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

[math]\displaystyle{ \frac{d^2y}{dt^2} + (\alpha^2 + \omega^2 \sgn \cos(t))y = 0 }[/math]

or

[math]\displaystyle{ \frac{d^2y}{dt^2} + ( 1 + r f(t;a,b) ) y = 0 }[/math]

where

[math]\displaystyle{ f(t;a,b) = -1 + 2 H_a( t \mod (a+b) ) }[/math]

and [math]\displaystyle{ H_c(t) }[/math] is the Heaviside function shifted to [math]\displaystyle{ c }[/math]. Another version is

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

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 [math]\displaystyle{ a = b = 1 }[/math], the Floquet exponents are roots of the quadratic equation

[math]\displaystyle{ \lambda^2 - 2 \lambda \cosh(\sqrt{r}) \cos(\sqrt{r}) + 1 = 0 . }[/math]

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