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 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 ${\displaystyle a=b=1}$, 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.

References

1. Richards, J. A. (1983). Analysis of periodically time-varying systems. Springer-Verlag. ISBN 9783540116899. LCCN 82005978.
2. E. Meissner (1918). "Ueber Schüttelerscheinungen in Systemen mit periodisch veränderlicher Elastizität". Schweiz. Bauzeit. 72 (11). pp. 95–98.