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The Ince equation, Painlevé-VI, and the elliptic vortex instability

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Date and time:Ìý

Thursday, March 19, 2015 - 2:00pm

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ECCR 257

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Hill's equation is a periodic differential equation that arises in the modeling of parametric resonance.Ìý Cases include the Mathieu and Ince equations.Ìý In their parameter spaces, they display 'resonance tongues': regions of exponential instability.Ìý The instability (or more generally the monodromy) of the Ince equation has been studied, and conditions for the absence of tongues are known.Ìý Systems modeled by the Ince equation include the elastic pendulum, where the instability transfers energy from the vertical oscillatory mode to the swinging mode.Ìý Also, in the 1980s Bayly and Waleffe proposed as a route to turbulence, in inviscid flow, the 3-D instability of a 2-D elliptical vortex; this instability turns out also to be governed by an Ince equation.Ìý Subsequently, Craik pointed out that a generalization holds when Coriolis forces are present, as in a rotating frame.Ìý We show that in this case, the Ince equation generalizes to one the monodromy of which is computable by integrating a Painlevé-VI equation. This allows the effect of Coriolis forces on instability region boundaries to be computed.