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On oscillator death in the Winfree model I: the one-dimensional case

Seung-Yeon Ryoo

TL;DR

This work proves a rigorous oscillator-death regime for the one-dimensional Winfree model: when the coupling κ exceeds a threshold tied to the intrinsic frequencies, Lebesgue almost every initial condition yields bounded, convergent dynamics with stationary speeds. The authors develop a bootstrapping approach based on the order parameter, combined with concentration-of-measure and large-deviation arguments, to show that oscillator death occurs for almost all initial data and persists under broad choices of interaction functions I,S. They further establish a gradient-flow structure via the Łojasiewicz framework for analytic interactions, extend results to general I,S with explicit κ-bounds, and provide a polynomial description that bounds the number of equilibria. Collectively, the results illuminate the asymptotic dynamics, quantify the prevalence of death, and lay groundwork for higher-dimensional generalizations explored in the sequel, with implications for understanding collective behavior in coupled oscillator systems.

Abstract

We show that for the standard sinusoidal Winfree model, a coupling strength exceeding twice the maximal magnitude of the intrinsic frequencies guarantees the convergence of the system for Lebesgue almost every initial data. This is proven by first showing, via an order parameter bootstrapping argument, that the pathwise critical coupling strength is upper bounded by a function of the order parameter, and then showing by a volumetric argument that for Lebesgue almost every data the order parameter cannot stay below and be bounded away from 1 for all time; this is a Winfree model counterpart of the analysis of Ha and the author (2020) performed for the Kuramoto model. Using concentration of measure and the aforementioned volumetric argument, we show that, except possibly on a set of very small measure, oscillator death is observed in finite time; this rigorously demonstrates the existence of the oscillator death regime numerically observed by Ariaratnam and Strogatz (2001). These results are robust under many other choices of interaction functions often considered for the Winfree model. We demonstrate that the asymptotic dynamics described in this paper are sharp by analyzing the equilibria of the Winfree model, and we bound the total number of equilibria using a polynomial description. The sequel to this paper will deal with the Winfree model on higher-dimensional manifolds, such as the Euclidean spheres and the unitary groups.

On oscillator death in the Winfree model I: the one-dimensional case

TL;DR

This work proves a rigorous oscillator-death regime for the one-dimensional Winfree model: when the coupling κ exceeds a threshold tied to the intrinsic frequencies, Lebesgue almost every initial condition yields bounded, convergent dynamics with stationary speeds. The authors develop a bootstrapping approach based on the order parameter, combined with concentration-of-measure and large-deviation arguments, to show that oscillator death occurs for almost all initial data and persists under broad choices of interaction functions I,S. They further establish a gradient-flow structure via the Łojasiewicz framework for analytic interactions, extend results to general I,S with explicit κ-bounds, and provide a polynomial description that bounds the number of equilibria. Collectively, the results illuminate the asymptotic dynamics, quantify the prevalence of death, and lay groundwork for higher-dimensional generalizations explored in the sequel, with implications for understanding collective behavior in coupled oscillator systems.

Abstract

We show that for the standard sinusoidal Winfree model, a coupling strength exceeding twice the maximal magnitude of the intrinsic frequencies guarantees the convergence of the system for Lebesgue almost every initial data. This is proven by first showing, via an order parameter bootstrapping argument, that the pathwise critical coupling strength is upper bounded by a function of the order parameter, and then showing by a volumetric argument that for Lebesgue almost every data the order parameter cannot stay below and be bounded away from 1 for all time; this is a Winfree model counterpart of the analysis of Ha and the author (2020) performed for the Kuramoto model. Using concentration of measure and the aforementioned volumetric argument, we show that, except possibly on a set of very small measure, oscillator death is observed in finite time; this rigorously demonstrates the existence of the oscillator death regime numerically observed by Ariaratnam and Strogatz (2001). These results are robust under many other choices of interaction functions often considered for the Winfree model. We demonstrate that the asymptotic dynamics described in this paper are sharp by analyzing the equilibria of the Winfree model, and we bound the total number of equilibria using a polynomial description. The sequel to this paper will deal with the Winfree model on higher-dimensional manifolds, such as the Euclidean spheres and the unitary groups.
Paper Structure (29 sections, 45 theorems, 279 equations)

This paper contains 29 sections, 45 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. Gallery of new results
  3. The pathwise critical coupling strength is bounded by a function of the order parameter
  4. The Łojasiewicz gradient theorem
  5. General interaction functions
  6. Equilibria of the prototypical Winfree model
  7. Previous work and directions for possible future research
  8. Oscillator death versus oscillator locking
  9. Winfree's original work: a perturbative linear periodic system
  10. Ariaratnam and Strogatz: rich nonlinear dynamics
  11. Rigorous nonlinear dynamics: oscillator death
  12. Rigorous nonlinear dynamics: oscillator locking
  13. Analogy with the Kuramoto model
  14. An order parameter bootstrapping argument
  15. Two toy models
  16. ...and 14 more sections

Key Result

Theorem 1

Fix parameters $\{\omega_i\}_{i=1}^N\in \mathbb{R}^N$ and $\kappa>0$ such that Then for Lebesgue almost every initial data $\{\theta_i^0\}_{i=1}^N$, the solution $\{\theta_i(t)\}_{i=1}^N$ to Winfree satisfies the following properties.

Theorems & Definitions (91)

  • Theorem 1
  • Remark 2
  • Corollary 3
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Definition 8
  • Theorem 9
  • Conjecture 10
  • Conjecture 11
  • ...and 81 more