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.[1] [2] 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 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 , 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.

References[edit]

  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). pp. 95–98.