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

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

or

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

where

$f(t;a,b)=-1+2H_a(tmod(a+b))$

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

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

The Meissner equation was first studied as a toy model of oscillations observed in the rod gear of electric trains 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 $a=b=1$, 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.