Fig. 6. Example 1: ring of 8 violin players.

For delay δ < 1 s or 3 < δ < 4 s, the network of violin players can be represented as undirected (a–d), and for 1 s < δ < 3 s as a directed network with “arrowhead” topology, oriented either clockwise (as in f, g, i, j) or counterclockwise. a Undirected network with minimal balanced coloring (“in-phase” global synchronization pattern). b Corresponding quotient network. c Undirected network with (non-minimal) balanced coloring. d Corresponding quotient network. e Basins of attraction of the two stable solutions admitted by the quotient network (d): “in-phase” global synchronization (yellow region) and “out-of-phase” synchronization with two clusters oscillating in antiphase (green region). f Directed network with “arrowhead” topology and balanced coloring. g Corresponding quotient network. h Asymptotic phase differences Δφ_i1 vs. δ: it is apparent that Δφ_i1 grows linearly with δ. i Directed network with “arrowhead” topology and minimal balanced coloring. j Corresponding quotient network. k MLE λ corresponding to this synchronization pattern, showing stability as λ(δ) < 0 for any δ ∈ (1, 3).