The impact of anticipation in dynamical systems

We have just submitted a manuscript that investigates the role of prediction in models of collective behaviour. The idea is quite simple: take a model where animals attract/repel each other based on a pairwise potential, and adjust it so that the animals act not on current, but on future positions (including their own). These anticipated or predicted position are assumed to be simple linear extrapolations some time T into the future. In other words, instead of using current positions x to calculate forces, we use x+T*v, where v is the velocity.

This seemingly simple modification changes the dynamics dramatically. For a typical interaction potential (e.g. Morse potential) the case of no prediction yields no pattern formation, simply particles attracting and colliding. But for an intermediate range of T we observe rapid formation of a milling structure. This means that prediction induces pattern formation and stabilises the dynamics.

Abstract:
The flocking of animals is often modelled as a dynamical system, in which individuals are represented as particles whose interactions are determined by the current state of the system. Many animals, however, including humans, have predictive capabilities, and presumably base their behavioural 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 extrapolation of the positions some time $\tau$ into the future. Our analysis shows that for intermediate values of $\tau$ the particles rapidly form milling structures, induced by velocity alignment that emerges from the prediction. We also show that for $\tau > 0$, any dynamical system governed by an even potential becomes dissipative. These results suggest that anticipation could play an important role in collective behaviour, since it induces pattern formation and stabilises the dynamics of the system. 

arXiv: http://arxiv.org/abs/1611.03637