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Asymptotic formation and orbital stability of phase-locked states in Kuramoto--Lohe type synchronization models on Lie groups

Seung-Yeon Ryoo

TL;DR

This work introduces a generalized Kuramoto--Lohe synchronization model on a Lie group $G$, formulated as $\dot{X}_i=(dR_{X_i})_e\big(H_i+\frac{\kappa}{N}\sum_j\phi(X_j X_i^{-1})\big)$ with phase-locked states defined by constant pairwise ratios $X_i(t)X_j(t)^{-1}$. It proves two main results: (i) global existence and exponential synchronization when all intrinsic Hamiltonians are identical and the initial dispersion is small for sufficiently large coupling $\kappa$, and (ii) the existence, uniqueness (up to right-multiplication), and orbital stability of phase-locked states when $\kappa$ dominates the intrinsic speeds $\|H_i\|$, with exponential convergence to a right-multiplication of a phase-locked configuration. The proofs proceed by reducing the group dynamics to a Lie-algebra system via $Y_{ij}=\log(X_i X_j^{-1})$, and then establishing two Gronwall-type Lyapunov functionals that control the diameter and the relative flows, yielding local stability and convergence of the flows and hence asymptotic phase-locking. The results generalize classical Kuramoto and Lohe models and provide a unified framework for phase-locking analysis on broad classes of Lie groups where the interaction $\phi$ satisfies $\phi\in C^1$, $\phi(e)=0$, and $(d\phi)_e$ has eigenvalues with positive real parts.

Abstract

Some mathematical models of synchronization, such as the Kuramoto model (1975) and its generalizations pioneered by Lohe (2009), are formulated as ordinary differential equations describing populations of particles on Lie groups with locally attractive interactions. We suggest a model of synchronization on Lie groups and present a framework to understand the formation of phase-locked states and their orbital stability. This is a sequel to a previous joint work with Ha and Ko (2017).

Asymptotic formation and orbital stability of phase-locked states in Kuramoto--Lohe type synchronization models on Lie groups

TL;DR

This work introduces a generalized Kuramoto--Lohe synchronization model on a Lie group , formulated as with phase-locked states defined by constant pairwise ratios . It proves two main results: (i) global existence and exponential synchronization when all intrinsic Hamiltonians are identical and the initial dispersion is small for sufficiently large coupling , and (ii) the existence, uniqueness (up to right-multiplication), and orbital stability of phase-locked states when dominates the intrinsic speeds , with exponential convergence to a right-multiplication of a phase-locked configuration. The proofs proceed by reducing the group dynamics to a Lie-algebra system via , and then establishing two Gronwall-type Lyapunov functionals that control the diameter and the relative flows, yielding local stability and convergence of the flows and hence asymptotic phase-locking. The results generalize classical Kuramoto and Lohe models and provide a unified framework for phase-locking analysis on broad classes of Lie groups where the interaction satisfies , , and has eigenvalues with positive real parts.

Abstract

Some mathematical models of synchronization, such as the Kuramoto model (1975) and its generalizations pioneered by Lohe (2009), are formulated as ordinary differential equations describing populations of particles on Lie groups with locally attractive interactions. We suggest a model of synchronization on Lie groups and present a framework to understand the formation of phase-locked states and their orbital stability. This is a sequel to a previous joint work with Ha and Ko (2017).

Paper Structure

This paper contains 8 sections, 7 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Motivation and significance of the generalized Kuramoto--Lohe model and Theorem \ref{['orbitalstabilitymainthm']}.
  3. Directions for further research.
  4. Notation and preliminaries on Lie groups and Lie algebras
  5. Proof of Theorems \ref{['idmainthm']} and \ref{['orbitalstabilitymainthm']}
  6. Reduction of Theorem \ref{['orbitalstabilitymainthm']}.
  7. Reducing from a model on the Lie group to a model on the Lie algebra.
  8. Two Gronwall inequalities.

Key Result

Proposition 1.1

\newlabelProp:pls A solution $\{X_i\}_{i=1}^N$ to genLohe is a phase-locked state if and only if $X_i(t)=X_i^\infty \exp(\Lambda t)$, where $\exp$ is the Lie group exponential and $X_i^\infty\in G$, $i=1,\cdots,N$, and $\Lambda\in \mathfrak{g}$ satisfy where $\operatorname{Ad}_g= (dL_g)_{g^{-1}}\circ (dR_{g^{-1}})_e$ is the adjoint.

Theorems & Definitions (18)

  • Definition 1.1
  • Proposition 1.1: Characterization of phase-locked states, lohe2009non
  • Theorem 1.1
  • Theorem 1.2
  • Example 1.1
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.1: First Gronwall inequality
  • ...and 8 more