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Resolution of the Quinn-Rand-Strogatz constant of nonlinear physics. (English) Zbl 1200.11098

In an analysis of coupled Winfree oscillators, D. Quinn, R. Rand, and S. Strogatz [Phys. Rev. E 75, 036218-1-10 (2007)] developed a certain \(N\)-oscillator scenario whose bifurcation phase offset \(\phi=\phi_N\) is implicitly defined, with a conjectured asymptotic behavior \(s_N=\sin\phi_N \sim 1 - c_1/N\); here \(s=s_N\) satisfies the algebraic equation \[ \sum_{n=0}^{N-1}\biggl(2\sqrt{1-s^2(1-2n/(N-1))^2} -\frac1{\sqrt{1-s^2(1-2n/(N-1))^2}}\biggr)=0. \] The article under review is aimed at proving this observation and even the general asymptotic formula \[ s_N=1-\frac{c_1}N-\frac{c_2}{N^2}-\frac{c_3}{N^3}-\dotsb \qquad\text{as}\quad N\to\infty, \] but also at developing computational and theoretic techniques for estimating the (QRS) constants \(c_1=0.605443657\dots\) and \(c_2=-0.104685459\dots\) . For example, \(c_1\) happens to be a unique real zero of the Hurwitz zeta function \(f(z)=\zeta(1/2,z/2)\) on the interval \(0<z<2\). Another curious constant \(C\) from the Quinn–Rand–Strogatz paper is rigorously identified with \(-f'(c_1)\).

MSC:

11Y60 Evaluation of number-theoretic constants
11M41 Other Dirichlet series and zeta functions
11Z05 Miscellaneous applications of number theory

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