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Floquet Analysis


Given a system of ordinary differential equations of the form

 d/(dt)[x; y; v_x; v_y]=-[0 0 -1 0; 0 0 0 -1; Phi_(xx)(t) Phi_(yx)(t) 0 0; Phi_(xy)(t) Phi_(yy)(t) 0 0][x; y; v_x; v_y]
(1)

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

 [x(t); y(t); v_x(t); v_y(t)]=[x_0; y_0; v_(x0); v_(y0)]e^(mut)P_mu(t),
(2)

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

 x^..+g(t)x=0,
(3)

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

x(t)=w(t)e^(ipsi(t))
(4)
x^.=(w^.+iwpsi^.)e^(ipsi)
(5)
x^..=[w^..+iw^.psi^.+i(w^.psi^.+wpsi^..+iwpsi^.^2)]e^(ipsi)
(6)
=[(w^..-wpsi^.^2)+i(2w^.psi^.+wpsi^..)]e^(ipsi).
(7)

Plugging these into (◇) gives

 w^..+2iw^.psi^.+w(g+ipsi^..-psi^.^2)=0,
(8)

so the real and imaginary parts are

 w^..+w(g-psi^.^2)=0
(9)
 2w^.psi^.+wpsi^..=0.
(10)

From (◇),

(2w^.)/w+(psi^..)/(psi^.)=2d/(dt)(lnw)+d/(dt)[ln(psi^.)]
(11)
=d/(dt)ln(psi^.w^2)
(12)
=0.
(13)

Integrating gives

 psi^.=C/(w^2),
(14)

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

 psi=int_(t_0)^t(dt)/(w^2).
(15)

The real solution is then

 x(t)=w(t)cos[psi(t)],
(16)

so

x^.=w^.cospsi-wpsi^.sinpsi
(17)
=w^.x/w-wpsi^.sinpsi
(18)
=w^.x/w-w1/(w^2)sinpsi
(19)
=w^.x/w-1/wsinpsi
(20)

and

1=cos^2psi+sin^2psi
(21)
=x^2w^(-2)+[w(w^.x/w-x^.)]^2
(22)
=x^2w^(-2)+(w^.x-wx^.)^2
(23)
=I(x,x^.,t),
(24)

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

 w^..+g(t)w-1/(w^3)=0,
(25)

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


See also

Floquet's Theorem, Hill's Differential Equation

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 727, 1972.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.Margenau, H. and Murphy, G. M. The Mathematics of Physics and Chemistry, 2 vols. Princeton, NJ: Van Nostrand, 1956-64.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 556-557, 1953.

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Floquet Analysis

Cite this as:

Weisstein, Eric W. "Floquet Analysis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FloquetAnalysis.html

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