Learning the Chaotic and Regular Nature of Trajectories in Hamiltonian Systems with Lagrangian descriptors

Abstract: In this paper, we explore the application of Machine Learning techniques, specifically Support Vector Machines (SVM), to unveil the chaotic and regular nature of trajectories in Hamiltonian systems using Lagrangian descriptors. Traditional chaos indicators, while effective, are computationally expensive and require an exhaustive study of the parameter space to establish the classification thresholds. By using SVMs trained on a dataset obtained from the analysis of the dynamics of the double pendulum Hamiltonian system, we aim at reducing the complexity of this process. Our trained SVM models demonstrate high accuracy when it comes to classifying trajectories in diverse Hamiltonian systems, such as for example in the four-well Hamiltonian, the H\'enon-Heiles system and the Chirikov Standard Map. The results indicate that SVMs, when combined with Lagrangian descriptors, offer a robust and efficient method for chaos classification across different dynamical systems. Our approach not only simplifies the classification process but also is highlighting the potential of Machine Learning algorithms in the study of nonlinear dynamics and chaos.