Meissner equation explained

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. Oneis as

(d ^{2y})/(dt ^{2}) + (\alpha ^{2} + \omega ^{2} sgn \cos(t))y = 0

or

(d ^{2y})/(dt ^{2}) + ( 1 + r f(t;a,b) ) y = 0

where

f(t;a,b) = -1 + 2 H _{a(} t \mod (a + b) ) and H _{c(t)} is the Heaviside function shifted to c . Another version is(d ^{2y})/(dt ^{2}) + \left( 1 + r (\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

a = b = 1 , the Floquet exponents are roots of the quadratic equationλ ^{2} - 2 λ \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.

Notes and References

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