Interaction between a robot and Bunimovich stadium billiards.

The robot-environment-task triad provides many opportunities to revisit physical problems with fresh eyes. Hence, we develop a simple experiment to observe chaos in classical billiards with a macroscopic 3.38-m long setup. Using a digital video camera, one records the dynamic time evolution of the interaction between a robot and Bunimovich stadium billiards with specular reflection. From the experimental time series, we calculate the Lyapunov exponent [Formula: see text] as a function of a geometric parameter. The results are in concordance with theoretical predictions. In addition, we determine the Poincaré surface of section from the experimental data and check its sensitivity to the initial conditions as a function of time.

Lyapunov exponent in the Vicsek model.

The well-known Vicsek model describes the dynamics of a flock of self-propelled particles (SPPs). Surprisingly, there is no direct measure of the chaotic behavior of such systems. Here we discuss the dynamical phase transition present in Vicsek systems in light of the largest Lyapunov exponent (LLE), which is numerically computed by following the dynamical evolution in tangent space for up to two million SPPs. As discontinuities in the neighbor weighting factor hinder the computations, we propose a smooth form of the Vicsek model. We find a chaotic regime for the collective behavior of the SPPs based on the LLE. The dependence of LLE with the applied noise, used as a control parameter, changes sensibly in the vicinity of the well-known transition points of the Vicsek model.