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# The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations

• In our previous work , we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on convergent graph sequences. There, we derived and rigorously justified the mean field limit for the KM on graphs. Using linear stability analysis, we identified the critical values of the coupling strength, at which the incoherent state looses stability, thus, determining the onset of synchronization in this model.

In the present paper, we study the corresponding bifurcations. Specifically, we show that similar to the original KM with all-to-all coupling, the onset of synchronization in the KM on graphs is realized via a pitchfork bifurcation. The formula for the stable branch of the bifurcating equilibria involves the principal eigenvalue and the corresponding eigenfunctions of the kernel operator defined by the limit of the graph sequence used in the model. This establishes an explicit link between the network structure and the onset of synchronization in the KM on graphs. The results of this work are illustrated with the bifurcation analysis of the KM on Erdős-Rényi, small-world, as well as certain weighted graphs on a circle.

Mathematics Subject Classification: Primary: 34C15, 35B32, 47G20, 45L05; Secondary: 74A25, 92B25.

 Citation: • • Figure 1.  Deformation of the integral path for the Laplace inversion formula

Figure 2.  Formation of partially phase-locked solutions near a bifurcation with two-dimensional null space. The KM with intrinsic frequencies from the standard normal distribution, graphon (6.21), and random initial condition was for suffiently large time to reach a stationary regime. The values of $K$ are a) $3.5$, b) $4$, and c) $5$. The asymptotic state in (a) combines oscillators grouped around a $1$-twisted state with those distributed randomly around $\mathbb{S}$. For increasing values of $K$, the noisy twisted states become more distinct (b, c)

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