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Kassabov M, Strogatz SH, Townsend A. A global synchronization theorem for oscillators on a random graph. CHAOS (WOODBURY, N.Y.) 2022; 32:093119. [PMID: 36182402 DOI: 10.1063/5.0090443] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2022] [Accepted: 08/19/2022] [Indexed: 06/16/2023]
Abstract
Consider n identical Kuramoto oscillators on a random graph. Specifically, consider Erdős-Rényi random graphs in which any two oscillators are bidirectionally coupled with unit strength, independently and at random, with probability 0 ≤ p ≤ 1. We say that a network is globally synchronizing if the oscillators converge to the all-in-phase synchronous state for almost all initial conditions. Is there a critical threshold for p above which global synchrony is extremely likely but below which it is extremely rare? It is suspected that a critical threshold exists and is close to the so-called connectivity threshold, namely, p ∼ log ( n ) / n for n ≫ 1. Ling, Xu, and Bandeira made the first progress toward proving a result in this direction: they showed that if p ≫ log ( n ) / n, then Erdős-Rényi networks of Kuramoto oscillators are globally synchronizing with high probability as n → ∞. Here, we improve that result by showing that p ≫ log ( n ) / n suffices. Our estimates are explicit: for example, we can say that there is more than a 99.9996 % chance that a random network with n = 10 and p > 0.011 17 is globally synchronizing.
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Affiliation(s)
- Martin Kassabov
- Department of Mathematics, Cornell University, Ithaca, New York 14853, USA
| | - Steven H Strogatz
- Department of Mathematics, Cornell University, Ithaca, New York 14853, USA
| | - Alex Townsend
- Department of Mathematics, Cornell University, Ithaca, New York 14853, USA
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Zahedinejad M, Awad AA, Muralidhar S, Khymyn R, Fulara H, Mazraati H, Dvornik M, Åkerman J. Two-dimensional mutually synchronized spin Hall nano-oscillator arrays for neuromorphic computing. NATURE NANOTECHNOLOGY 2020; 15:47-52. [PMID: 31873287 DOI: 10.1038/s41565-019-0593-9] [Citation(s) in RCA: 51] [Impact Index Per Article: 12.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/24/2019] [Accepted: 11/08/2019] [Indexed: 06/10/2023]
Abstract
In spin Hall nano-oscillators (SHNOs), pure spin currents drive local regions of magnetic films and nanostructures into auto-oscillating precession. If such regions are placed in close proximity to each other they can interact and may mutually synchronize. Here, we demonstrate robust mutual synchronization of two-dimensional SHNO arrays ranging from 2 × 2 to 8 × 8 nano-constrictions, observed both electrically and using micro-Brillouin light scattering microscopy. On short time scales, where the auto-oscillation linewidth [Formula: see text] is governed by white noise, the signal quality factor, [Formula: see text], increases linearly with the number of mutually synchronized nano-constrictions (N), reaching 170,000 in the largest arrays. We also show that SHNO arrays exposed to two independently tuned microwave frequencies exhibit the same synchronization maps as can be used for neuromorphic vowel recognition. Our demonstrations may hence enable the use of SHNO arrays in two-dimensional oscillator networks for high-quality microwave signal generation and ultra-fast neuromorphic computing.
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Affiliation(s)
- Mohammad Zahedinejad
- Physics Department, University of Gothenburg, Gothenburg, Sweden
- NanOsc AB, Kista, Sweden
| | - Ahmad A Awad
- Physics Department, University of Gothenburg, Gothenburg, Sweden
- NanOsc AB, Kista, Sweden
| | | | - Roman Khymyn
- Physics Department, University of Gothenburg, Gothenburg, Sweden
- NanOsc AB, Kista, Sweden
| | - Himanshu Fulara
- Physics Department, University of Gothenburg, Gothenburg, Sweden
- NanOsc AB, Kista, Sweden
| | - Hamid Mazraati
- NanOsc AB, Kista, Sweden
- Material and Nanophysics, School of Engineering Sciences, KTH Royal Institute of Technology, Kista, Sweden
| | - Mykola Dvornik
- Physics Department, University of Gothenburg, Gothenburg, Sweden
- NanOsc AB, Kista, Sweden
| | - Johan Åkerman
- Physics Department, University of Gothenburg, Gothenburg, Sweden.
- NanOsc AB, Kista, Sweden.
- Material and Nanophysics, School of Engineering Sciences, KTH Royal Institute of Technology, Kista, Sweden.
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