In this work, the relationship between Chaos and Machine Learning is studied using an Echo State Network (ESN) to predict classic chaotic series. Through the Lorenz, Rössler, and Mackey-Glass attractors, the concept of chaos and the definition and importance of Lyapunov exponents are studied, and then these findings are applied to analyze the internal dynamics of the ESN. It is demonstrated how an ESN reaches an optimal prediction peak at the critical threshold of its activation dynamics. Additionally, the effect of implementing synaptic plasticity rules on the reservoir dynamics is analyzed. These studies are further accompanied by a recurrence analysis that allows the graphical visualization of the network's dynamics.