Performance of chaos diagnostics based on Lagrangian descriptors. Application to the 4D standard map

Abstract

We investigate the ability of simple diagnostics based on Lagrangian descriptor (LD) computations of initially nearby orbits to detect chaos in conservative dynamical systems with phase space dimensionality higher than two. In particular, we consider the recently introduced methods of the difference and the ratio of the LDs of neighboring orbits, as well as a quantity related to the finite-difference second spatial derivative of the LDs, and use them to determine the chaotic or regular nature of ensembles of orbits of a prototypical area-preserving map model, the 4-dimensional symplectic standard map. We compare the characterization obtained by these LDs-based diagnostics against that achieved by the Smaller Alignment Index method of chaos detection, by recording the percentage agreement PA between the two classifications. We also study the influence of the final number of orbit iterations T, the order n of the indices, as well as the distance σ of neighboring orbits on the performance of these methods, and find appropriate T, n and σ values which allow the efficient use of the three indices as short time and computationally cheap chaos diagnostics achieving PA≳90%. Our findings clearly indicate the capability of LDs to efficiently identify chaos in systems whose phase space is difficult to visualize, due to its high dimensionality, without knowing the variational equations (tangent map) of continuous (discrete) time systems needed by traditional chaos indicators.