Bifurcations and patterns in the Kuramoto model with inertia

Abstract

In this work, we analyze the Kuramoto model (KM) with inertia on a convergent family of graphs. It is assumed that the intrinsic frequencies of the individual oscillators are sampled from a probability distribution. In addition, a given graph, which may also be random, assigns network connectivity. As in the original KM, in the model with inertia, the weak coupling regime features mixing, the state of the network when the phases (but not velocities) of all oscillators are distributed uniformly around the unit circle. We study patterns, which emerge when mixing loses stability under the variation of the strength of coupling. We identify a pitchfork (PF) and an Andronov-Hopf (AH) bifurcations in the model with multimodal intrinsic frequency distributions. To this effect, we use a combination of the linear stability analysis and Penrose diagrams, a geometric technique for studying stability of mixing. We show that the type of a bifurcation and a nascent spatiotemporal pattern depend on the interplay of the qualitative properties of the intrinsic frequency distribution and network connectivity.

arXiv comment20 pages, 8 figures. Revised version accepted to Journal of Nonlinear Science. arXiv admin note: text overlap with arXiv:2105.07541

References in corpus (12)

• The Kuramoto model in complex networks
• On the Dynamics of Large Particle Systems in the Mean Field Limit
• Synchronization in populations of globally coupled oscillators with inertial effects
• A proof of the Kuramoto conjecture for a bifurcation structure of the infinite dimensional Kuramoto model
• Stability and bifurcation for the Kuramoto model
• The continuum limit of the Kuramoto model on sparse random graphs
• Nonequilibrium first-order transition in coupled oscillator systems with inertia and noise
• Low-dimensional behavior of Kuramoto model with inertia in complex networks
• Bifurcations in the Kuramoto model on graphs
• Stability and control of power grids with diluted network topology
• Stability of clusters in the second-order Kuramoto model on random graphs
• Chimeras unfolded