Multiple-scale analysis: Difference between revisions

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Revision as of 04:30, 7 May 2022

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===Straightforward perturbation-series solution===

A regular [[perturbation theory|perturbation-series approach]] to the problem proceeds by writing <math display="inline">y(t) = y_0(t) + \varepsilon y_1(t) + \mathcal{O}(\varepsilon^2)</math> and substituting this into the undamped Duffing equation. Matching powers of <math display="inline">\varepsilon</math> gives the system of equations

<math display="block">\begin{align}

<math>\frac{d^2 y_0}{dt^2} + y_0 &= 0,</math>\\

<math>\frac{d^2 y_1}{dt^2} + y_1 &= - y_0^3</math>.

\end{align}</math>

<math>\frac{d^2 y_1}{dt^2} + y_1 = - y_0^3</math>.

Solving these subject to the initial conditions yields

<math display="block">

:<math>

y(t) = \cos(t)

+ \varepsilon \left[ \tfrac{1}{32} \cos(3t) - \tfrac{1}{32} \cos(t) - \underbrace{\tfrac38tfrac 3 8\, t\, \sin(t)}_\text{secular} \right]

+ \mathcal{O}(\varepsilon^2).

</math>

Note that the last term between the square braces is secular: it grows without bound for large |''t''|. In particular, for <math>t = O(\epsilonvarepsilon^{-1})</math> 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===