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Parametric resonance is growth caused by periodic variation of a system coefficient, rather than an additive forcing term. In the Mathieu equationthe stiffness varies near twice the natural angular frequency. Put $T=ϵt$ and $u_0=a(T)\cos t+b(T)\sin t$. The order-$ϵ$ correction has resonant coefficients $-2b_T-a$ in the cosine component and $2a_T+b$ in the sine component. Eliminating them givesThe slow eigenvalues are $\pm 1/2$, so an initial component in the growing direction is amplified as $e^{ϵt/2}$ 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.^[ref]