Chaos and Regularity in the Double Pendulum with Lagrangian Descriptors

In this paper, we apply the method of Lagrangian descriptors as an indicator to study the chaotic and regular behaviors of trajectories in the phase space of the classical double pendulum system. To successfully quantify the degree of chaos with this tool, we first derive Hamilton’s equations of motion for the problem in nondimensional form and demonstrate how they can be expressed compactly using matrix algebra. Once the dynamical equations are obtained, we conduct a parametric study based on the system’s total energy and other key parameters such as the lengths and masses of the pendulums, as well as gravity, to determine the extent of the chaotic and regular regions in the phase space. Our numerical results reveal that, for a given mass ratio, the maximum chaotic fraction of phase space trajectories is attained when the pendulums have equal lengths. Furthermore, we characterize how chaos grows and decays within the system in terms of the model parameters, and explore the hypothesis that the chaotic fraction follows an exponential law over different energy regimes.