Fig. 5.

Limit Cycle phase boundaries and onset. (A) $f_{nm,sat}$ vs. γ for $\mu =0.05$. (B) $f_{nm,sat}$ vs. μ for $\gamma =0.025$. (A and B) Squares: individual realizations. Solid line: ensemble average. (C) Ensemble averaged ${\overline{f}}_{nm,sat}$ vs. γ at $\mu =0.05$, varying the number of grains N. (D) Average number of shear cycles needed to establish a limit cycle $\overline{n}*$ vs. N for $\mu =0.05$. Lines show fits to $\overline{n}*=aN$ with (dashed line, $\gamma =0.0125$) $\gamma a=3.4\times {10}^{-3}\pm 5.6\times {10}^{-4}$, (dotted line, $\gamma =0.025$) $\gamma a=3.5\times {10}^{-3}\pm 6.9\times {10}^{-4}$, and (solid line, $\gamma =0.03$) $\gamma a=4.6\times {10}^{-3}\pm 1.4\times {10}^{-3}$. For ensemble averages, we used 20 realizations for $N\le \mathrm{2,000}$, 12 realizations for $N=\mathrm{4,000}$, and 5 realizations for $N=\mathrm{8,000}$. Error bars in D show the SD within an ensemble.