Fixed points and stability in the two-network frustrated Kuramoto model

• We model two networked groups internally synchronising and externally competing.
• Fixed points describe the two groups phase locked or one group in two fragments.
• We analytically solve for thresholds for loss of synchrony leading to fragmentation.
• We numerically solve for one group on a tree graph and the other on a random graph.
• Numerical and analytical results for thresholds agree across a range of couplings.

We examine a modification of the Kuramoto model for phase oscillators coupled on a network. Here, two populations of oscillators are considered, each with different network topologies, internal and cross-network couplings and frequencies. Additionally, frustration parameters for the interactions of the cross-network phases are introduced. This may be regarded as a model of competing populations: internal to any one network phase synchronisation is a target state, while externally one or both populations seek to frequency synchronise to a phase in relation to the competitor. We conduct fixed point analyses for two regimes: one, where internal phase synchronisation occurs for each population with the potential for instability in the phase of one population in relation to the other; the second where one part of a population remains fixed in phase in relation to the other population, but where instability may occur within the first population leading to ‘fragmentation’. We compare analytic results to numerical solutions for the system at various critical thresholds.