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Dances And Escape Of The Vortex Quartet, Brandon Behring

Dissertations

This dissertation considers the linear stability of a one-parameter family of periodic solutions of the four-vortex problem known as 'leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices, were known to W. Gröbli (1877) and A. E. H. Love (1883) and can be parameterized by a dimensionless parameter related to the geometry of the initial configuration. Simulations by Acheson and numerical Floquet analysis by Tophøj and Aref both indicate, to many digits, that the bifurcation occurs at a value related to the inverse square of the golen ratio. Acheson observed that, after an initial period …