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

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 [14] developed a certain N -oscillator scenario whose bifurcation phase offset φ is implicitly defined, with a conjectured asymptotic behavior: sinφ ∼ 1 − c1/N , with experimental estimate c1 = 0.605443657 . . .. We are able to derive the exact theoretical value of this “QRS constant” c1 as a real zero of a particular Hurwitz zeta function. This discovery enables, for example, the rapid resolution of c1 to extreme precision. Results and conjectures are provided in regard to higher-order terms of the sinφ asymptotic, and to yet more physics constants emerging from the original QRS work. ∗Lawrence Berkeley National Laboratory, Berkeley, CA 94720, [email protected]. Supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. †Faculty of Computer Science, Dalhousie University, Halifax, NS, B3H 2W5, Canada, [email protected]. Supported in part by NSERC and the Canada Research Chair Programme. ‡Center for Advanced Computation, Reed College, Portland OR, [email protected].