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.
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.
Relying on the fractal character of the largest clusters at criticality, we employ a finite-size scaling analysis to obtain an accurate phase-diagram of the percolation transition in chains with bond concentration decaying as a power-law on the form 1/ r(1+sigma) . For the particular case of sigma=1, no percolation transition is observed to occur at a finite dilution, in contrast with the finite temperature Kosterlitz-Thouless transition exhibited in Ising and Potts chains with inverse square-law couplings. The fractal dimension of the critical percolation cluster is found to follow distinct dependencies on the decay exponent being numerically fitted by d(f) =0.35+4sigma/5 for 0
