Generic Poincar\'{e}-Bendixson Theorem for singularly perturbed monotone systems with respect to cones of rank-$2$

TL;DR

This paper extends the Poincaré-Bendixson theorem to singularly perturbed monotone systems with cones of rank 2, showing that generic bounded invariant sets tend to closed orbits if they contain no equilibria.

Contribution

It establishes a generic Poincaré-Bendixson theorem for singularly perturbed monotone systems with cones of rank 2, a significant generalization in dynamical systems theory.

Findings

Existence of an open dense subset where omega-limit sets are closed orbits

Extension of Poincaré-Bendixson theorem to a new class of systems

Characterization of limit sets in singularly perturbed monotone systems

Abstract

We investigate the singularly perturbed monotone systems with respect to cones of rank $2$ and obtain the so called Generic Poincar\'{e}-Bendixson theorem for such perturbed systems, that is, for a bounded positively invariant set, there exists an open and dense subset $\mathcal{P}$ such that for each $z\in \mathcal{P}$, the $\omega$-limit set $\omega(z)$ that contains no equilibrium points is a closed orbit.