Floquet multiplier
= Floquet multiplier
{c}
{title2=$Nv=mv$}
{wiki}
A <Floquet multiplier> is an <eigenvalue> of a one-period linear evolution map. For a homogeneous <linear differential equation> with continuous coefficients of period $T>0$, this map is the <monodromy matrix of a periodic linear system> $N$. If $Nv=mv$ for a nonzero <eigenvector> $v$, the associated <solution of a differential equation> with $Y(0)=v$ satisfies $Y(t+T)=mY(t)$. Its exponential rate is $T^{-1}\log|m|$, the corresponding <Floquet growth rate>. A <Floquet multiplier> equal to one gives a $T$-<periodic solution> in that <eigenvector> direction; a nontrivial <Jordan block> at a unit-modulus <Floquet multiplier> can still produce polynomial growth.
For a nonlinear <periodic orbit>, the <Floquet multipliers> are <eigenvalues> of the one-period map of its <variational equation>. A nonconstant <periodic orbit> of a smooth <autonomous differential equation> has a <Floquet multiplier> equal to one in its tangent direction: differentiating the <solution of a differential equation> with respect to its starting phase gives a <periodic solution> of the <variational equation>. The derivative of a transverse <Poincaré map> removes this phase direction and retains the transverse <Floquet multipliers>. Transverse <Floquet multipliers> strictly inside the <unit circle> give local attraction, while one outside gives instability; a transverse <Floquet multiplier> on the <unit circle> requires further analysis. For a planar <periodic orbit>, the transverse <Floquet multiplier> is given by the <divergence test for a planar periodic orbit>.