  We study semiflows satisfying a certain squeezing condition with respect to a
quadratic functional in some Banach space. Under certain compactness
assumptions from our previous results it follows that there exists an invariant
manifold, which is under more restrictive conditions is an inertial manifold.
In the case of a two-dimensional manifold we obtain an analog of the
Poincar\'{e}-Bendixson theorem on the trichotomy of $\omega$-limit sets.
Moreover, we obtain conditions for the existence of an orbitally stable
periodic orbit. Our approach unifies a series of papers by R.~A.~Smith,
establishes their connection with the theory of inertial manifolds and opens a
new perspective of applications. To verify the squeezing property in
applications we use recently developed versions of the frequency theorem, which
guarantee the existence of the required quadratic functional if some
frequency-domain condition is satisfied. We present applications of our results
for nonlinear delay equations in $\mathbb{R}^{n}$ and semilinear parabolic
equations and discuss perspectives of applications to parabolic problems with
delay and boundary controls.
