event
Monday 12 Mar 2012: ****POSTPONED***** MCS colloquium: Coupled Oscillators and the Mobius Group
Prof. Ian Stewart - University of Warwick
Harrison 203 15:00-16:00
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POSTPONED DUE TO ILLNESS BUT TEA AND CAKES WILL GO AHEAD AT 4pm
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In 1991 KY Tsang and others observed numerically that equations modelling an arbitrarily large array of Josephson junctions seemed to exhibit low-dimensional dynamics. In 1994 S Watanabe and SH Strogatz explained these ?ndings analytically, by deriving rather mysterious coordinate transformations that reduce the equations to a 3-dimensional system. In 2009 SA Marvel, RE Mirollo, and Strogatz generalised the results to "sinusoidally coupled" phase oscillators, obtaining an explicit reduction of the dynamics to a parametrised family of ODEs on the 3-dimensional Mobius group. (This differs from the usual reduction to dynamics on the orbit space of a symmetry group.)
The appearance of the Mobius group suggests that projective geometry might provide a simpler explanation. We sketch how that viewpoint leads to an alternative and simpler proof that trajectories lie on orbits of the Mobius group, by considering the familiar cross-ratio invariant of projective geometry. We derive a different explicit form for the reduced ODE, by exploiting homogeneous coordinates to linearise the action of the Mobius group.
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After the colloquium, tea coffee and cakes will be served in the Harrison Foyer.
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