= Instability tongue
An instability tongue is a tongue-shaped region in parameter space where a periodically forced linear system has an exponentially growing mode. In the <Mathieu equation>
$$
u''+(a-2q\cos2t)u=0,
$$
<Floquet theory> detects instability through a Floquet multiplier of modulus greater than one. The exactly zero solution remains zero, and initial data on a decaying mode need not grow.
Near the principal resonance $a=1$ with small $q$, multiple-scale analysis gives the leading growth rate
$$
\lambda\approx\frac12\sqrt{q^2-(a-1)^2}.
$$
It is real and nonzero inside the leading region $|a-1|<|q|$. The exact boundary has corrections beyond this approximation. The width therefore opens from zero at the resonance point, giving the tongue shape.
Unlike additive resonance, the drive here changes a coefficient of the equation. Detuning $a-1$ competes with that parametric coupling; damping and nonlinear saturation can further change the instability region and long-time behavior.<Asymptotic Analysis and Singular Perturbation Theory by John K. Hunter>[Section5.2, parametrically forced Mathieu oscillator and detuning]{ref}
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