The Poincar\'e-Bendixson theory for certain compact semi-flows in Banach spaces

Abstract

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.