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Synchronization of second-order Kuramoto model with frustration on strongly connected digraph

Tingting Zhu, Xiongtao Zhang

TL;DR

This work addresses synchronization of the second-order Kuramoto model with inertia and frustration on strongly connected digraphs, where network asymmetry invalidates standard energy methods. The authors develop time-dependent weighted $\ell^1$-type energies and four convex-combination functions $Q(t),P(t),A(t),B(t)$ to control phase, frequency, acceleration, and jerk diameters, establishing hypo-coercivity of the frequency diameter. They prove exponential convergence of the frequency differences to zero under large coupling and small inertia/frustration, after ensuring phase cohesiveness and finite-time entrance to a small-region regime. The results provide explicit parametric conditions and a constructive energy-based framework that extends to asymmetric networks and has potential implications for power-grid stability and directed-network synchronization.

Abstract

We study the emergent behavior of a second-order Kuramoto-type model with frustration effect on a strongly connected digraph. The main challenge arises from the lack of symmetry in this system, which renders standard approaches for symmetric models, such as the gradient-flow method and classical $\ell^p$ or $\ell^\infty$-type energy estimates, ineffective. To address these difficulties, our primary contribution is the development of time-dependent weighted $\ell^1$-type energy estimates to establish the hypo-coercivity of the frequency diameter. Specifically, we construct novel energy functions incorporating convex combinations of phases, frequencies, accelerations, and jerks, which are shown to be dissipative and capable of bounding both phase and frequency diameters. This framework enables us to demonstrate the emergence of frequency synchronization with an exponential convergence rate.

Synchronization of second-order Kuramoto model with frustration on strongly connected digraph

TL;DR

This work addresses synchronization of the second-order Kuramoto model with inertia and frustration on strongly connected digraphs, where network asymmetry invalidates standard energy methods. The authors develop time-dependent weighted -type energies and four convex-combination functions to control phase, frequency, acceleration, and jerk diameters, establishing hypo-coercivity of the frequency diameter. They prove exponential convergence of the frequency differences to zero under large coupling and small inertia/frustration, after ensuring phase cohesiveness and finite-time entrance to a small-region regime. The results provide explicit parametric conditions and a constructive energy-based framework that extends to asymmetric networks and has potential implications for power-grid stability and directed-network synchronization.

Abstract

We study the emergent behavior of a second-order Kuramoto-type model with frustration effect on a strongly connected digraph. The main challenge arises from the lack of symmetry in this system, which renders standard approaches for symmetric models, such as the gradient-flow method and classical or -type energy estimates, ineffective. To address these difficulties, our primary contribution is the development of time-dependent weighted -type energy estimates to establish the hypo-coercivity of the frequency diameter. Specifically, we construct novel energy functions incorporating convex combinations of phases, frequencies, accelerations, and jerks, which are shown to be dissipative and capable of bounding both phase and frequency diameters. This framework enables us to demonstrate the emergence of frequency synchronization with an exponential convergence rate.
Paper Structure (11 sections, 11 theorems, 134 equations, 1 figure)

This paper contains 11 sections, 11 theorems, 134 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic concepts
  4. Construction of $Q(t), P(t), A(t)$ and $B(t)$
  5. Main result
  6. Phase cohesinveness
  7. Energy function and its dynamics
  8. Entrance to small region
  9. Frequency synchronization
  10. Simulations
  11. Summary

Key Result

Lemma 2.1

Let $z(t) = (z_1(t),\ldots,z_N(t))$ be the state quantity of oscillators associated to system KMI at time $t$. Then, we have where $D_z(t) = \max\limits_{1 \le i \le N} z_i(t) - \min\limits_{1 \le i \le N} z_i(t)$.

Figures (1)

  • Figure 1: Complete synchronization for the second-order model \ref{['KMI']} on the network \ref{['P-1']}.

Theorems & Definitions (25)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 15 more