Minimal Reservoir Computing

How small can a reservoir be and still forecast chaos? A systematic phase diagram of Echo State Networks over the (sparsity, spectral-radius) plane, scoring each configuration by how many Lyapunov times its forecast stays valid — the horizon normalised by the largest Lyapunov exponent computed directly from the dynamics (Benettin method), not a tabulated constant. The same pipeline is benchmarked across two continuous flows (Lorenz-63, Rössler) and a discrete map (Hénon), locating the minimal reservoir that still tracks each attractor and the boundary beyond which prediction collapses. Built in Julia (interactive Pluto notebook) with a mirrored Python implementation.