FIGURE 2. Spontaneous synchronization in a network of limit-cycle oscillators with distributed natural frequencies.

The state of each oscillator is represented geometrically as a dot in the complex plane. The amplitude and phase of the oscillation correspond to the radius and angle of the dot in polar coordinates. Colours code the oscillators' natural frequencies, running from slowest (red) to fastest (violet). In the absence of coupling, each oscillator would settle onto its limit cycle (circle) and rotate at its natural frequency. However, here all the oscillators are also pulled towards the mean field that they generate collectively (shown as an asterisk at the centre of the population). Time increases from left to right, and from top to bottom. Starting from a random initial condition, the oscillators self-organize by collapsing their amplitudes; then they sort their phases so that the fastest oscillators are in the lead. Ultimately they all rotate as a synchronized pack, with locked amplitudes and phases. The governing equations describe a mean-field model of a laser array²³. (Simulation provided by R. Oliva.)