14.1 Regular perturbation and secular terms
For \(x' = f(x, \varepsilon )\) with \(f\) smooth, the solution has an expansion \(x = x_0 + \varepsilon x_1 + \varepsilon ^2x_2 + \cdots \), obtained by substituting and equating powers of \(\varepsilon \): \(x_0\) solves the unperturbed problem, and each \(x_k\) solves a linear equation, the variational equation of Theorem 3.5.2 with a forcing term. By the dependence-on-parameters results of Section 3.5 and Gronwall’s inequality, the truncated expansion is accurate to \(O(\varepsilon ^{N+1})\) on any fixed interval \([0, T]\). It can fail badly on long intervals.
Example 14.1.1. For the Duffing oscillator \(x'' + x + \varepsilon x^3 = 0\), \(x(0) = a\), \(x'(0) = 0\), the zeroth order is \(x_0 = a\cos t\), and the first order satisfies \[ x_1'' + x_1 = -a^3\cos ^3t = -\tfrac 34a^3\cos t - \tfrac 14a^3\cos 3t, \qquad x_1(0) = x_1'(0) = 0 . \] The forcing \(\cos t\) is resonant (Example 5.5.1) and produces the term \(-\frac 38a^3t\sin t\) in \(x_1\). This secular term grows without bound, so the expansion \(x_0 + \varepsilon x_1\) is useless once \(\varepsilon t\) is of order one, although the true solution is periodic and bounded (the energy \(\frac 12x'^2 + \frac 12x^2 + \frac 14\varepsilon x^4\) is conserved). The secular term is an artefact: the true solution oscillates at a slightly different frequency, and expanding \(\cos (\omega t)\) with \(\omega = 1 + \varepsilon \omega _1\) in powers of \(\varepsilon \) produces \(-\varepsilon \omega _1t\sin t\).
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