Abstract for Roy Goodman
### Roy Goodman (New Jersey Institute of Technology)

Friday, March 31, 2023

11:00-11:40am

### 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 TophÃ¸j 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.