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Equilibria in Kuramoto Oscillator Networks: An Algebraic Approach

https://doi.org/10.1137/21M1457321

Nguyen, Tung T. ; Budzinski, Roberto C. ; Ðoàn, Jacqueline ; Pasini, Federico W. ; Mináč, Ján ; Muller, Lyle E. ( June 2023 , SIAM Journal on Applied Dynamical Systems)

Free, publicly-accessible full text available June 30, 2024
Analytical prediction of specific spatiotemporal patterns in nonlinear oscillator networks with distance-dependent time delays

https://doi.org/10.1103/PhysRevResearch.5.013159

Budzinski, Roberto C. ; Nguyen, Tung T. ; Benigno, Gabriel B. ; Đoàn, Jacqueline ; Mináč, Ján ; Sejnowski, Terrence J. ; Muller, Lyle E. ( March 2023 , Physical Review Research)

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Geometry unites synchrony, chimeras, and waves in nonlinear oscillator networks

https://doi.org/10.1063/5.0078791

Budzinski, Roberto C. ; Nguyen, Tung T. ; Đoàn, Jacqueline ; Mináč, Ján ; Sejnowski, Terrence J. ; Muller, Lyle E. ( March 2022 , Chaos: An Interdisciplinary Journal of Nonlinear Science)

One of the simplest mathematical models in the study of nonlinear systems is the Kuramoto model, which describes synchronization in systems from swarms of insects to superconductors. We have recently found a connection between the original, real-valued nonlinear Kuramoto model and a corresponding complex-valued system that permits describing the system in terms of a linear operator and iterative update rule. We now use this description to investigate three major synchronization phenomena in Kuramoto networks (phase synchronization, chimera states, and traveling waves), not only in terms of steady state solutions but also in terms of transient dynamics and individual simulations. These results provide new mathematical insight into how sophisticated behaviors arise from connection patterns in nonlinear networked systems.
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