Resolution of the Quinn–Rand–Strogatz Constant of Nonlinear Physics

Abstract

Herein we develop connections between zeta functions and some recent ``mysterious'' constants of nonlinear physics. In an important analysis of coupled Winfree oscillators, Quinn, Rand, and Strogatz developed a certain $N$-oscillator scenario whose bifurcation phase offset $\phi$ is implicitly defined, with a conjectured asymptotic behavior $\sin \phi \sim 1 - c_1/N$, with experimental estimate $c_1 = 0.605443657\dotsc$. We are able to derive the exact theoretical value of this ``QRS constant'' $c_1$ as a real zero of a particular Hurwitz zeta function. This discovery enables, for example, the rapid resolution of $c_1$ to extreme precision. Results and conjectures are provided in regard to higher-order terms of the $\sin \phi$ asymptotic, and to yet more physics constants emerging from the original QRS work.