= Floquet growth rate
{c}
{title2=$\gamma=P^{-1}\log\rho(N)$}
The <Floquet growth rate> measures the exponential trend of a periodic <linear differential equation> over many coefficient periods. A <Floquet multiplier> $m$ measures the factor per period $T$; the corresponding rate per unit time is
$$
\gamma_m=\frac{\log|m|}{T}=\operatorname{Re}\rho_m,\qquad e^{T\rho_m}=m,
$$
where $\rho_m$ is a <Floquet exponent>. Taking the <real part> removes the ambiguity in the complex <logarithm>.
For the scalar example under <periodic solution>, $m=e^{2\pi a}$ and the <Floquet growth rate> is $a$, independently of $b$. Indeed,
$$
\frac{1}{t}\log\frac{y(t)}{y(0)}=a+b\frac{\sin t}{t}\longrightarrow a,\qquad \frac{y'(t)}{y(t)}=a+b\cos t.
$$
The instantaneous <logarithmic derivative> keeps oscillating even though its average over complete periods is exactly $a$. Sampling at the same phase in each period removes the modulation altogether: $y(nT)=m^ny(0)$.
\Image[/floquet-stroboscopic-multiplier-and-rate.png]
{title=The multiplier gives a factor per period and its logarithm gives a rate per unit time}
{description=Left: sampling $y'=\left(a+\cos t\right)y$ at $t=nT$, $T=2\pi$, gives $y(nT)=m^n$; straight lines on the logarithmic scale have slopes $\log m$ per period. Right: the instantaneous logarithmic derivatives $a+\cos t$ oscillate around the constant long-term rates $a=-0.2,0,+0.2$.}
For the general finite-dimensional homogeneous <linear differential equation> $x'=A(t)x$, assume that $A$ is continuous on the real line and $T$-periodic, with $T>0$. Normalize its <fundamental matrix> by $\Phi(0)=I$ and set $M=\Phi(T)$. The maximal asymptotic exponential rate, using any induced <matrix norm>, is
$$
\gamma=\lim_{t\to\infty}\frac{\log\|\Phi(t)\|}{t}=\frac{\log\rho(M)}{T},\qquad \rho(M)=\max_j|m_j|.
$$
Here $\rho(M)$ is the <spectral radius>, $m_j$ are the <Floquet multipliers>, and invertibility gives $\rho(M)>0$. This concerns growth of the solution operator; it does not say that every <initial condition> attains the maximum.<Ordinary Differential Equations and Dynamical Systems by Gerald Teschl>[§3.6, printed pp. 91–93, equation (3.120), Theorem 3.15 and the multiplier/exponent relation]{ref}
To prove the formula, write $t=nT+s$ with $0\leq s<T$. Periodicity and uniqueness give
$$
\Phi(t)=\Phi(s)M^n.
$$
Both $\Phi(s)$ and $\Phi(s)^{-1}$ have uniformly bounded <matrix norms> on a period. Consequently $\|\Phi(t)\|$ differs from $\|M^n\|$ by at most fixed multiplicative factors. The <spectral radius formula> gives $\|M^n\|^{1/n}\to\rho(M)$, proving the claimed limit without choosing a <matrix logarithm>.
For a particular nonzero <initial condition> $v$, decompose $v$ into the <generalized eigenspaces> of $M$ over the complex numbers. The rate of $\Phi(t)v$ is the largest $T^{-1}\log|m_j|$ among components that are nonzero. Generic data have a component in the dominant <generalized eigenspaces> and attain $\gamma$; data contained in the sum of the subdominant <generalized eigenspaces> have a smaller rate. For example, with $A=\operatorname{diag}(0.2,-0.3)$, the maximal rate is $0.2$, while $v=(0,1)$ decays at rate $-0.3$. These <invariant subspaces> describe initial data at phase zero. Their transported versions $E(t)=\Phi(t)E(0)$ can vary with time and satisfy $E(t+T)=E(t)$.
A negative <Floquet growth rate> allows temporary amplification. For the constant <non-normal matrix>
$$
A=\begin{pmatrix}-1&10\\0&-1\end{pmatrix},\qquad e^{tA}=e^{-t}\begin{pmatrix}1&10t\\0&1\end{pmatrix},
$$
the <initial condition> $v=(0,1)$ has Euclidean <norm> $e^{-t}\sqrt{100t^2+1}$, which rises from $1$ to about $3.697$ at $t=1$ before eventually decaying. Its exponential rate is still $-1$: the <Jordan block> contributes a polynomial factor, whose <logarithm> divided by $t$ tends to zero. At a unit-modulus <Floquet multiplier>, such polynomial growth can even be unbounded with zero exponential rate; $A=\left(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\right)$ gives $e^{tA}v=(t,1)$. Thus zero <Floquet growth rate> alone proves neither boundedness nor a <periodic solution>.
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