= Floquet theory
{c}
{title2=$X(t)=P(t)e^{tB}$}
{wiki}
For a continuous periodic coefficient matrix $A(t+L)=A(t)$, a <fundamental matrix> $X$ for $X'=A(t)X$, 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.
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