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Overdetermined ODEs and rigid periodic states in network dynamics
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Stewart, Ian (2022) Overdetermined ODEs and rigid periodic states in network dynamics. Portugaliae Mathematica, 79 (1/2). pp. 85-161. doi:10.4171/pm/2080 ISSN 0032-5155.
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Official URL: https://doi.org/10.4171/pm/2080
Abstract
We consider four long-standing Rigidity Conjectures about synchrony and phase patterns for hyperbolic periodic orbits of admissible ODEs for networks. Proofs of stronger local versions of these conjectures, published in 2010–12, are now known to have a gap, but remain valid for a broad class of networks. Using different methods we prove local versions of the conjectures under a stronger condition, ‘strong hyperbolicity’, which is related to a network analogue of the Kupka–Smale Theorem. Under this condition we also deduce global versions of the conjectures and an analogue of the H/KH/K Theorem in equivariant dynamics.We prove the Rigidity Conjectures for all 1- and 2-colourings and all 2- and 3-node networks by showing that strong hyperbolicity is generic in these cases.
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software |
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| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| SWORD Depositor: | Library Publications Router | ||||||
| Library of Congress Subject Headings (LCSH): | Synchronization, Dynamics, Differentiable dynamical systems, Neural networks (Computer science) | ||||||
| Journal or Publication Title: | Portugaliae Mathematica | ||||||
| Publisher: | European Mathematical Society - EMS - Publishing House GmbH | ||||||
| ISSN: | 0032-5155 | ||||||
| Official Date: | 27 July 2022 | ||||||
| Dates: |
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| Volume: | 79 | ||||||
| Number: | 1/2 | ||||||
| Page Range: | pp. 85-161 | ||||||
| DOI: | 10.4171/pm/2080 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||
| Copyright Holders: | © 2022 Sociedade Portuguesa de Matemática | ||||||
| Date of first compliant deposit: | 2 September 2022 | ||||||
| Date of first compliant Open Access: | 2 September 2022 |
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