Floquet Analysis -- from Wolfram MathWorld

Floquet Analysis

(1)

that are periodic in , the solution can be written as a linear combination of functions of the form

(2)

where is a function periodic with the same period as the equations themselves. Given an ordinary differential equation of the form

(3)

where is periodic with period , the ODE has a pair of independent solutions given by the real and imaginary parts of

Plugging these into (◇) gives

(8)

(9)

(10)

From (◇),

Integrating gives

(14)

where is a constant which must equal 1, so is given by

(15)

The real solution is then

(16)

and

which is an integral of motion. Therefore, although is not explicitly known, an integral always exists. Plugging (◇) into (◇) gives

(25)

which, however, is not any easier to solve than (◇).

References

Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 175, 1987.

Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, p. 32, 1983.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 556-557, 1953.

Floquet Analysis