The Meissner equation is a linear

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...

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 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 In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where a and q are parameters. They were first introduced by Émile Léonard Mathieu, ...

. 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

{{reflist Ordinary differential equations