Figure 1. Limit cycles.

Repetitive particle trajectories in a period two limit cycle. (a) The entire system, (b) local environment and the trajectories that each particle is undergoing and (c) the trajectory of one particle. (d) The strain applied using the Lees–Edwards boundary conditions (purple arrows show how the Lees–Edwards boundaries move with respect to the simulation square when the system is sheared in the positive direction). Since the limit cycle has period two the trajectories repeat themselves only after two shearing cycles (the blue and green lines in d). The particle starts from the orange initial point and moves to the right on the blue trajectory, due to the external strain that shears the material to the right, then moves back to the centre and to the left, when the strain is changed accordingly (blue curve on d). When the strain is set back to zero, the particle reaches the purple point. Then, when the strain is applied again to the right, the particle moves accordingly, but this time on the green trajectory. The particle then moves to the centre and to the left due to the applied strain (green curve in d). Eventually, the particle comes back to the orange point, the initial condition. The same two trajectories repeat in the next two cycles and following cycles.