The Meissner equation is a linear ordinary differential equation that is a special case of Hill's equation with the periodic function given as a square wave.[1][2] There are many ways to write the Meissner equation. One is as
or
where
and is the Heaviside function shifted to . Another version is
The Meissner equation was first studied as a toy problem for certain resonance problems. It is also useful for understand resonance problems in evolutionary biology.
Because the time-dependence is piecewise linear, many calculations can be performed exactly, unlike for the Mathieu equation. When , the Floquet exponents are roots of the quadratic equation
The determinant of the Floquet matrix is 1, implying that origin is a center if and a saddle node otherwise.