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 equationFloquet 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 rateIt 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.^[ref]