7.5 Mathieu’s equation and parametric resonance

The simplest Hill equation is Mathieu’s equation \[ y'' + (\delta + \varepsilon \cos t)y = 0 , \] with period \(T = 2\pi \). For \(\varepsilon = 0\) and \(\delta = \omega ^2 > 0\), \(\Delta = 2\cos 2\pi \omega \), which equals \(\pm 2\) exactly when \(\omega = n/2\), that is \(\delta = n^2/4\). From each of these points a region of instability, an Arnold tongue, opens as \(\varepsilon \) grows (Figure 10). Inside the tongues the oscillator is driven to grow by the periodic variation of its own stiffness.

inverted pendulum
stabilised in the
δ𝜀−00120011nststnnn0.2.2.4.8.2.6araa===.555rbbowlele123 white strip — Figure 10: Stability chart of Mathieu’s equation \(y'' + (\delta + \varepsilon \cos t)y = 0\), computed from the monodromy matrix on a \(301 \times 161\) grid. Shaded: \(|\Delta | > 2\), unstable. The tongues touch the axis at \(\delta = n^2/4\); the first is wide (its edges are \(\delta \approx \frac 14 \pm \frac \varepsilon 2\)), the higher ones are narrow (width of order \(\varepsilon ^n\)). For \(\delta < 0\) the unstretched oscillator is the inverted pendulum, unstable, but a thin stable region extends to the left of \(\delta = 0\) (the boundary \(\delta \approx -\varepsilon ^2/2\) is dashed); the dot marks the case \(\delta = -0.03\), \(\varepsilon = 0.3\) of Software 7.6.3.

The first tongue by harmonic balance

Near \(\delta = \frac 14\) the free oscillation has frequency \(\frac 12\), half the forcing frequency, and the boundary solutions are \(4\pi \)-periodic. Seek \(y \approx a\cos \frac t2 + b\sin \frac t2\). Since \(\cos t\cos \frac t2 = \frac 12\bigl (\cos \frac {3t}2 + \cos \frac t2\bigr )\) and \(\cos t\sin \frac t2 = \frac 12\bigl (\sin \frac {3t}2 - \sin \frac t2\bigr )\), substituting and keeping only the terms at frequency \(\frac 12\) gives \[ \Bigl (\delta - \frac 14 + \frac \varepsilon 2\Bigr )a = 0, \qquad \Bigl (\delta - \frac 14 - \frac \varepsilon 2\Bigr )b = 0 . \] A non-zero solution exists only on the lines \(\delta = \frac 14 \pm \frac \varepsilon 2\), which are the edges of the first tongue to first order in \(\varepsilon \). This is the rule of parametric resonance: a parameter varying at twice the natural frequency destabilises the oscillation most strongly. A child on a swing pumps twice per cycle, standing at the bottom and squatting at the top, which varies the effective length at twice the swing frequency.

The inverted pendulum

For \(\delta < 0\), the equation without forcing describes an inverted pendulum, \(y'' = |\delta |y\), which falls. Kapitza (1951) showed that vibrating the support vertically can hold it upright. With support displacement \(a\cos \omega t\), pendulum length \(\ell \) and time \(\tau = \omega t\), the linearised equation for the angle from the upright position is Mathieu’s equation with \[ \delta = -\frac {g}{\ell \omega ^2} < 0, \qquad \varepsilon = \frac a\ell . \] For small \(\varepsilon \) the stable strip in Figure 10 is \(-\varepsilon ^2/2 < \delta < 0\), and the condition for stability becomes \(a^2\omega ^2 > 2g\ell \): the support must be shaken fast enough. The method of averaging, which explains the strip by an effective potential, is described in the last section.

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