  We introduce new machine-learning techniques for analyzing chaotic dynamical
systems. The primary objectives of the study include the development of a new
and simple method for calculating the Lyapunov exponent using only two
trajectory data points unlike traditional methods that require an averaging
procedure, the exploration of phase transition graphs from regular periodic to
chaotic dynamics to identify "almost integrable" trajectories where conserved
quantities deviate from whole numbers, and the identification of "integrable
regions" within chaotic trajectories. These methods are applied and tested on
two dynamical systems: "Two objects moving on a rod" and the "Henon-Heiles"
systems.
