Meissner equation - Alchetron, The Free Social Encyclopedia

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

or

d 2 y d t 2 + ( 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

d 2 y d t 2 + ( 1 + r sin ⁡ ( ω t ) | sin ⁡ ( ω t ) | ) 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

λ 2 − 2 λ cosh ⁡ ( r ) cos ⁡ ( r ) + 1 = 0.

The determinant of the Floquet matrix is 1, implying that origin is a center if | cosh ⁡ ( r ) cos ⁡ ( r ) | < 1 and a saddle node otherwise.

References

Meissner equation Wikipedia

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