Meissner equation

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. There are many ways to write the Meissner equation. One is as


 * $$ \frac{d^2y}{dt^2} + (\alpha^2 + \omega^2 \sgn \cos(t))y = 0 $$

or


 * $$ \frac{d^2y}{dt^2} + ( 1 + r f(t;a,b) ) y = 0 $$

where
 * $$ f(t;a,b) = -1 + 2 H_a( t \mod (a+b) ) $$

and $$ H_c(t) $$ is the Heaviside function shifted to $$c$$. Another version is


 * $$ \frac{d^2y}{dt^2} + \left(  1 + r \frac{\sin( \omega t)}{|\sin(\omega t)|} \right) y = 0. $$

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 $$ a = b = 1$$, the Floquet exponents are roots of the quadratic equation


 * $$ \lambda^2 - 2 \lambda \cosh(\sqrt{r}) \cos(\sqrt{r}) + 1 = 0 .$$

The determinant of the Floquet matrix is 1, implying that origin is a center if $$ |\cosh(\sqrt{r}) \cos(\sqrt{r})| < 1 $$ and a saddle node otherwise.