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The '''Meissner equation''' is a linear [[ordinary differential equation]] that is a special case of [[Hill differential equation|Hill's equation]] with the periodic function given as a square wave.<ref>{{cite book |
title=Analysis of periodically time-varying systems |
author=Richards, J. A. |
isbn=9783540116899 |
lccn=82005978 |
year=1983 |
publisher=Springer-Verlag
}} </ref> <ref>
{{cite article | author=E. Meissner |
title=Ueber Schüttelerscheinungen in Systemen mit periodisch veränderlicher Elastizität |
journal=Schweiz. Bauzeit. |
volume=72 |
number=11 |
year=1918 |
pages=95–98
}}
</ref> There are many ways to write the Meissner equation. One
is as

: <math> \frac{d^2y}{dt^2} + (\alpha^2 + \omega^2 \sgn \cos(t)) = 0 </math>

or

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

where
: <math> f(t;a,b) = -1 + 2 H_a( t \mod (a+b) ) </math>
and <math> H_c(t) </math> is the Heaviside function shifted to <math>c</math>. Another version is

: <math> \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> a = b = 1</math>, the [[Floquet theory|Floquet exponents]] are roots of the quadratic equation

: <math> \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> |\cosh(\sqrt{r}) \cos(\sqrt{r})| < 1 </math>
and a saddle node otherwise.

== References ==
{{reflist}}

[[Category:Ordinary differential equations]]

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160.45.33.182 at 05:16, 11 April 2013