Learning to Forecast and Control Chaotic Trajectories: A Billiard Testbed for Cascade-Prone Dynamical Systems

Forecasting and controlling the evolution of nonlinear, chaotic dynamical systems is a central challenge across physical, biological, and infrastructural domains, where small perturbations can propagate into large, hard-to-predict cascades. Billiard dynamics offer a canonical, mathematically transparent testbed for this problem: even simple geometries generate trajectories with sensitive dependence on initial conditions, positive Lyapunov exponents, and mixing behavior representative of far more complex cascading systems, while remaining fully controllable and reproducible in simulation. We use Takens delay-coordinate embedding to reconstruct the effective phase space of a billiard trajectory from scalar observations, and train a radial basis function (RBF) forecasting model on the reconstructed dynamics to predict short-horizon future states. Building on this forecast, we implement an Ott–Grebogi–Yorke (OGY)-type control scheme that applies small, targeted perturbations to stabilize the trajectory near a desired unstable periodic orbit or to steer it away from undesired escape or collision events. Numerical experiments across billiard geometries of varying chaoticity show that the RBF forecaster achieves accurate short-term trajectory prediction from limited scalar history, and that the resulting control strategy substantially reduces trajectory divergence and stabilizes target orbits at low control cost compared to uncontrolled dynamics. These results establish the billiard system as a validated, minimal dynamical-systems foundation for forecast-driven control, and motivate its extension to higher-dimensional, network-coupled cascading systems, where analogous forecasting and targeted-intervention strategies can inform the identification and control of critical nodes.