= Monodromy matrix of a periodic linear system
{title2=$N=X(P)X(0)^{-1}$}
For the homogeneous <linear differential equation> $\dot Y=K(t)Y$ with continuous coefficient matrix of period $P>0$, the <monodromy matrix of a periodic linear system> is the one-period map from $Y(0)$ to $Y(P)$. If $X$ is any <fundamental matrix>,
$$
N=X(P)X(0)^{-1},\qquad Y(P)=NY(0).
$$
Its <eigenvalues> are the <Floquet multipliers>. Multipliers are nonzero, since a <fundamental matrix> is invertible. Changing the starting phase conjugates $N$, so the <Floquet multipliers> and <Jordan blocks> are independent of that choice.<Ordinary Differential Equations and Dynamical Systems by Gerald Teschl>[§3.6, printed pp. 91–93, equations (3.119), (3.122) and (3.127)]{ref}
This turns a continuous-time problem into repeated <matrix multiplication>:
$$
Y(nP)=N^nY(0).
$$
If $Nv=mv$, the associated <solution of a differential equation> obeys $Y(t+P)=mY(t)$ at every time, not merely at the sampled times. In particular, $Nv=v$ gives a nonzero $P$-<periodic solution>, and every nonzero $P$-<periodic solution> arises this way. If $Nv=-v$, the state changes sign after one period and repeats after two. A unit-modulus <Floquet multiplier> different from $1$ need not give repetition after one period, and a <generalized eigenvector> in a nontrivial <Jordan block> need not even give bounded motion.
For the scalar example under <periodic solution>, $N$ is the one-by-one matrix $(e^{2\pi a})$. Numerically solving a matrix problem uses the same idea: integrate one <initial value problem> for each column of $X(0)=I$ through one period, then compute the <eigenvalues> of $X(P)$. This is useful when an explicit <solution of a differential equation> is unavailable: one period determines the <Floquet growth rate> over arbitrarily many periods.
For piecewise constant coefficients, which are also covered by the one-period construction, $N$ is the time-ordered product of the segment <matrix exponentials>, with the earliest segment on the right. For example, take period $P=2$ and alternate
$$
K_1=\begin{pmatrix}0&1\\0&0\end{pmatrix}\quad(0\leq t<1),\qquad K_2=\begin{pmatrix}0&0\\1&0\end{pmatrix}\quad(1\leq t<2).
$$
Since $K_1^2=K_2^2=0$,
$$
N=e^{K_2}e^{K_1}=\begin{pmatrix}1&1\\1&2\end{pmatrix},\qquad m_\pm=\frac{3\pm\sqrt5}{2},\qquad \gamma=\frac12\log\frac{3+\sqrt5}{2}\approx0.4812.
$$
Replacing the coefficient by its mean would instead give $e^{K_1+K_2}=\left(\begin{smallmatrix}\cosh1&\sinh1\\\sinh1&\cosh1\end{smallmatrix}\right)$ and the incorrect rate $0.5$. The <matrices> do not commute. The scalar mean-coefficient formula therefore cannot simply be transferred to matrix systems. Nevertheless the <Liouville formula for a fundamental matrix> gives the correct <determinant> even in the noncommuting case:
$$
\det N=\exp\left(\int_0^P\operatorname{tr}K(t)\,dt\right).
$$
In the example, this is $1=m_+m_-$; it fixes the product of the <Floquet multipliers>, rather than either multiplier separately.
In Sur les équations différentielles linéaires à coefficients périodiques (1883), <Gaston Floquet> calls a function satisfying $f(t+P)=mf(t)$ periodic of the second kind, and distinguishes the genuinely <periodic function> obtained when $m=1$. This historical terminology captures precisely the difference between repetition of the coefficient and repetition of the state.<Sur les équations différentielles linéaires à coefficients périodiques by Gaston Floquet (1883)>[Printed p. 48, definition of a function periodic of the second kind and its multiplier; p. 49, transformation of a fundamental solution system after one period]{ref}
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