  We show that the presence of a two-dimensional inertial manifold for an
ordinary differential equation in ${\mathbb R}^{n}$ permits reducing the
problem of determining asymptotically orbitally stable limit cycles to the
Poincare--Bendixson theory. In the case $n=3$ we implement such a scenario for
a model of a satellite rotation around a celestial body of small mass and for a
biochemical model.
