Deciphering complexity: machine learning insights into the chaos

We introduce new machine learning techniques for analyzing chaotic dynamical systems. The main goal of this study is to develop a simple method for calculating the Lyapunov exponent using only two trajectory data points, in contrast to traditional methods that require averaging procedures. Additionally, we explore phase transition graphs to analyze the shift from regular periodic to chaotic dynamics, focusing on identifying “almost integrable” trajectories where conserved quantities deviate from whole numbers. Furthermore, we identify “integrable regions” within chaotic trajectories. These methods are tested on two dynamical systems: “two objects moving on a rod” and the “Henon–Heiles” system.