A Poincar\'e-Bendixson theorem for flows with arbitrarily many singular points

TL;DR

This paper extends the Poincaré-Bendixson theorem to flows with many singular points on non-compact surfaces, enabling analysis of complex dynamical behaviors like Lakes of Wada attractors.

Contribution

It introduces a generalized Poincaré-Bendixson theorem applicable to flows with arbitrarily many singular points on non-compact surfaces, using foliation theory and topology.

Findings

Generalization of the Poincaré-Bendixson theorem to complex flows

Framework for analyzing Lakes of Wada attractors

Applicable to non-compact surfaces with multiple singularities

Abstract

The Poincar\'{e}-Bendixson theorem is one of the most fundamental tools to capture the limit behaviors of orbits of flows. It was generalized and applied to various phenomena in dynamical systems, differential equations, foliations, group actions, translation lines, and semi-dynamical systems. On the other hand, though the no-slip boundary condition is a fundamental condition in differential equations and appears in various fluid phenomena, and Lakes of Wada attractors naturally occur in discrete and continuous real dynamical systems and complex dynamics, no generalizations of the Poincar\'{e}-Bendixson theorem can be applied to any differential equations with no-slip boundary condition on surfaces with boundary and flows with Lakes of Wada attractors. To analyze them, we generalize the Poincar\'{e}-Bendixson theorem into one for flows with arbitrarily many singular points on possibly non-compact surfaces by introducing some concepts to describe limit behaviors and using methods of foliation theory and general topology.