= Primary instability tongue of a weak Mathieu oscillator
{title2=$\omega=\frac12+\epsilon k,\quad |k|<\frac12$}
For $y''+(\omega^2+\epsilon\cos t)y=0$, use the <method of multiple scales> with $T=\epsilon t$ and $y_0=a(T)\cos(t/2)+b(T)\sin(t/2)$. Removing <secular terms> gives $a_T=(k-1/2)b$ and $b_T=-(k+1/2)a$. The slow <eigenvalues> satisfy $\lambda^2=1/4-k^2$, so an exponentially growing mode exists inside the displayed leading instability interval. At its leading endpoints the slow matrix is nilpotent; higher-order corrections are needed to determine the exact edges. <Floquet theory> distinguishes this exponential instability from algebraic growth at a repeated multiplier.
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