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
or
where
and is the Heaviside function shifted to. Another version is
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, the Floquet exponents are roots of the quadratic equation
The determinant of the Floquet matrix is 1, implying that origin is a center if
and a saddle node otherwise.