Floquet theory ($X(t)=P(t)e^{tB}$)

For a continuous periodic coefficient matrix $A(t+L)=A(t)$, a fundamental matrix $X$ for $X^{\prime }=A(t)X$, normalized by $X(0)=I$, obeys $X(t+L)=X(t)X(L)$. The invertible matrix $X(L)$ has a complex matrix logarithm; set $B=L^{-1}\log X(L)$ and $P(t)=X(t)e^{-tB}$. Since $X(L)=e^{LB}$ commutes with $B$, direct substitution gives $P(t+L)=P(t)$ and proves the factorization. A real logarithm is not required for this complex representation.

The eigenvalues of $X(L)$ are Floquet multipliers. All solutions are bounded for forward time when every multiplier has modulus at most one and those on the unit circle have no nontrivial Jordan blocks. In a real undamped second-order scalar equation with no first-derivative term, the monodromy determinant is one. Distinct conjugate unit-modulus multipliers give bounded solutions; a nontrivial Jordan block at multiplier $1$ or $-1$ gives an unbounded second solution even when one periodic or antiperiodic solution exists.

The maximal asymptotic growth rate of a finite-dimensional periodic linear differential equation is $P^{-1}\log \rho (N)$, where $P$ is the coefficient period and $N$ the monodromy matrix of a periodic linear system. The spectral radius selects the dominant Floquet multiplier. Generic initial data attain this rate; data lying entirely in other invariant subspaces can grow or decay at different rates.

For $\dot{Y}=K(t)Y$ with coefficient period $P$, the monodromy matrix maps $Y(0)$ to $Y(P)$. Its eigenvalues are the Floquet multipliers. For piecewise constant coefficients it is the time-ordered product of the segment matrix exponentials, with the earliest segment on the right. Its determinant is $\exp ({\int }_0^P\mathrm{tr}K(t)dt)$.