Cycle flows and multistability in oscillatory networks

Area of Science:

• Complex Systems
• Network Science
• Power Systems Engineering

Background:

• Phase locked states are crucial for understanding synchronized behavior in coupled oscillator networks.
• The Kuramoto and swing equation models are widely used to study these dynamics, particularly in electrical power grids.
• Geometrical frustration can arise in these systems, complicating the analysis of steady states.

Purpose of the Study:

• To investigate multistability in phase locked states within networks of phase oscillators.
• To establish the existence of geometrically frustrated states and analyze stable fixed points.
• To develop a formalism for bounding and computing these states in various network topologies.

Main Methods:

• Analysis of Kuramoto and swing equation dynamics for phase oscillator networks.
• Introduction of the cycle flow formalism to describe stable fixed points.
• Derivation of bounds and scaling relations for fixed point counts in ring and planar networks.
• Development of an algorithm for computing all phase locked states.

Main Results:

• Demonstrated the existence of geometrically frustrated states where steady flow patterns lack dynamical fixed points.
• Characterized stable fixed points using cycle flows, with phase differences limited to π/2.
• Established that network topology (long cycles), edge weights, and parameter distribution (frequencies/injections) increase the number of fixed points.
• Derived accurate bounds and scaling relations for fixed point counts in planar networks.

Conclusions:

• The cycle flow formalism provides a robust method for analyzing multistability in phase locked systems.
• Network properties significantly influence the number and stability of phase locked states in power grid models.
• An efficient algorithm is presented for identifying all phase locked states in planar networks, aiding in grid stability assessment.