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

: $\frac + (\alpha^2 + \omega^2 \sgn \cos(t))y = 0$

or

: $\frac + ( 1 + r f(t;a,b) ) y = 0$

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

: $\frac + \left( 1 + r \frac \right) y = 0.$

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

: $\lambda^2 - 2 \lambda \cosh(\sqrt) \cos(\sqrt) + 1 = 0 .$

The determinant of the Floquet matrix is 1, implying that origin is a center if $, \cosh(\sqrt) \cos(\sqrt), < 1$
and a saddle node otherwise.

References

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Ordinary differential equations