Floquet multiplier

Remarks

A 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 $t$. Every multiplier is nonzero, and by Liouville's formula their product is $\det M = \exp\left(\int_0^T \tr A(t)\, dt\right)$. A number $\lambda$ with $e^{\lambda T} = \rho$ is called a Floquet exponent. All solutions tend to $0$ if and only if every multiplier has absolute value less than $1$.

Example: 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 $A$. For 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 $e^{2\pi}$.

Non-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 $t$.

For a $T$-periodic orbit $\gamma$ of an autonomous system $x' = f(x)$, the multipliers of $\gamma$ are those of the linear system $y' = Df(\gamma(t))\, y$. One of them is always $1$, with the eigenvector $f(\gamma(0))$.