Periodic differential equations with self-adjoint monodromy operator
V. I. Yudovich · Sbornik Mathematics · 2001
A linear differential equation .u=A(t)u with p-periodic (generally speaking, unbounded) operator coefficient in a Euclidean or a Hilbert space H is considered. It is proved under natural constraints that the monodromy operator U{sub p} is self-adjoint and strictly positive if A*(-t)=A(t) for all t element of R. It is shown that Hamiltonian systems in the class under consideration are usually unstable and, if they are stable, then the operator U{sub 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{sub r}=a{sub z}=0, a{sub {theta}}={lambda}c(t)r{sup {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.