这是一份liverpool利物浦大学MATH433的成功案例

and hence the characteristic equation for the monodromy matrix
$$
M\left(t_{0}\right)=\left(\begin{array}{cc}
c\left(t_{0}+T, t_{0}\right) & s\left(t_{0}+T, t_{0}\right) \
\dot{c}\left(t_{0}+T, t_{0}\right) & \dot{s}\left(t_{0}+T, t_{0}\right)
\end{array}\right),
$$
is given by
$$
\rho^{2}-2 \Delta \rho+1=0
$$
where
$$
\Delta=\frac{\operatorname{tr}\left(M\left(t_{0}\right)\right)}{2}=\frac{c\left(t_{0}+T, t_{0}\right)+\dot{s}\left(t_{0}+T, t_{0}\right)}{2} .
$$
If $\Delta^{2}>1$ we have two different real eigenvalues
$$
\rho_{\pm}=\Delta \pm \sqrt{\Delta^{2}-1}=\sigma \mathrm{e}^{\pm T \gamma},
$$
with corresponding eigenvectors
$$
u_{\pm}\left(t_{0}\right)=\left(\begin{array}{c}
1 \
m_{\pm}\left(t_{0}\right)
\end{array}\right)
$$
where
$$
m_{\pm}\left(t_{0}\right)=\frac{\rho_{\pm}-c\left(t_{0}+T, t_{0}\right)}{s\left(t_{0}+T, t_{0}\right)}=\frac{\dot{s}\left(t_{0}+T, t_{0}\right)}{\rho_{\pm}-\dot{c}\left(t_{0}+T, t_{0}\right)}
$$

MATH433 COURSE NOTES ：

If $\Delta^{2}=1$ we have $\rho_{\pm}=\Delta$ and either two solutions
$$
p_{\pm}(t), \quad p_{\pm}(t+T)=\sigma p_{\pm}(t)
$$
or two solutions
$$
p_{+}(t), \quad p_{-}(t)+t p_{+}(t), \quad p_{\pm}(t+T)=\sigma p_{\pm}(t)
$$
where $\sigma=\operatorname{sgn}(\Delta)=\Delta$.
Since a periodic equation is called stable if all solutions are bounded, we have shown