# Resolution of the Quinn-Rand-Strogatz constant of nonlinear physics

Bailey, David H. and Borwein, Jonathan M. and Crandall, Richard E. (2009) Resolution of the Quinn-Rand-Strogatz constant of nonlinear physics. Experimental Mathematics, 18 . pp. 107-116.

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## 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 $c_1 = 0.605443657\ldots$. 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.

Item Type: Article pubdom FALSE 65-xx Numerical analysis41-xx Approximations and expansions UNSPECIFIED lingyun ye 05 Jun 2007 05 Jan 2015 16:16 https://docserver.carma.newcastle.edu.au/id/eprint/345

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