14.3 Averaging and multiple scales
For a weakly perturbed oscillator \(x'' + x = \varepsilon h(x, x')\), write \(x = r\cos \theta \), \(x' = -r\sin \theta \) with \(\theta = t + \varphi \). The exact equations for \(r\) and \(\varphi \) are \[ r' = -\varepsilon h\sin \theta , \qquad r\varphi ' = -\varepsilon h\cos \theta , \] with \(h\) evaluated at \((r\cos \theta , -r\sin \theta )\). Since \(r\) and \(\varphi \) change slowly, at rate \(O(\varepsilon )\), while \(\theta \) goes round once per period, one may average the right-hand sides over \(\theta \) at fixed \(r\): \[ r' = -\frac {\varepsilon }{2\pi }\int _0^{2\pi }h(r\cos \theta , -r\sin \theta )\sin \theta \,d\theta , \qquad \varphi ' = -\frac {\varepsilon }{2\pi r}\int _0^{2\pi }h(r\cos \theta , -r\sin \theta )\cos \theta \,d\theta . \] The averaging theorem states that the solutions of the averaged equations stay within \(O(\varepsilon )\) of the true \(r\) and \(\varphi \) for times of order \(1/\varepsilon \), and that a hyperbolic equilibrium of the averaged \(r\)-equation corresponds to a hyperbolic limit cycle of the original system. The method of multiple scales, introducing the slow time \(s = \varepsilon t\) explicitly, gives the same equations.
Example 14.3.1. For van der Pol’s equation, \(h = (1 - x^2)x'\) and \[ r' = -\frac \varepsilon {2\pi }\int _0^{2\pi }(1 - r^2\cos ^2\theta )(-r\sin \theta )\sin \theta \,d\theta = \frac {\varepsilon r}{2}\Bigl (1 - \frac {r^2}{4}\Bigr ) = \frac \varepsilon 8r(4 - r^2), \qquad \varphi ' = 0 . \] The averaged equation has the stable equilibrium \(r = 2\), the amplitude found by the energy argument of Section 12.4, and its linearisation \(r' \approx -\varepsilon (r - 2)\) gives the rate at which orbits approach the cycle. For Duffing’s \(h = -x^3\), the \(r\)-equation is \(r' = 0\) and \(\varphi ' = \frac 38\varepsilon r^2\), recovering Theorem 14.2.1. Averaging is also the method that explains Kapitza’s pendulum: averaging over the fast vibration produces an effective potential with a minimum at the upright position when \(a^2\omega ^2 > 2g\ell \).
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