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  "body": "\n## Definition\n\nLet $T \u003e 0$ and let $A : \\R \\to \\C^{n \\times n}$ be continuous and [[periodic|089KWVYD69]] with period $T$.\n*Floquet theory* is the theory of the [[linear system|0A6QZ367GX]] $x' = A(t)\\, x$, based on the following objects.\n\n1. For a [[fundamental matrix|0MQ24RBG93]] $\\Phi$, the matrix $t \\mapsto \\Phi(t + T)$ is again a fundamental matrix, so $\\Phi(t + T) = \\Phi(t)\\, M$ for a constant invertible matrix $M = \\Phi(0)^{-1} \\Phi(T)$, called the monodromy matrix.\n2. A Floquet representation of $\\Phi$ is a factorization $\\Phi(t) = P(t)\\, e^{tB}$ with a constant $B \\in \\C^{n \\times n}$ and the [[matrix exponential|0K665H5V1S]] $e^{tB}$. Here $P : \\R \\to \\C^{n \\times n}$ is continuously differentiable, $T$-periodic and invertible at every $t$.\n\nFloquet's theorem states that every fundamental matrix has a Floquet representation, with $B$ any matrix such that $e^{TB} = M$.\n\n## Remarks\n\nThe eigenvalues of $M$ are the characteristic multipliers; they do not depend on the choice of $\\Phi$, since another choice replaces $M$ by a similar matrix.\nThe change of variables $x = P(t)\\, z$ turns the system into $z' = B z$ with constant coefficients.\nAll solutions tend to $0$ if and only if all multipliers have modulus less than $1$; there is a nonzero $T$-periodic solution if and only if $1$ is a multiplier.\nExample: for $n = 1$ and $A(t) = 1 + \\cos t$, $T = 2\\pi$, the solution is $\\Phi(t) = e^{t + \\sin t}$, so $M = e^{2\\pi}$, $B = 1$ and $P(t) = e^{\\sin t}$, which is $2\\pi$-periodic.\nNon-example: if $A$ is not periodic, the solutions need not have this form.\nFor $A(t) = 2t$ the solution $e^{t^2}$ is not $P(t) e^{tB}$ with $P$ periodic, since then $\\abs{\\Phi(t)} \\le C e^{t \\operatorname{Re} B}$ would grow at most exponentially.\nOver $\\R$ one can take $B$ real if one allows $P$ to have period $2T$.\n",
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