Periodic differential equations with self-adjoint monodromy operator

A linear differential equation (u)over dot = A(t)u with p-periodic (general ly speaking, unbounded) operator coefficient in a Euclidean or a Hilbert sp ace H is considered. It is proved under natural constraints that the monodr omy operator Up is self-adjoint and strictly positive if A*(-t) = A(t) for all t is an element of R. It is shown that Hamiltonian systems in the class under consideration are u sually unstable and, if they are stable, then the operator Up reduces to th e identity and all solutions are p-periodic. For higher frequencies average d equations are derived. Remarkably, high-frequency modulation may double t he number of critical values. General results are applied to rotational flows with cylindrical components of the velocity a(r) = a(z) = 0 and a(theta) = lambdac(t)r(beta), where be ta < -1 and c(t) is an even p-periodic function, and also to several proble ms of free gravitational convection of fluids in periodic fields.