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  "body": "\n## Definition\n\nLet $T \u003e 0$ and let $A \\colon \\R \\to \\R^{n \\times n}$ be continuous and [[periodic|0ATF6Q0633]] of period $T$.\nConsider the linear system $x' = A(t)\\, x$ on $\\R^n$.\nLet $\\Phi$ be its [[fundamental matrix|0H1GN6P9R5]] with $\\Phi(0) = I$.\nThe *monodromy matrix* of the system is $M = \\Phi(T)$.\nA *Floquet multiplier* of the system is an [[eigenvalue|0BQAA2BK3Q]] of $M$, counted with its algebraic multiplicity.\n\n## Remarks\n\nA complex number $\\rho$ is a Floquet multiplier if and only if the system has a nonzero solution with $x(t + T) = \\rho\\, x(t)$ for all $t$.\nEvery multiplier is nonzero, and by Liouville's formula their product is $\\det M = \\exp\\left(\\int_0^T \\tr A(t)\\, dt\\right)$.\nA number $\\lambda$ with $e^{\\lambda T} = \\rho$ is called a Floquet exponent.\nAll solutions tend to $0$ if and only if every multiplier has absolute value less than $1$.\n\nExample: for a constant matrix $A$, one has $\\Phi(t) = e^{tA}$, so $M = e^{TA}$ and the multipliers are $e^{\\lambda T}$ for the eigenvalues $\\lambda$ of $A$.\nFor the scalar equation $x' = (1 + \\cos t)\\, x$ with $T = 2\\pi$, one has $\\Phi(t) = e^{t + \\sin t}$, and the only multiplier is $e^{2\\pi}$.\n\nNon-example: for $x' = t\\, x$, the coefficient is not periodic, so the definition does not apply; the ratio $x(t + T)/x(t) = e^{tT + T^2/2}$ depends on $t$.\n\nFor a $T$-periodic orbit $\\gamma$ of an autonomous system $x' = f(x)$, the multipliers of $\\gamma$ are those of the linear system $y' = Df(\\gamma(t))\\, y$.\nOne of them is always $1$, with the eigenvector $f(\\gamma(0))$.\n",
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