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Periodic differential equations with self-adjoint monodromy operator
V. I. Yudovich Rostov State University
Abstract:
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.
UDC:
517.98
MSC: 34G10,
34A30,
76D05,
76E06 Received: 14.11.1999 and 24.08.2000
DOI:
10.4213/sm554
Fulltext:
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