Impact of anticipation in dynamical systems

Many animals, including humans, have predictive capabilities and, presumably, base their behavioral decisions-at least partially-upon an anticipated state of their environment. We explore a minimal version of this idea in the context of particles that interact according to a pairwise potential. Anticipation enters the picture by calculating the interparticle forces from linear extrapolations of the particle positions some time tau in the future. Simulations show that for intermediate values of t, compared to a transient time scale defined by the potential and the initial conditions, the particles form rotating clusters in which the particles are arranged in a hexagonal pattern. Analysis of the system shows that anticipation induces energy dissipation and we show that the kinetic energy asymptotically decays as 1/t. Furthermore, we show that the angular momentum is not necessarily conserved for tau > 0, and that asymmetries in the initial condition therefore can cause rotational movement. These results suggest that anticipation could play an important role in collective behavior, since it may induce pattern formation and stabilizes the dynamics of the system.