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

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

or

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

where

$f(t;a,b)=-1+2H_{a}(t\mod (a+b))$

and $H_{c}(t)$ is the Heaviside function shifted to $c$. Another version is

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

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.