because the final training and testing accuracies are the same. Our code is developed in Keras and is made publicly available (see Code and data availability).

Results {#sec_results}
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Short-term prediction: Comparison of the RC-ESN, ANN, and RNN-LSTM performances
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The short-term prediction skills of the three deep learning methods for the same training/testing sets are compared in Figure \[esn\_time\]. Given the chaotic nature of the system, the performance of the methods depends on the initial condition from which the prediction is conducted. To give the readers a comprehensive view of the performance of these methods, Figures \[esn\_time\](A) and (B) show examples of the predicted trajectories (for one element of $X(t)$) versus the true trajectory for two specific initial conditions (from the $100$ initial conditions we used): the one for which RC-ESN shows the best performance (Figure \[esn\_time\](A)), and the one for which RC-ESN shows the worst performance (Figure \[esn\_time\](B)).

![Comparison of the short-term forecast skills among the three deep learning methods. The lines show truth (black), RC-ESN (red), ANN (blue), and RNN-LSTM (cyan). (A) and (B) show examples (differing in initial condition) for which RC-ESN yields the longest and shortest prediction horizons, respectively. Vertical dashed lines approximately show the 