An S = 0 equilibrium with five groups, and the corresponding equilateral pentagon. |ασ | = r 0 < 1 and σ = 1, 2, 3, 4, and 5.
An equilibrium with S ≠ 0. The group order parameters lie on a straight line through the origin, and their magnitudes are all solutions to Eq. (17) .
(Color) A sampling of oscillator phases from a numerical simulation with M = 3. Individual oscillator phases appear as solid dots on the unit circle, and the colors (red, green, and blue) correspond to different oscillator groups. Group order parameters ασ appear as larger open circles.
(Color) Results for a case with three groups (M = 3), obtained by integrating Eq. (7) with , and defined as in Sec. IV A and the perturbations ωσ , δσ , and given by Eq. (63) . (a) The initial conditions, α 1(0) (red dot), α 2(0) (green dot), and α 3(0) (blue dot); (b) the evolution of |S|; (c) evolution of rσ with respect to r 0; (d) evolution of ψ 1, ψ 2, and ψ 3 (in radians); (e) evolution of , where . In (e), the horizontal line denotes the value w = wω + wk given in Eq. (57) .
(Color) Numerical results for three cases with four groups (M = 4). ((a), (c), and (e)) The evolution of ψ 1, ψ 2, ψ 3, and ψ 4 (in radians); ((b), (d), and (f)) the corresponding evolution of T. The pair switches in each simulation are possible only when the value of T nears zero. In (a) and (b), this occurs at t ≈ 130 and t ≈ 860. In (c) and (d), the pairs switch in a periodic manner throughout the simulation. In (e) and (f), we see that T ≈ 0 several times in the simulation but only one pair switch occurs, at t ≈ 35.
(Color) Numerical results for a case with five groups (M = 5). (a) The evolution of ψ 1, ψ 2, ψ 3, ψ 4, and ψ 5 (in radians) and (b) the corresponding evolution of T. The value of T remains large throughout the simulation, in contrast to the cases with M = 4 in Fig. 5 .
(Color) Numerical results for a case with six groups (M = 6). (a) The evolution of ψ 1, ψ 2, ψ 3, ψ 4, ψ 5, and ψ 6 (in radians) and (b) the corresponding evolution of T.
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