Abstract for Roy Goodman

Roy Goodman (New Jersey Institute of Technology)

Friday, March 31, 2023

New insights into the leapfrogging vortex problem

Abstract: We investigate the 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 Grobli (1877) and Love (1883), and can be parameterized by a dimensionless parameter related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Tophøj and Aref (2012) both indicate, to many digits, that the bifurcation occurs when $\alpha=\phi^{-2}$, where $\phi$ is the golden ratio. These numerical studies indicated a sequence of behaviors that emerge as this parameter is further decreased, leading to the disintegration of the leapfrogging orbit into a pair of dipoles that escape to infinity along transverse rays.

This study has two objectives. The first is to rigorously explain the origin of this remarkable bifurcation value and to generalize this analysis to the leapfrogging of unequal vortex pairs. The second is to understand the sequence of transitions in the phase space of the system that allows for the emergence of the various behaviors. While the first objective is essentially linear, finding the answer requires applying several tricks from the classical mechanics toolkit. The second objective is inherently nonlinear, and our approach involves both analysis and numerics. In particular, we make use of the recently developed technique of Lagrangian descriptors to visualize the phase space structures, including invariant manifolds.