Chaos and Regularity in the Double Pendulum with Lagrangian Descriptors

TL;DR

This study uses Lagrangian descriptors to analyze chaos and regularity in the double pendulum, deriving equations, conducting parametric analysis, and characterizing chaos growth with respect to system parameters.

Contribution

It introduces a novel application of Lagrangian descriptors to quantify chaos in the double pendulum and provides detailed parametric insights into chaotic behavior.

Findings

Maximum chaos occurs at equal pendulum lengths for a given mass ratio.

Chaotic fraction varies with energy and system parameters.

Chaos growth may follow an exponential law across energy regimes.

Abstract

In this paper we apply the method of Lagrangian descriptors as an indicator to study the chaotic and regular behavior of trajectories in the phase space of the classical double pendulum system. In order to successfully quantify the degree of chaos with this tool, we first derive Hamilton's equations of motion for the problem in non-dimensional form, showing that they can be written compactly using matrix algebra. Once the dynamical equations are obtained, we carry out a parametric study in terms of the system's total energy and the other model parameters (lengths and masses of the pendulums, and gravity), to determine the extent of the chaotic and regular regions in the phase space. Our numerical results show that for a given mass ratio, the maximum chaotic fraction of phase space trajectories is attained when the pendulums have equal lengths. Moreover, we give a characterization of the growth and decay of chaos in the system in terms of the model parameters, and explore the hypothesis that the chaotic fraction follows an exponential law over different energy regimes.