Leveraging temporal features of the divergence quantifier of recurrence plot to detect chaos in conservative systems

The recurrence-based divergence quantifier ($DIV$), traditionally applied to dissipative systems, is shown here to be an effective finite-time chaos indicator for conservative dynamics. We benchmark its performances against the well-established fast Lyapunov indicator (FLI), focusing on the standard map, a canonical model of Hamiltonian chaos. Through extensive numerical simulations on moderately long orbits, we find strong agreement between $DIV$ and FLI, supporting the reported correlation between the divergence of recurrences and positive Lyapunov exponents. Additionally, our study sheds more light into asymptotic time properties of $DIV$ by revealing distinct power laws on regular and chaotic components, both in the original and reconstructed phase spaces. In particular, on a regular component, the space average of $DIV$ decays with the time $N$ as $1/N$, mirroring the decay rate of the maximal Lyapunov exponent. On chaotic components, the space average of $DIV$ decreases at a much slower rate, close to $1/\sqrt{N}$. This scaling insight opens new avenues for characterizing chaos from time series. Our numerical results thus demonstrate $DIV$ to be a computationally viable and theoretically rich tool for chaos detection in conservative systems.

💡 Research Summary

The paper investigates the use of the recurrence‑plot based divergence quantifier (DIV), traditionally employed for dissipative systems, as a finite‑time chaos indicator in conservative (Hamiltonian) dynamics. The authors focus on the standard map, a prototypical area‑preserving map, and benchmark DIV against the fast Lyapunov indicator (FLI), a well‑established variational chaos detector.

Methodologically, recurrence plots are constructed with a fixed recurrence rate of 5 % and a Theiler window of two iterations to avoid trivial recurrences. The longest non‑trivial diagonal line ℓ_max in each plot is identified, and DIV is defined as 1/ℓ_max. Since a diagonal of length ℓ corresponds to a sequence of ℓ successive recurrences, its length is directly related to the rate at which nearby trajectories separate; thus ℓ_max is expected to be inversely proportional to the maximal Lyapunov exponent.

A large ensemble of initial conditions (200 points) is sampled uniformly in the torus