Lyapunov, Adaptive, and Optimal Design Techniques for Cooperative Systems on Directed Communication Graphs

A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation (\(\eta\)) added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, \(| {\bf v}_a | =0\)) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous since \(| {\bf v}_a |\) is found to scale as \((\eta_c-\eta)^\beta\) with \(\beta\simeq 0.45\).