Non-Oscillation Principle for Eventually Competitive and Cooperative Systems

TL;DR

This paper introduces the Non-oscillation Principle for eventually competitive and cooperative systems, establishing non-ordering of limit sets and proving key theorems like Poincaré-Bendixson and structural stability in three dimensions.

Contribution

It presents a novel Non-oscillation Principle for these systems and extends classical results such as the Poincaré-Bendixson Theorem to higher-dimensional cases.

Findings

Non-ordering of limit sets in such systems

Poincaré-Bendixson Theorem established for 3D systems

Structural stability proven for these classes

Abstract

A nonlinear dynamical system is called eventually competitive (or cooperative) provided that it preserves a partial order in backward (or forward) time only after some reasonable initial transient. We presented in this paper the Non-oscillation Principle for eventually competitive or cooperative systems, by which the non-ordering of (both $\omega$- and $\alpha$-) limit sets is obtained for such systems; and moreover, we established the Poincar\'{e}-Bendixson Theorem and structural stability for three-dimensional eventually competitive and cooperative systems.