Multistability of Phase-Locking in Equal-Frequency Kuramoto Models on Planar Graphs

Abstract

The number $\mathcal{N}$ of stable fixed points of locally coupled Kuramoto models depends on the topology of the network on which the model is defined. It has been shown that cycles in meshed networks play a crucial role in determining $\mathcal{N}$, because any two different stable fixed points differ by a collection of loop flows on those cycles. Since the number of different loop flows increases with the length of the cycle that carries them, one expects $\mathcal{N}$ to be larger in meshed networks with longer cycles. Simultaneously, the existence of more cycles in a network means more freedom to choose the location of loop flows differentiating between two stable fixed points. Therefore, $\mathcal{N}$ should also be larger in networks with more cycles. We derive an algebraic upper bound for the number of stable fixed points of the Kuramoto model with identical frequencies, under the assumption that angle differences between connected nodes do not exceed $\pi /2$. We obtain $\mathcal{N}\le {\prod }_{k=1}^c\left[2\cdot \mathrm{Int}(n_k/4)+1\right]$, which depends both on the number $c$ of cycles and on the spectrum of their lengths $\{n_k\}$. We further identify network topologies carrying stable fixed points with angle differences larger than $\pi /2$, which leads us to conjecture an upper bound for the number of stable fixed points for Kuramoto models on any planar network. Compared to earlier approaches that give exponential upper bounds in the total number of vertices, our bounds are much lower and therefore much closer to the true number of stable fixed points.

arXiv comment10 pages, 3 figures

References in corpus (7)

• Synchronization in Complex Oscillator Networks and Smart Grids
• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• There are no non-zero Stable Fixed Points for dense networks in the homogeneous Kuramoto model
• Multistability of Phase-Locking and Topological Winding Numbers in Locally Coupled Kuramoto Models on Single-Loop Networks
• Graphical notation reveals topological stability criteria for collective dynamics in complex networks
• Multistable behavior above synchronization in a locally coupled Kuramoto model
• Topologically Protected Loop Flows in High Voltage AC Power Grids

Cited by in corpus (19)

• Cycle flows and multistabilty in oscillatory networks: an overview
• Synchronization and Stability for Quantum Kuramoto
• The Size of the Sync Basin Revisited
• Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian
• Stability and control of power grids with diluted network topology
• A Theory of Solvability for Lossless Power Flow Equations -- Part II: Conditions for Radial Networks
• Noise-Induced Desynchronization and Stochastic Escape from Equilibrium in Complex Networks
• A Matrix Valued Kuramoto Model
• Multistability in lossy power grids and oscillator networks
• When is sync globally stable in sparse networks of identical Kuramoto oscillators?
• Nonlinear consensus on networks: equilibria, effective resistance and trees of motifs
• Synchronization conditions in the Kuramoto model and their relationship to seminorms
• Twisted states in low-dimensional hypercubic lattices
• Clusterization and phase diagram of the bimodal Kuramoto model with bounded confidence
• Synchronized states of power grids and oscillator networks by convex optimization
• Reconstructing Network Structures from Partial Measurements
• Large and small fluctuations in oscillator networks from heterogeneous and correlated noise
• Directed acyclic decomposition of Kuramoto equations
• Order parameter allows classification of planar graphs based on balanced fixed points in the Kuramoto model