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A Poincaré-Bendixson theorem for flows with arbitrarily many singular points

Tomoo Yokoyama

Abstract

The Poincaré-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 Poincaré-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 Poincaré-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.

A Poincaré-Bendixson theorem for flows with arbitrarily many singular points

Abstract

The Poincaré-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 Poincaré-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 Poincaré-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.

Paper Structure

This paper contains 60 sections, 70 theorems, 18 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Statements of main results
  3. Preliminaries
  4. Topological notion
  5. Curves and loops
  6. Essential property
  7. Notion of dynamical systems
  8. Topological equivalence
  9. Separatrices
  10. Orbit arcs
  11. Recurrence and invariance
  12. Topological properties of orbits
  13. Flow boxes
  14. Topological properties of Q-sets
  15. Circuits
  16. ...and 45 more sections

Key Result

Theorem 1

The following statements hold for a flow with arbitrarily many singular points on a compact surface: (a) The $\omega$-limit set of any non-closed orbit is one of the following exclusively: $(1)$ A nowhere dense subset of singular points. $(2)$ A semi-attracting limit cycle. $(3)$ A quasi-semi-attrac

Figures (20)

  • Figure 1: Left, a trivial flow box; middle, the intersection of a flow box and its small neighborhood of the boundary, right; a Cherry flow box.
  • Figure 2: A circuit that consists of a degenerate singular point and infinitely many connecting separatrices, and its neighborhood which consists of a singular point, periodic orbits, and non-recurrent orbits.
  • Figure 3: A transverse closed arc $I$ and the first return map $f_I$ on $I$.
  • Figure 4: A flow box $D_i$ whose boundary is the union of $I_i \cup C_i \cup I_{i+1} \cup C_{i+1}$.
  • Figure 5: A flow box with a non-arcwise-connected connected invariant subset.
  • ...and 15 more figures

Theorems & Definitions (226)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Theorem 4
  • Definition 1
  • Lemma 2.1
  • proof
  • Definition 2
  • Definition 3
  • Lemma 2.2
  • ...and 216 more