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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Kuramoto Oscillators: Dynamical Systems meet Compu
tational Algebraic Geometry - Henry Schenck (Aubur
n University)
DTSTART;TZID=Europe/London:20240126T141500
DTEND;TZID=Europe/London:20240126T151500
UID:TALK208090AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/208090
DESCRIPTION:When does a system of coupled oscillators synchron
ize? This central question in dynamical systems ar
ises in applications ranging from power grids to n
euroscience to biology: why do fireflies sometimes
begin flashing in harmony? Perhaps the most studi
ed model is due to Kuramoto (1975)\; we \;ana
lyze the Kuramoto model from the perspectives of a
lgebra and topology. Translating dynamics into a s
ystem of algebraic equations enables us to identif
y classes of network topologies that exhibit unexp
ected behaviors. Many previous studies focus on sy
nchronization of networks having high connectivity
\, or of a specific type (e.g. circulant networks)
\; our work also tackles more general situations.\
nWe introduce the Kuramoto ideal\; an algebraic an
alysis of this ideal allows us to identify feature
s beyond synchronization\, such as positive dimens
ional components in the set of potential solutions
(e.g. curves instead of points). We prove suffici
ent conditions on the network structure for such s
olutions to exist. The points lying on a positive
dimensional component of the solution set can neve
r correspond to a linearly stable state. We apply
this framework to give a complete analysis of line
ar stability for all networks on at most eight ver
tices. The talk will include a surprising (at leas
t to us!) connection to Segre varieties\, and clos
e with examples of computations using the Macaulay
2 software package "Oscillator"\nJoint work with H
eather Harrington (Oxford/Dresden) and Mike Stillm
an (Cornell).\n \;
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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