The impact of anticipation in dynamical systems

TL;DR

This paper investigates how anticipation, modeled as future position extrapolation, influences collective motion in particle systems, leading to pattern formation, energy dissipation, and potential rotational behaviors.

Contribution

It introduces a minimal model incorporating anticipation into particle interactions, revealing its effects on pattern formation and system stabilization.

Findings

Anticipation causes particles to form rotating hexagonal clusters.

Energy dissipates over time, with kinetic energy decaying as 1/t.

Anticipation can induce rotational movement even without initial angular momentum.

Abstract

Collective motion in biology 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 extrapolations of the particle positions some time $\tau$ into the future. Simulations show that for intermediate values of $\tau$, 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 behaviour, since it induces pattern formation and stabilises the dynamics of the system.