Dissertations

Dissertation

12-31-2020

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

Roy Goodman

Denis L. Blackmore

Michael Siegel

Panayotis G. Kevrekidis

Richard O. Moore

Abstract

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 of aperiodic leapfrogging, the perturbed solutions could transition into one of two behaviors: a bounded orbit he called `\emph{walkabout}' and an unbounded orbit he called “disintegration”.' In the walkabout orbit, two like-signed vortices couple together, and the motion resembles a three-vortex system. In disintegration, four vortices separate into two pairs---each pair consisting one negative and one positive vortex---that escape to infinity along two transverse rays.

Two goals are addressed in this dissertation:Goal 1: To rigorously demonstrate, without numerics, the exact algebraic value for which the Hamiltonian pitchfork bifurcation occurs. Goal 2:Understand how, as the parameter is decreased, the dynamics transitions between the various regimes and escape become first possible and then almost inevitable, as well as identifying the structures in phase-space that are responsible for the transition between these regimes.

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