Mathieu equation ($u^{\prime \prime }+(a-2q\cos 2t)u=0$)

The Mathieu equation is the displayed linear differential equation with periodic coefficients. Floquet theory describes its bounded and exponentially growing solutions, while its periodic Mathieu characteristic values mark the edges of parametric resonance gaps. The forcing normalization matters: replacing $t$ by $t/2$ rescales both the constant and periodic coefficients.

The Mathieu characteristic values are the parameter values $a$ for which the Mathieu equation has the even periodic mode traditionally denoted ${\mathrm{ce}}_n$, or the odd periodic mode ${\mathrm{se}}_n$. They are eigenvalues of $-d^2/d{t}^2+2q\cos 2t$ with the corresponding periodic or antiperiodic boundary conditions. Their small-$q$ expansions can be derived by projecting onto a Fourier series and perturbing the resulting tridiagonal matrix.