} + \frac{\partial Y_0}{d\tau}\frac{d \tau}{dt} + \epsilon\left(\frac{\partial Y_1}{\partial t}+\frac{\partial Y_1}{d\tau}\frac{d \tau}{dt}\right) +\cdots$ Since $$d\tau/dt=\epsilon$$ then $\tag{10} \frac{dy}{dt}=\frac{\partial Y_0}{\partial t} + \epsilon\left(\frac{\partial Y_0}{\partial \tau}+\frac{\partial Y_1}{\partial t}\right) +\cdots$

Substituting (9) into (1) using (10) and equating terms of like powers of $$\epsilon$$ gives $\tag{11} \frac{\partial^2Y_0}{\partial t^2}+Y_0=0$

$\tag{12} \frac{\partial^2Y_1}{\partial t^2}+Y_1=-Y_0^3-2\frac{\partial^2Y_0}{\partial\tau\partial t}$

The general real solution to (11) is $\tag{13} Y_0=a(\tau)e^{it}+a^*(\tau)e^{-it}$

Substituting (13) into (12) gives $\tag{14} \frac{\partial^2Y_1}{\partial t^2}+Y_1 = -a^3e^{3it}-a^{*3}e^{-3it} - (2i\frac{\partial a}{\partial \tau} + 3 a^2a^{*})e^{it}+ (2i\frac{\partial a^*}{\partial \tau} -3aa^{*2})e^{-it}$

To ensure that there are no secular terms in $$Y_1(t)\ ,$$ the resonant terms on the right hand side of (14) are forced to be zero, i.e. $\tag{15} 2i\frac{\partial a}{\partial \tau} + 3 |a|^2a=0$

$\tag{16} 2i\frac{\partial a^*}{\partial \tau} -3|a|^2a^{*}=0$

Set $$a(\tau)=r(\tau)e^{i\theta(\tau)}\ ,$$ insert into (15) or (16) and equate real and imaginary parts to obtain $\tag{17} \begin{array}{lcl} \frac{dr}{d \tau}&=&0\\ \frac{d\theta}{d \tau}&=&\frac{3}{2}r^2 \end{array}$

Hence $$r=r(0)$$ and $$a(\tau)=a(0)e^{i\theta(0)+3ir^2(0)\tau/2}$$ giving $Y_0=2r(0) \cos(\theta(0)+3r^2(0)\tau/2+t)$ Applying the initial conditions gives $$r(0)=1/2$$ and $$\theta(0)=0$$ giving the approximation $y(t)=\cos(t+3\epsilon t/8) + O(\epsilon).$

Higher order terms in the expansion can be generated similarly by systematically eliminating secular terms at each order to produce a uniform perturbation expansion. A more general two-scale scheme uses for the fast time scale the form $$t^+=(1+\epsilon^2\omega_2 + \cdots)t\ .$$ This more general scheme will only differ from the simpler scheme when going beyond first order.

## References

• C. M. Bender and S. A. Orzag,Advanced Mathematical Methods for Scientists and Engineers, Springer-Verlag, New York, 1999.
• J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York, 1996.
• M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, 1975.

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