Quantifying chaos using Lagrangian descriptors

TL;DR

This paper introduces simple, efficient methods using Lagrangian descriptors to quantify chaos in low-dimensional dynamical systems, achieving high accuracy with less computational effort than established techniques.

Contribution

The paper proposes two novel chaos indicators based on Lagrangian descriptors, validated against SALI, and demonstrates their effectiveness on short time scales with reduced computational cost.

Findings

Indicators correctly classify chaos with over 90% accuracy

Short time, coarse grid LD computations are sufficient

Methods reveal detailed phase space structure

Abstract

We present and validate simple and efficient methods to estimate the chaoticity of orbits in low dimensional dynamical systems from computations of Lagrangian descriptors (LDs) on short time scales. Two quantities are proposed for determining the chaotic or regular nature of orbits in a system's phase space, which are based on the values of the LDs of these orbits and of nearby ones: The difference (DNLD) and ratio (RNLD) of neighboring orbits' LDs. We find that, typically, these indicators are able to correctly characterize the chaotic or regular nature of orbits to better than 90 % agreement with results obtained by implementing the Smaller Alignment Index (SALI) method, which is a well established chaos detection technique. Further investigating the performance of the two introduced quantities we discuss the effects of the total integration time and of the spacing between the used neighboring orbits on the accuracy of the methods, finding that even typical short time, coarse grid LD computations are sufficient to provide a reliable quantification of a system's chaotic component, using less CPU time than the SALI. In addition to quantifying chaos, the introduced indicators have the ability to reveal details about the systems' local and global chaotic phase space structure. Our findings clearly suggest that LDs can also be used to quantify and investigate chaos in continuous and discrete low dimensional dynamical systems.