## Experimental Mathematics

- Experiment. Math.
- Volume 18, Issue 1 (2009), 107-116.

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

D. H. Bailey, J. M. Borwein, and R. E. Crandall

#### 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.

#### Article information

**Source**

Experiment. Math., Volume 18, Issue 1 (2009), 107-116.

**Dates**

First available in Project Euclid: 27 May 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1243430534

**Mathematical Reviews number (MathSciNet)**

MR2548991

**Zentralblatt MATH identifier**

1200.11098

**Subjects**

Primary: 11Y60: Evaluation of constants

Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$

**Keywords**

Winfree oscillators high-precision arithmetic Hurwitz zeta Richardson extrapolation

#### Citation

Bailey, D. H.; Borwein, J. M.; Crandall, R. E. Resolution of the Quinn–Rand–Strogatz Constant of Nonlinear Physics. Experiment. Math. 18 (2009), no. 1, 107--116. https://projecteuclid.org/euclid.em/1243430534