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SUMMARY:Kuramoto Oscillators: Dynamical Systems meet Computational Algebra
 ic Geometry - Henry Schenck (Auburn University)
DTSTART:20240126T141500Z
DTEND:20240126T151500Z
UID:TALK208090@talks.cam.ac.uk
DESCRIPTION:When does a system of coupled oscillators synchronize? This ce
 ntral question in dynamical systems arises in applications ranging from po
 wer grids to neuroscience to biology: why do fireflies sometimes begin fla
 shing in harmony? Perhaps the most studied model is due to Kuramoto (1975)
 \; we &nbsp\;analyze the Kuramoto model from the perspectives of algebra a
 nd topology. Translating dynamics into a system of algebraic equations ena
 bles us to identify classes of network topologies that exhibit unexpected 
 behaviors. Many previous studies focus on synchronization of networks havi
 ng high connectivity\, or of a specific type (e.g. circulant networks)\; o
 ur work also tackles more general situations.\nWe introduce the Kuramoto i
 deal\; an algebraic analysis of this ideal allows us to identify features 
 beyond synchronization\, such as positive dimensional components in the se
 t of potential solutions (e.g. curves instead of points). We prove suffici
 ent conditions on the network structure for such solutions to exist. The p
 oints lying on a positive dimensional component of the solution set can ne
 ver 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. The talk will include a surprising (at least to us!) connection
  to Segre varieties\, and close with examples of computations using the Ma
 caulay2 software package "Oscillator"\nJoint work with Heather Harrington 
 (Oxford/Dresden) and Mike Stillman (Cornell).\n&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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