Semiflows Strongly Focusing Monotone with Respect to High-Rank Cones: I. Generic Dynamics

We consider a smooth semiflow strongly focusing monotone with respect to a cone of rank k on a Banach space. We obtain its generic dynamics, that is, semiorbits with initial data from an open and dense subset of any open bounded set either are pseudo-ordered or convergent to an equilibrium. For the case \(k=1\) $k=1$ , it is the celebrated Hirsch’s Generic Convergence Theorem. For the case \(k=2\) $k=2$ , we obtain the generic Poincaré-Bendixson Theorem.