Quantifying chaos using Lagrangian descriptors

Abstract

We present and validate simple and efficient methods to estimate the chaoticity of orbits in low-dimensional conservative dynamical systems, namely, autonomous Hamiltonian systems and area-preserving symplectic maps, 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 and ratio of neighboring orbits’ LDs. Using as generic test models the prototypical two degree of freedom Hénon–Heiles system and the two-dimensional standard map, we find that 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 reliable quantification of the systems’ 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 conservative dynamical systems.