Multiple-scale analysis explained

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] $$\backslash frac\; +\; y\; +\; \backslash varepsilon\; y^3\; =\; 0,$$ $$y(0)=1,\; \backslash qquad\; \backslash frac(0)=0,$$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 < ε ≪ 1. The undamped Duffing equation is known to be a Hamiltonian system:$$\backslash frac=-\backslash frac,\; \backslash qquad\; \backslash frac=+\backslash frac,\; \backslash quad\; \backslash text\; \backslash quad\; H\; =\; \backslash tfrac12\; p^2\; +\; \backslash tfrac12\; q^2\; +\; \backslash tfrac14\; \backslash varepsilon\; q^4,$$with q = y(t) and p = dy/dt. Consequently, the Hamiltonian H(p, q) is a conserved quantity, a constant, equal to H =  +  ε for the given initial conditions. This implies that both y and dy/dt have to be bounded:$$\backslash left|\; y(t)\; \backslash right|\; \backslash le\; \backslash sqrt\; \backslash quad\; \backslash text\; \backslash quad\; \backslash left|\; \backslash frac\; \backslash right|\; \backslash le\; \backslash sqrt\; \backslash qquad\; \backslash text\; t.$$

Straightforward perturbation-series solution

A regular perturbation-series approach to the problem proceeds by writing $y(t)\; =\; y\_0(t)\; +\; \backslash varepsilon\; y\_1(t)\; +\; \backslash mathcal(\backslash varepsilon^2)$ and substituting this into the undamped Duffing equation. Matching powers of $\backslash varepsilon$ gives the system of equations

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

Note that the last term between the square braces is secular: it grows without bound for large |t|. In particular, for

t = O(\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

t = O(\epsilon ^{-1} ) , the method of multiple-scale analysis is used. Introduce the slow scale t₁:$$t\_1\; =\; \backslash varepsilon\; t$$and assume the solution y(t) is a perturbation-series solution dependent both on t and t₁, treated as:$$y(t)\; =\; Y\_0(t,t\_1)\; +\; \backslash varepsilon\; Y\_1(t,t\_1)\; +\; \backslash cdots.$$

So:using dt₁/dt = ε. Similarly:$$\backslash frac\; =\; \backslash frac\; +\; \backslash varepsilon\; \backslash left(2\; \backslash frac\; +\; \backslash frac\; \backslash right)\; +\; \backslash mathcal(\backslash varepsilon^2).$$

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

Solution

The zeroth-order problem has the general solution:$$Y\_0(t,t\_1)\; =\; A(t\_1)\backslash ,\; e^\; +\; A^\backslash ast(t\_1)\backslash ,\; e^,$$with A(t₁) a complex-valued amplitude to the zeroth-order solution Y₀(t, t₁) and i² = −1. Now, in the first-order problem the forcing in the right hand side of the differential equation is$$\backslash left[-3\backslash ,\; A^2\backslash ,\; A^\backslash ast\; -\; 2\backslash ,\; i\backslash ,\; \backslash frac\{dA\}\{dt\_1\}\; \backslash right]\backslash ,\; e^\; -\; A^3\backslash ,\; e^\; +\; c.c.$$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(t₁) the solvability condition$$-3\backslash ,\; A^2\backslash ,\; A^\backslash ast\; -\; 2\backslash ,\; i\backslash ,\; \backslash frac\; =\; 0.$$

The solution to the solvability condition, also satisfying the initial conditions and, is:$$A\; =\; \backslash tfrac\; 1\; 2\backslash ,\; \backslash exp\; \backslash left(\backslash tfrac\; 3\; 8\backslash ,\; i\; \backslash ,\; t\_1\; \backslash right).$$

As a result, the approximate solution by the multiple-scales analysis is$$y(t)\; =\; \backslash cos\; \backslash left[\backslash left(1\; +\; \backslash tfrac38\backslash ,\; \backslash varepsilon\; \backslash right)\; t\; \backslash right]\; +\; \backslash mathcal(\backslash varepsilon),$$using and valid for . This agrees with the nonlinear frequency changes found by employing the Lindstedt–Poincaré method.

This new solution is valid until

t = O(\epsilon ^{-2} ) . Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, i.e.,,, etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see ;).^[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, as described next.

A solution

y ≈ r\cos\theta is sought in new coordinates (r,\theta) where the amplitude r(t) varies slowly and the phase \theta(t) varies at an almost constant rate, namely d\theta/dt ≈ 1 . Straightforward algebra finds the coordinate transform$$y=r\backslash cos\backslash theta\; +\backslash frac1\backslash varepsilon\; r^3\backslash cos3\backslash theta\; +\backslash frac1\backslash varepsilon^2r^5(-21\backslash cos3\backslash theta+\backslash cos5\backslash theta)+\backslash mathcal\; O(\backslash varepsilon^3)$$transforms Duffing's equation into the pair that the radius is constant dr/dt =0 and the phase evolves according to$$\backslash frac\; =\; 1\; +\; \backslash frac\; 3\; 8\; \backslash varepsilon\; r^2\; -\backslash frac\backslash varepsilon^2r^4\; +\backslash mathcal\; O(\backslash varepsilon^3).$$

That is, Duffing's oscillations are of constant amplitude

r but have different frequencies d\theta/dt depending upon the amplitude.

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.

See also

• Method of matched asymptotic expansions
• WKB approximation
• Method of averaging
• Krylov–Bogoliubov averaging method

Notes and References

1. This example is treated in: Bender & Orszag (1999) pp. 545–551.
2. Bender & Orszag (1999) p. 551.