Prevalent behavior and almost sure Poincare-Bendixson Theorem for smooth flows with invariant k-cones

TL;DR

This paper establishes that in smooth flows with invariant cones of rank k, most orbits tend to equilibria or are pseudo-ordered, extending classical theorems to high-dimensional systems from a measure-theoretic viewpoint.

Contribution

It generalizes the Poincare-Bendixson theorem and Hirsch's convergence theorem to higher-dimensional flows with invariant cones, providing a measure-theoretic framework.

Findings

Almost all orbits are pseudo-ordered or converge to equilibria.

Extends Poincare-Bendixson theorem to high-dimensional differential equations.

Provides measure-theoretic conditions for global dynamics.

Abstract

We investigate the global dynamics from a measure-theoretic perspective for smooth flows with invariant cones of rank k. For such systems, it is shown that prevalent (or equivalently, almost all) orbits will be pseudo-ordered or convergent to equilibria. This reduces to Hirsch's prevalent convergence Theorem if the rank k=1; and implies an almost-sure Poincare-Bendixson Theorem for the case k=2. These results are then applied to obtain an almost sure Poincare-Bendixson theorem for high-dimensional differential equations.