Algebraic aspects of homogeneous Kuramoto oscillators
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- by Heather A. Harrington, Hal Schenck and Mike Stillman;
- Math. Comp.
- DOI: https://doi.org/10.1090/mcom/4072
- Published electronically: February 18, 2025
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Abstract:
We investigate algebraic characteristics of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of network topologies that exhibit unexpected behaviors. Many previous studies focus on synchronization of networks having high connectivity, or of a specific type (e.g. circulant networks). We introduce the Kuramoto ideal; an algebraic analysis of this ideal allows us to identify features beyond synchronization, such as positive dimensional components in the set of potential solutions (e.g. curves instead of points). We prove sufficient conditions on the network structure for such solutions to exist. The points lying on a positive dimensional component of the solution set can never correspond to a linearly stable state. We apply this framework to give a complete analysis of linear stability for all networks on at most eight vertices. Furthermore, we describe a construction of networks on an arbitrary number of vertices having linearly stable states that are not twisted stable states.References
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Bibliographic Information
- Heather A. Harrington
- Affiliation: Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK; Wellcome Centre for Human Genetics, University of Oxford, Oxford OX3 7BN, UK; Max Planck Institute of Molecular Cell Biology and Genetics, 01307 Dresden, Germany; Centre for Systems Biology Dresden, 01307 Dresden, Germany; and Faculty of Mathematics, Technische Universitat Dresden, 01062 Dresden, Germany
- ORCID: 0000-0002-1705-7869
- Email: harrington@maths.ox.ac.uk
- Hal Schenck
- Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849; and Mathematical Institute, University of Oxford, Oxford OX3 7BN, UK
- MR Author ID: 621581
- ORCID: 0000-0002-1692-7500
- Email: hks0015@auburn.edu
- Mike Stillman
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14850; and Mathematical Institute, University of Oxford, Oxford OX3 7BN, UK
- MR Author ID: 167420
- ORCID: 0000-0003-0078-3116
- Email: mike@math.cornell.edu
- Received by editor(s): May 13, 2024
- Received by editor(s) in revised form: November 18, 2024, and December 27, 2024
- Published electronically: February 18, 2025
- Additional Notes: The first author was supported by EPSRC EP/R018472/1, EP/R005125/1, EP/T001968/1, RGF 201074, UF150238, and a Royal Society University Research Fellowship. The second author was supported by NSF DMS 2006410 and a Leverhulme Visiting Professorship. The third author was supported by NSF DMS 2001367 and a Simons Fellowship.
- © Copyright 2025 American Mathematical Society
- Journal: Math. Comp.
- MSC (2020): Primary 90C26, 90C35, 34D06, 35B35
- DOI: https://doi.org/10.1090/mcom/4072