Multiple-scale analysis

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

Here the differences between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:^[1] Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2 y}{d t^2} + y + \varepsilon y^3 = 0,} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(0)=1, \qquad \frac{dy}{dt}(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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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,} with q = y(t) and p = dy/dt. Consequently, the Hamiltonian H(p, q) is a conserved quantity, a constant, equal to H = ½ + ¼ ε for the given initial conditions. This implies that both y and dy/dt have to be bounded: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.}

Straightforward perturbation-series solution

A regular perturbation-series approach to the problem proceeds by writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \varepsilon} gives the system of equations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{d^2 y_0}{dt^2} + y_0 &= 0,\\ \frac{d^2 y_1}{dt^2} + y_1 &= - y_0^3. \end{align}}

Solving these subject to the initial conditions yields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = O(\epsilon^{-1})} , the method of multiple-scale analysis is used. Introduce the slow scale t₁: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1 = \varepsilon t} and assume the solution y(t) is a perturbation-series solution dependent both on t and t₁, treated as: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t) = Y_0(t,t_1) + \varepsilon Y_1(t,t_1) + \cdots.}

So: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align}} using dt₁/dt = ε. Similarly: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2 y}{d t^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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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}}

Solution

The zeroth-order problem has the general solution: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_0(t,t_1) = A(t_1)\, e^{+it} + A^\ast(t_1)\, e^{-it},} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 of the preceding terms. The occurrence of secular terms can be prevented by imposing on the – yet unknown – amplitude A(t₁) the solvability condition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(t) = \cos \left[ \left( 1 + \tfrac38\, \varepsilon \right) t \right] + \mathcal{O}(\varepsilon),} using $t 1 = εt$ and valid for $εt = O(1)$ . This agrees with the nonlinear frequency changes found by employing the Lindstedt–Poincaré method.

This new solution is valid until Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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; 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\approx r\cos\theta} is sought in new coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (r,\theta)} where the amplitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r(t)} varies slowly and the phase Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta(t)} varies at an almost constant rate, namely Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\theta/dt\approx 1.} Straightforward algebra finds the coordinate transform^{[citation needed]} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)} transforms Duffing's equation into the pair that the radius is constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dr/dt=0} and the phase evolves according to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d\theta}{dt} = 1 + \frac 3 8 \varepsilon r^2 -\frac{15}{256}\varepsilon^2r^4 +\mathcal O(\varepsilon^3).}

That is, Duffing's oscillations are of constant amplitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} but have different frequencies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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]

Notes

^ This example is treated in: Bender & Orszag (1999) pp. 545–551.
^ Bender & Orszag (1999) p. 551.
^ Lamarque, C.-H.; Touze, C.; Thomas, O. (2012), "An upper bound for validity limits of asymptotic analytical approaches based on normal form theory" (PDF), Nonlinear Dynamics, 70 (3): 1931–1949, doi:10.1007/s11071-012-0584-y, hdl:10985/7473
^ Roberts, A.J., Modelling emergent dynamics in complex systems, retrieved 2013-10-03
^ Roberts, A.J., Construct centre manifolds of ordinary or delay differential equations (autonomous), retrieved 2013-10-03

References

Kevorkian, J.; Cole, J. D. (1996), Multiple scale and singular perturbation methods, Springer, ISBN 978-0-387-94202-5
Bender, C.M.; Orszag, S.A. (1999), Advanced mathematical methods for scientists and engineers, Springer, pp. 544–568, ISBN 978-0-387-98931-0
Nayfeh, A.H. (2004), Perturbation methods, Wiley–VCH Verlag, ISBN 978-0-471-39917-9

External links

Carson C. Chow (ed.). "Multiple scale analysis". Scholarpedia.

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

[2] Bender & Orszag (1999) p. 551.

[3] Lamarque, C.-H.; Touze, C.; Thomas, O. (2012), "An upper bound for validity limits of asymptotic analytical approaches based on normal form theory" (PDF), Nonlinear Dynamics, 70 (3): 1931–1949, doi:10.1007/s11071-012-0584-y, hdl:10985/7473

[4] Roberts, A.J., Modelling emergent dynamics in complex systems, retrieved 2013-10-03

[5] Roberts, A.J., Construct centre manifolds of ordinary or delay differential equations (autonomous), retrieved 2013-10-03

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