Meissner equation

Last updated October 24, 2020

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. One is as

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

or

{\frac {d^{2}y}{dt^{2}}}+(1+rf(t;a,b))y=0

where

f(t;a,b)=-1+2H_{a}(t\mod (a+b))

and $H_{c}(t)$ is the Heaviside function shifted to $c$ . Another version is

{\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 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 equation

\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 $|\cosh({\sqrt {r}})\cos({\sqrt {r}})|<1$ and a saddle node otherwise.

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References

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

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[1] Richards, J. A. (1983). Analysis of periodically time-varying systems. Springer-Verlag. ISBN 9783540116899. LCCN 82005978.

[2] E. Meissner (1918). "Ueber Schüttelerscheinungen in Systemen mit periodisch veränderlicher Elastizität". Schweiz. Bauzeit. 72 (11): 95–98.