Floquet theory

Remarks

The 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. The change of variables $x = P(t)\, z$ turns the system into $z' = B z$ with constant coefficients. All 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. Example: 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. Non-example: if $A$ is not periodic, the solutions need not have this form. For $A(t) = 2t$ 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. Over $\R$ one can take $B$ real if one allows $P$ to have period $2T$.