Semiflows strongly focusing monotone with respect to high-rank cones: II. Pseudo-ordered principle

TL;DR

This paper establishes a pseudo-ordered principle for semiflows focusing monotone with respect to high-rank cones, leading to generalized Poincaré-Bendixson theorems and solid dynamics results in Banach spaces.

Contribution

It introduces the pseudo-ordered principle for high-rank cone monotone semiflows and derives new Poincaré-Bendixson and solid dynamics theorems based on this principle.

Findings

Omega-limit set of pseudo-ordered semiorbit is ordered.

Solid Poincaré-Bendixson theorem for rank 2 semiflows.

Generic solid dynamics theorem for arbitrary rank k.

Abstract

We consider a semiflow strongly focusing monotone with respect to a cone of rank k on a Banach space. We prove that the omega-limit set of a pseudo-ordered semiorbit is ordered, which is called as pseudo-ordered principle. Based on this principle, we obtain the solid Poincar\{'}e-Bendixson theorem with the rank k=2, that is, the omega-limit set of a pseudo-ordered semiorbit is either a nontrivial periodic orbit or a set consisting of equilibria with their potential connected orbits. The generic solid dynamics theorem with a general rank k and the generic solid Poincar\{'}e-Bendixson theorem with the rank k=2 are also obtained.