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Stability and bifurcation of mixing in the Kuramoto model with inertia
Dynamical Systems · Submitted 14 Jun 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation.
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The abstract, sentence by sentence
The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erd\H{o}s--R\'enyi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. | contribution 0.93 | 0.86 | 0.03 | asserted 0.98 |
| s1 | The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. | background 0.91 | 0.09 | 0.02 | asserted 1.00 |
| s2 | The connectivity of the coupled system is assigned by a graph which may be random as well. | background 0.61 | 0.20 | 0.02 | hedged possibility 0.77 |
| s3 | In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. | method 0.71 | 0.37 | 0.05 | asserted 1.00 |
| s4 | We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. | contribution 0.99 | 0.87 | 0.07 | asserted 0.99 |
| s5 | Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. | result 0.98 | 0.94 | 0.50 | asserted 1.00 |
| s6 | It corresponds to the loss of stability of mixing and marks the onset of synchronization. | interpretation 0.92 | 0.70 | 0.38 | asserted 1.00 |
| s7 | As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. | background 0.92 | 0.11 | 0.03 | asserted 1.00 |
| s8 | To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. | method 0.99 | 0.82 | 0.03 | asserted 1.00 |
| s9 | The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erd\H{o}s--R\'enyi and small-world graphs. | method 0.60 | 0.94 | 0.05 | asserted 1.00 |
| s10 | Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. | background 0.84 | 0.06 | 0.03 | asserted 1.00 |
| s11 | The analysis in this paper provides a mathematical description of the onset of synchronization in these systems. | interpretation 0.87 | 0.78 | 0.24 | asserted 1.00 |
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Original abstract
The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erd\H{o}s--R\'enyi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.
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