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
d 2 y d t 2 + ( α 2 + ω 2 sgn cos ( t ) ) y = 0 {\displaystyle {\frac {d^{2}y}{dt^{2}}}+(\alpha ^{2}+\omega ^{2}\operatorname {sgn} \cos(t))y=0}
or
d 2 y d t 2 + ( 1 + r f ( t ; a , b ) ) y = 0 {\displaystyle {\frac {d^{2}y}{dt^{2}}}+(1+rf(t;a,b))y=0}
where
f ( t ; a , b ) = − 1 + 2 H a ( t mod ( a + b ) ) {\displaystyle f(t;a,b)=-1+2H_{a}(t\mod (a+b))}
and H c ( t ) {\displaystyle H_{c}(t)} is the Heaviside function shifted to c {\displaystyle c} . Another version is
d 2 y d t 2 + ( 1 + r sin ( ω t ) | sin ( ω t ) | ) y = 0. {\displaystyle {\frac {d^{2}y}{dt^{2}}}+\left(1+r{\frac {\sin(\omega t)}{|\sin(\omega t)|}}\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 {\displaystyle a=b=1} , the Floquet exponents are roots of the quadratic equation
λ 2 − 2 λ cosh ( r ) cos ( r ) + 1 = 0. {\displaystyle \lambda ^{2}-2\lambda \cosh({\sqrt {r}})\cos({\sqrt {r}})+1=0.}
The determinant of the Floquet matrix is 1, implying that origin is a center if | cosh ( r ) cos ( r ) | < 1 {\displaystyle |\cosh({\sqrt {r}})\cos({\sqrt {r}})|<1}
and a saddle node otherwise.
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