Periodic differential equations with self-adjoint monodromy operator

A linear differential equation $\dot u=A(t)u$ with $p$-periodic (generally speaking, unbounded) operator coefficient in a Euclidean or a Hilbert space $\mathbb H$ is considered. It is proved under natural constraints that the monodromy operator $U_p$ is self-adjoint and strictly positive if $A^*(-t)=A(t)$ for all $t\in\mathbb R$.
It is shown that Hamiltonian systems in the class under consideration are usually unstable and, if they are stable, then the operator $U_p$ reduces to the identity and all solutions are $p$-periodic.
For higher frequencies averaged equations are derived. Remarkably, high-frequency modulation may double the number of critical values.
General results are applied to rotational flows with cylindrical components of the velocity $a_r=a_z=0$, $a_\theta=\lambda c(t)r^\beta$, $\beta<-1$,   $c(t)$ is an even $p$-periodic function, and also to several problems of free gravitational convection of fluids in periodic fields.