Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Traveling Waves in the McKean-Vlasov Equation under Sakaguchi-Kuramoto Interaction with Phase Frustration

Jesenko Vukadinovic

TL;DR

This work analyzes the McKean-Vlasov equation for weakly coupled oscillators under the Sakaguchi-Kuramoto interaction with phase frustration $α$. By reducing the one-mode coupling problem to a finite-dimensional system and introducing an asymmetrically extended von Mises distribution (AvMPDF), the authors derive a traveling-wave McKean-Vlasov equation whose coherence is captured by a pair $(r,k)$ of order-parameter and wave speed. The main result proves a continuous global phase transition at $μ_α = 2\sec α$, from incoherence to a unique traveling-wave coherent state for $α∈(0,π)$, with special cases: a coherent stationary state for $α=0$ (or $π$) when $μ>2$, and no traveling waves at $α=π/2$. The analysis hinges on an asymmetrical extension of the modified Bessel functions and a detailed monotonicity study, providing a rigorous framework for traveling waves in non-symmetric, phase-frustrated coupling. This advances mathematical understanding of synchronization with delays in neural-like networks and broadens the spectrum of phase transitions beyond symmetric Kuramoto-type interactions.

Abstract

We study the McKean-Vlasov equation for weakly coupled oscillators subject to the Sakaguchi-Kuramoto interaction. While the Kuramoto interaction provides a good approximation for small, densely connected networks, time delays in larger networks lead to symmetry-breaking phase offsets (frustrations). The Sakaguchi-Kuramoto interaction is the simplest such generalization, featuring a single frustration parameter. We establish the existence of a continuous global phase transition from incoherence to coherence, in the form of a propagating asymmetrically extended von Mises probability distribution function (AvMPDF). The corresponding traveling wave equation reduces to a system of two equations in two unknowns: the order parameter for the AvMPDF and the wave speed. The analysis relies on an appropriate asymmetrical extension of the modified Bessel function.

Traveling Waves in the McKean-Vlasov Equation under Sakaguchi-Kuramoto Interaction with Phase Frustration

TL;DR

This work analyzes the McKean-Vlasov equation for weakly coupled oscillators under the Sakaguchi-Kuramoto interaction with phase frustration . By reducing the one-mode coupling problem to a finite-dimensional system and introducing an asymmetrically extended von Mises distribution (AvMPDF), the authors derive a traveling-wave McKean-Vlasov equation whose coherence is captured by a pair of order-parameter and wave speed. The main result proves a continuous global phase transition at , from incoherence to a unique traveling-wave coherent state for , with special cases: a coherent stationary state for (or ) when , and no traveling waves at . The analysis hinges on an asymmetrical extension of the modified Bessel functions and a detailed monotonicity study, providing a rigorous framework for traveling waves in non-symmetric, phase-frustrated coupling. This advances mathematical understanding of synchronization with delays in neural-like networks and broadens the spectrum of phase transitions beyond symmetric Kuramoto-type interactions.

Abstract

We study the McKean-Vlasov equation for weakly coupled oscillators subject to the Sakaguchi-Kuramoto interaction. While the Kuramoto interaction provides a good approximation for small, densely connected networks, time delays in larger networks lead to symmetry-breaking phase offsets (frustrations). The Sakaguchi-Kuramoto interaction is the simplest such generalization, featuring a single frustration parameter. We establish the existence of a continuous global phase transition from incoherence to coherence, in the form of a propagating asymmetrically extended von Mises probability distribution function (AvMPDF). The corresponding traveling wave equation reduces to a system of two equations in two unknowns: the order parameter for the AvMPDF and the wave speed. The analysis relies on an appropriate asymmetrical extension of the modified Bessel function.
Paper Structure (8 sections, 5 theorems, 69 equations)

This paper contains 8 sections, 5 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Stationary and traveling-waves McKean-Vlasov equations
  3. Finite-dimensional reduction
  4. Asymmetrical extension of the von Mises distribution and the modified Bessel functions
  5. The statement of the main result
  6. Properties of the asymmetrically extended modified Bessel functions
  7. The proof of the main result
  8. Proof of the main result

Key Result

Theorem 5.1

Consider TWMVE, trwaves, associated with the one-mode interaction potential with a phase shift $\alpha\in[0,\pi]$, For $\alpha\in[0,\pi/2)$ (or $\alpha\in(\pi/2,\pi])$), the system (OP4) undergoes a supercritical (or subcritical) pitchfork bifurcation in $\mu$ at the bifurcation point $\mu_\alpha=2\sec\alpha$. For $\mu\leq\mu_\alpha$ (or $\mu\geq\mu_\alpha$), the trivial solution $r=0$ is the uni

Theorems & Definitions (6)

  • Definition 4.1
  • Theorem 5.1
  • Proposition 6.1
  • Lemma 6.2
  • Lemma 7.1
  • Lemma 7.2