Semiflows “Monotone with Respect to High-Rank Cones" on a Banach Space

Abstract

We consider semiflows in general Banach spaces motivated by monotone cyclic feedback systems or differential equations with integer-valued Lyapunov functionals. These semiflows enjoy strong monotonicity properties with respect to “cones” of high ranks, which imply order-related structures on the $\omega$-limit sets of precompact semiorbits. We show that for a pseudoordered precompact semiorbit the $\omega$-limit set $\omega(x)$ either is ordered, or is contained in the set of equilibria, or possesses a certain ordered homoclinic property. In particular, we show that if $\omega(x)$ contains no equilibrium, then $\omega(x)$ is ordered, and hence the dynamics of the semiflow on $\omega(x)$ is topologically conjugate to a compact flow on $\mathbb{R}^k$ with $k$ being the rank. We also establish a Poincaré--Bendixson-type theorem in the case where $k=2$. All our results are established without the smoothness condition on the semiflow, allowing applications to many cellular or physiological feedback systems with piecewise linear vector fields and to infinite-dimensional systems where the $C^1$-closing lemma or smooth manifold theory has not been developed.