Stability of leapfrogging vortex pairs: A semi-analytic approach

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 W. Gröbli [Ph.D. thesis, Georg-August-Universität Göttingen, 1877] and A. E. H. Love [Proc. London Math. Soc. 1, 185 (1883)10.1112/plms/s1-25.1.185] and can be parameterized by a dimensionless parameter α related to the geometry of the initial configuration. Simulations by Acheson [Eur. J. Phys. 21, 269 (2000)10.1088/0143-0807/21/3/310] and numerical Floquet analysis by Tophøj and Aref [Phys. Fluids 25, 014107 (2013)10.1063/1.4774333] both indicate, to many digits, that the bifurcation occurs when 1/α=φ2, where φ is the golden ratio. This study aims to explain the origin of this remarkable value. Using a trick from the gravitational two-body problem, we change variables to render the Floquet problem in an explicit form that is more amenable to analysis. We then implement G. W. Hill's [Acta Math. 8, 1 (1886)10.1007/BF02417081] method of harmonic balance to high order using computer algebra to construct a rapidly converging sequence of asymptotic approximations to the bifurcation value, confirming the value found earlier.