QRS Constant

Quinn et al. (2007) investigated a class of  coupled oscillators whose bifurcation phase offset had a conjectured asymptotic behavior of , with an experimental estimate for the constant  as  (OEIS A131329). Rather amazingly, Bailey et al. (2007) were able to find a closed form for  as the unique root of  in the interval , where  is a Hurwitz zeta function.

A related constant conjectured by Quinn et al. (2007) to exist was defined in terms of

(1)

and given by

(2)

(OEIS A131330). Even more amazingly, the exact value of this constant was also found by Bailey et al. (2007) without full proof, but with enough to indicate that such a proof could in principle be constructed, to have the exact value

(3)

REFERENCES:

Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Resolution of the Quinn-Rand-Strogatz Constant of Nonlinear Physics." Preprint. June 4, 2007. https://users.cs.dal.ca/~jborwein/QRS.pdf.

Quinn, D. ; Rand, R.; and Strogatz, S. "Singular Unlocking Transition in the Winfree Model of Coupled Oscillators." Phys. Rev. E 75, 036218-1-10, 2007.

Sloane, N. J. A. Sequences A131329 and A131330 in "The On-Line Encyclopedia of Integer Sequences."