Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The open dense conjecture on eventually slow oscillations of the differential equation with delayed negative feedback

Lirui Feng

TL;DR

The paper tackles the open dense conjecture on eventual slow oscillations for the delay differential equation $x'(t) = -\mu x(t) + f(x(t-1))$ under $\mu\ge0$ and $f'<0$, showing that the set of initial data in $X=C[-1,0]$ leading to ES is open and dense. It develops a framework based on strongly order-preserving semiflows with respect to high-rank (specifically, 2-cone) structures, proving invariance and monotonicity properties, a global attractor, and a robust flow-extension on $\omega$-limit sets. A key step is the construction of a complemented 2-cone structure and the SOP dynamics that preserve cones and map certain subsets into their interiors after sufficient time. The main theorem yields the open-dense result for $\mathcal{D}_{ES}$ and, as a corollary, resolves Kaplan and Yorke’s conjecture, providing a rigorous description of the asymptotic behavior of delayed negative-feedback systems with broad implications for the analysis of similar delay equations.

Abstract

In this paper, we show how to use the approach of the strongly order-preserving semiflow with respect to high-rank cones to solve the open dense conjecture on eventually slow oscillations of the differential equation with delayed negative feedback.

The open dense conjecture on eventually slow oscillations of the differential equation with delayed negative feedback

TL;DR

The paper tackles the open dense conjecture on eventual slow oscillations for the delay differential equation under and , showing that the set of initial data in leading to ES is open and dense. It develops a framework based on strongly order-preserving semiflows with respect to high-rank (specifically, 2-cone) structures, proving invariance and monotonicity properties, a global attractor, and a robust flow-extension on -limit sets. A key step is the construction of a complemented 2-cone structure and the SOP dynamics that preserve cones and map certain subsets into their interiors after sufficient time. The main theorem yields the open-dense result for and, as a corollary, resolves Kaplan and Yorke’s conjecture, providing a rigorous description of the asymptotic behavior of delayed negative-feedback systems with broad implications for the analysis of similar delay equations.

Abstract

In this paper, we show how to use the approach of the strongly order-preserving semiflow with respect to high-rank cones to solve the open dense conjecture on eventually slow oscillations of the differential equation with delayed negative feedback.
Paper Structure (9 sections, 26 theorems, 69 equations)

This paper contains 9 sections, 26 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The strongly order-preserving semiflow $\Phi_t$ on a Banach space
  4. The delayed equation and its generated semiflow $F_t$
  5. Fundamental properties of the semiflow $F_t$
  6. Properties related to $k$-cones
  7. The open dense conjecture on enventually slowly oscillating solutions
  8. Pseudo-ordered semiorbits and enventually slowly oscillating solutions
  9. Proof of the main theorem

Key Result

Lemma 2.1

Assume that $\Phi_t$ on $\tilde{X}$ is a semiflow SOP with respect to a solid $k$-cone $C$, which admits a flow extension on each nonempty omega-limit set. Let $O^+(x)$ be a precompact pseudo-ordered semiorbit. Then the closure of any full-orbit in $\omega(x)$ is ordered.

Theorems & Definitions (54)

  • Definition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 44 more