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We investigate the ability of simple diagnostics based on Lagrangiandescriptor (LD) computations of initially nearby orbits to detect chaos inconservative dynamical systems with phase space dimensionality higher than two.In particular, we consider the recently introduced methods of the difference($D_L^n$) and the ratio ($R_L^n$) of the LDs of neighboring orbits, as well asa quantity ($S_L^n$) related to the finite-difference second spatial derivativeof the LDs, and use them to determine the chaotic or regular nature ofensembles of orbits of a prototypical area-preserving map model, the4-dimensional (4D) symplectic standard map. Using the distributions of theindices' values we determine appropriate thresholds to discriminate betweenregular and chaotic orbits, and compare the obtained characterization againstthat achieved by the Smaller Alignment Index (SALI) method of chaos detection,by recording the percentage agreement $P_A$ between the two classifications. Westudy the influence of various factors on the performance of these indices, andshow that the increase of the final number of orbit iterations T and the ordern of the indices (i.e. the dimensionality of the space where the considerednearby orbits lie), as well as the decrease of the distance $\sigma$ ofneighboring orbits, increase the $P_A$ values along with the requiredcomputational effort. Balancing between these two factors we find appropriateT, n and $\sigma$ values, which allow the efficient use of the $D_L^n$, $R_L^n$and $S_L^n$ indices as short time and computationally cheap chaos diagnosticsachieving $P_A \gtrsim 90 \%$, with $D_L^n$ and $S_L^n$ having larger $P_A$values than $R_L^n$. Our results show that the three LDs-based indices performbetter for systems with large percentages of chaotic orbits.

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