Fig. 4

Phase diagrams show control can be achieved to arbitrary phase locking Δ modulating rotor properties. a–c Simulations with thermal fluctuations (with experiments as markers), and d–f theory without noise. The average state in the simulations is the angle of the average direction, \({\mathrm{arg}}\left( {\left\langle {{\sum} e^{i{\mathrm{\Delta }}}} \right\rangle } \right)\), while it is simply the position of the global minimum of potential Δ_min in the theory. The maps represent cuts of the phase diagram at h = 10 μm and δ = 3π/4 in a, d, R = 4.7 μm and δ = 3π/4 in b, e, and R = 4.7 μm and h = 10 μm in (c), f. The two types of markers in a–c show experiments with an average phase difference below () and above () Δ = 0.1 rad. The cut (line in a and d) is shown in detail in (g) and h for simulations and theory, respectively. Theory shows a first-order transition, which is smoothed out in the experiments and simulations. Error bars are standard deviation over all runs