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.^{[1] [2]} There are many ways to write the Meissner equation. One is as

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

or

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

where

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

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

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

The Meissner equation was first studied as a toy model of oscillations observed in the rod gear of electric trains ^[2] where the elasticity of the system could not reasonably be treated as a constant . It is also useful for understand resonance problems in the quantum mechanics of semiconductors and evolutionary biology under periodic environment switching.

Because the time-dependence is piecewise linear, many calculations can be performed exactly, unlike for the Mathieu equation. When ${\displaystyle a=b=1}$, the Floquet exponents are roots of the quadratic equation

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

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