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Meissner equation

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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 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 {\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.

References

  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.
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