Impact of weak generalized synchronization on time series forecasting using reservoir computers

We investigate the relationship between the performance of chaotic time series forecasting and the dynamical properties of echo state networks (ESNs) from the viewpoint of generalized synchronization (GS). By treating the ESN as a response system driven by chaotic input, we analyze the transversal stability of GS using conditional Lyapunov exponents and a replica synchronization error-based method. We distinguish between strong GS, where the reservoir state depends smoothly on the input, and weak GS, where the synchronization still holds but loses smoothness and becomes more sensitive to perturbations, often accompanied by bubbling-like dynamics. Our results show that forecasting performance does not simply peak at the edge of conditional stability. For noise-free time series, optimal performance is achieved near the parameter value at which the sufficient condition for strong GS breaks down, while GS itself remains intact. In contrast, in the presence of observational noise, the optimal forecasting point shifts toward the vicinity of the bubbling transition, where transversal contraction is reduced, although GS is still present. In both cases, a pronounced degradation of performance occurs once bubbling becomes dominant. These findings highlight the importance of transversal stability in determining forecasting performance.