I must find solutions for the system $$ \left( \begin{matrix} \dot{x_1}\\ \dot{x_2} \end{matrix} \right) = \left( \begin{matrix} \cos(t) & -\sin(t)\\ \sin(t) & \cos(t) \end{matrix} \right) \left( \begin{matrix} x_1\\ x_2 \end{matrix} \right) $$ and then, obtain the period advance map $\varphi(2\pi,0)=P$.
My problem is how to find the solutions of the non-autonomous system above.
Could someone help me or refer me to somewhere where I can find information about it?
Rewrite this as $\dot z(t)=\mathrm e^{\mathrm it}z(t)$ where $z(t)=x_1(t)+\mathrm ix_2(t)$, hence $$z(t)=\mathrm e^{\mathrm i(1-\exp(\mathrm it))}z(0),$$ in particular, $z(t+2\pi)=z(t)$ for every $t$, that is, $$(x_1(t+2\pi),x_2(t+2\pi))=(x_1(t),x_2(t)).$$ To get the solutions $(x_1(t),x_2(t))$, one determines the real and imaginary parts of the expression of $z(t)$ as a complex number above. Thus, one starts from $$x_1(t)+\mathrm ix_2(t)=\mathrm e^{\sin(t)}\mathrm e^{\mathrm i(1-\cos(t))}(x_1(0)+\mathrm ix_2(0)),$$ which yields $$x_1(t)=\mathrm e^{\sin(t)}(\cos(1-\cos(t))x_1(0)-\sin(1-\cos(t))x_2(0),$$ and $$x_2(t)=\mathrm e^{\sin(t)}(\cos(1-\cos(t))x_2(0)+\sin(1-\cos(t))x_1(0),$$