The Poincaré-Bendixson theory for certain semi-flows in Hilbert spaces

Mikhail Anikushin

Published 2020-01-23Version 1

We study semi-flows satisfying a certain squeezing condition with respect to the quadratic form of a bounded self-adjoint operator acting in a Hilbert space. Under certain compactness assumptions from our previous results it follows that there exists an invariant topological manifold that attracts all compact trajectories. In the case of a two-dimensional manifold we obtain an analog of the Poincare-Bendixson theorem on the trichotomy of $\omega$-limit sets. Moreover, we obtain the conditions of existence of an orbitally stable periodic trajectory. We present applications of our results to study certain nonlinear systems of delay equations and reaction-diffusion systems. For these the required operator is obtained as a solution to certain operator inequalities with the use of the Yakubovich-Likhtarnikov frequency theorem for $C_{0}$-semigroups and its properties are established from the Lyapunov inequality and dichotomy of the linear part.