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A Poincaré-Bendixson theorem for Bebutov shifts and applications to switched systems

Jairo Bochi, Ian D. Morris

TL;DR

This work proves a Poincaré-Bendixson–type theorem for switched Bebutov shifts on $S^2$ and $\mathbb{R}P^2$, showing that recurrent trajectories have preperiodic (and in the injective case, constant) representatives, a result that drives a PB-type theory for semiflows and switched vector fields on sphere-like spaces. It develops a robust topological framework based on plane graphs and polygonal approximations to obtain the core combinatorial lemmas, then leverages these to derive existence of periodic orbits in omega-limit sets and to resolve questions about periodic asymptotic stability versus global exponential stability for low-dimensional switched systems. As applications, it proves that periodic asymptotic stability implies global uniform exponential stability for real linear systems in up to three dimensions and for complex linear systems in dimension two, and it provides a comprehensive stability analysis for switched homogeneous systems in up to three real dimensions; it also demonstrates non-uniqueness of extremal trajectories in this setting. The results offer a natural continuous-time analogue of Lagarias–Wang finiteness, clarify the role of switching in non-smooth settings, and open avenues for extending PB-type conclusions to semiflows on non-manifold spaces and to dwell-time constrained switching in future work.

Abstract

We prove a version of the Poincaré-Bendixson theorem for certain classes of curves on the 2-sphere which are not required to be the trajectories of an underlying flow or semiflow on the sphere itself. Using this result we extend the Poincaré-Bendixson theorem to the context of continuous semiflows on compact subsets of the 2-sphere and the projective plane, give new sufficient conditions for the existence of periodic trajectories of certain low-dimensional affine control systems, and give a new criterion for the global uniform exponential stability of switched systems of homogeneous ODEs in dimension three. We prove in particular that periodic asymptotic stability implies global uniform exponential stability for real linear switched systems of dimension three and complex linear switched systems of dimension two. In combination with a recent result of the second author, this resolves a question of R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King and resolves a natural analogue of the Lagarias-Wang finiteness conjecture in continuous time.

A Poincaré-Bendixson theorem for Bebutov shifts and applications to switched systems

TL;DR

This work proves a Poincaré-Bendixson–type theorem for switched Bebutov shifts on and , showing that recurrent trajectories have preperiodic (and in the injective case, constant) representatives, a result that drives a PB-type theory for semiflows and switched vector fields on sphere-like spaces. It develops a robust topological framework based on plane graphs and polygonal approximations to obtain the core combinatorial lemmas, then leverages these to derive existence of periodic orbits in omega-limit sets and to resolve questions about periodic asymptotic stability versus global exponential stability for low-dimensional switched systems. As applications, it proves that periodic asymptotic stability implies global uniform exponential stability for real linear systems in up to three dimensions and for complex linear systems in dimension two, and it provides a comprehensive stability analysis for switched homogeneous systems in up to three real dimensions; it also demonstrates non-uniqueness of extremal trajectories in this setting. The results offer a natural continuous-time analogue of Lagarias–Wang finiteness, clarify the role of switching in non-smooth settings, and open avenues for extending PB-type conclusions to semiflows on non-manifold spaces and to dwell-time constrained switching in future work.

Abstract

We prove a version of the Poincaré-Bendixson theorem for certain classes of curves on the 2-sphere which are not required to be the trajectories of an underlying flow or semiflow on the sphere itself. Using this result we extend the Poincaré-Bendixson theorem to the context of continuous semiflows on compact subsets of the 2-sphere and the projective plane, give new sufficient conditions for the existence of periodic trajectories of certain low-dimensional affine control systems, and give a new criterion for the global uniform exponential stability of switched systems of homogeneous ODEs in dimension three. We prove in particular that periodic asymptotic stability implies global uniform exponential stability for real linear switched systems of dimension three and complex linear switched systems of dimension two. In combination with a recent result of the second author, this resolves a question of R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King and resolves a natural analogue of the Lagarias-Wang finiteness conjecture in continuous time.
Paper Structure (20 sections, 23 theorems, 37 equations, 2 figures)

This paper contains 20 sections, 23 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction and motivation
  2. A Poincaré-Bendixson Theorem for switched Bebutov shifts
  3. Principal definitions and results
  4. A foray into plane topology
  5. Overview
  6. Preliminaries on plane graphs
  7. A key lemma
  8. Polygonal approximation
  9. Derivation of Proposition \ref{['pr:three-discs-lemma']}
  10. Proof of Theorem \ref{['th:core']}
  11. Proof of Theorem \ref{['th:stronger']}
  12. Some Questions
  13. Applications
  14. A Poincaré-Bendixson theorem for semiflows
  15. Periodic limit trajectories for switched continuous vector fields
  16. ...and 5 more sections

Key Result

theorem 2.4

Let $Z$ be a compact topological space which is homeomorphic to a subset of either $S^2$ or $\mathbb{R}\mathrm{P}^2$, and let $\mathfrak{X}$ be a $Z$-valued switched Bebutov shift. If $\phi \in \mathfrak{X}$ is recurrent with respect to the shift semiflow on $\mathfrak{X}$, then there exists $\chi \

Figures (2)

  • Figure 1: The two pieces of trajectory produced by Theorem \ref{['th:core']}.
  • Figure 2: An example in the setting of Proposition \ref{['pr:three-discs-lemma']}. The existence of two arcs which begin in $D_1$, end in $D_2$ and do not pass through $D_3$ guarantees the existence of an arc making the same journey in the reverse direction which also does not pass through $D_3$.

Theorems & Definitions (50)

  • definition 2.1
  • remark 2.2
  • definition 2.3
  • theorem 2.4
  • corollary 2.5
  • proof
  • remark 2.6
  • remark 2.7
  • theorem 2.8
  • remark 2.9
  • ...and 40 more