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  "body": "\n## Summary\n\nFloquet theory studies linear systems $x' = A(t)\\, x$ whose coefficient matrix $A(t)$ is periodic in time.\nIts main result is that every [[fundamental matrix|0H1GN6P9R5]] is a periodic matrix times a matrix exponential $e^{tB}$.\nSo a periodic change of variables turns the system into one with constant coefficients.\nThe behaviour of solutions, such as stability, is read off from the eigenvalues of the monodromy matrix, the [[Floquet multipliers|FPS7YFGX32]].\nIt applies to the [[Hill equation|YE7H8DGDPY]], to the stability of periodic orbits and to waves in periodic media.\nSee the [[definition|GW8SBF37K1]].\n\n## Explanations\n\n### L1 human\n\nA child on a swing who pumps the legs periodically changes the equation of motion periodically in time.\nFloquet theory answers whether such a periodically forced linear system has growing or bounded motion.\nThe key is to look at the system once per period, like taking a photo every period.\nThe map from one photo to the next is a fixed matrix, the monodromy matrix, and its eigenvalues decide everything.\n",
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