Stability of infinite systems of coupled oscillators via random walks on weighted graphs
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- by Jason J. Bramburger PDF
- Trans. Amer. Math. Soc. 372 (2019), 1159-1192 Request permission
Abstract:
Weakly coupled oscillators are used throughout the physical sciences, particularly in mathematical neuroscience to describe the interaction of neurons in the brain. Systems of weakly coupled oscillators have a well-known decomposition to a canonical phase model, which forms the basis of our investigation in this work. Particularly, our interest lies in examining the stability of synchronous (phase-locked) solutions to this phase system: solutions with phases having the same temporal frequency but differing through time-independent phase-lags. The main stability result of this work comes from adapting a series of investigations into random walks on infinite weighted graphs. We provide an interesting link between the seemingly unrelated areas of coupled oscillators and random walks to obtain algebraic decay rates of small perturbations off the phase-locked solutions under some minor technical assumptions. We also provide some interesting and motivating examples that demonstrate the stability of phase-locked solutions, particularly that of a rotating wave solution arising in a well-studied paradigm in the theory of coupled oscillators.References
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Additional Information
- Jason J. Bramburger
- Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island, 02912
- MR Author ID: 1087209
- Received by editor(s): March 22, 2018
- Received by editor(s) in revised form: May 2, 2018
- Published electronically: March 20, 2019
- Additional Notes: This research was supported by an Ontario Graduate Scholarship held at the University of Ottawa.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 1159-1192
- MSC (2010): Primary 34Dxx; Secondary 05C81
- DOI: https://doi.org/10.1090/tran/7609
- MathSciNet review: 3968799