= Parametric resonance
{wiki}
<Parametric resonance> is growth caused by periodic variation of a system coefficient, rather than an additive forcing term. In the <Mathieu equation>
$$
u''+[1+2\epsilon\cos(2t)]u=0,
$$
the stiffness varies near twice the natural <angular frequency>. Put $T=\epsilon t$ and $u_0=a(T)\cos t+b(T)\sin t$. The order-$\epsilon$ correction has resonant coefficients $-2b_T-a$ in the cosine component and $2a_T+b$ in the sine component. Eliminating them gives
$$
a_T=-b/2,\qquad b_T=-a/2.
$$
The slow <eigenvalues> are $\pm1/2$, so an initial component in the growing direction is amplified as $e^{\epsilon t/2}$ at leading order. The exactly zero solution remains zero. Detuning competes with this coupling and produces an <instability tongue>, while <Floquet theory> gives the exact periodic-coefficient stability criterion. A weak-parameter growth approximation needs a stated time interval and can fail once neglected nonlinearities become important.<Asymptotic Analysis and Singular Perturbation Theory by John K. Hunter>[§5.2, parametrically forced Mathieu oscillator]{ref}
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