Trajectory arclength reveals chaos

Abstract

In this paper, we investigate the time-averaged phase space arclength of a trajectory, expressed through a normalized Lagrangian descriptor, as a trajectory-based diagnostic for distinguishing regular and chaotic dynamics. The proposed indicator, $\Omega$, is computed by integrating the equations of motion together with a single scalar arclength-accumulation equation and therefore does not require variational equations, neighboring trajectories, or spatial derivatives of a Lagrangian descriptor field. Using Birkhoff's ergodic theorem together with estimates for quasiperiodic motion and correlation-based arguments for chaotic ergodic dynamics, we show that regular and chaotic trajectories exhibit different finite-time behavior despite the convergence of $\Omega$ in both regimes. For quasiperiodic trajectories with Diophantine frequency vectors, deviations from the limiting average decay as $\mathcal{O}({\tau }^{-1})$, whereas sufficiently mixing chaotic dynamics exhibit root-mean-square fluctuations of order ${\tau }^{-1/2}$ under suitable correlation assumptions. This difference in finite-time convergence and fluctuation scaling provides the theoretical basis for using $\Omega$ to discriminate between the two dynamical regimes. We validate the proposed approach in the Hénon--Heiles and Fermi--Pasta--Ulam systems, comparing the resulting classification of initial conditions with SALI as a reference chaos indicator. The results support $\Omega$ as a complementary single-trajectory diagnostic for chaos detection and as a practical source of scalar features for the analysis of trajectory ensembles, particularly in settings involving large numbers of trajectories or high-dimensional systems.