Beyond the Largest Lyapunov Exponent: Entropy-Based Diagnostics of Chaos in Henon-Heiles and N-Body Dynamics

TL;DR

This paper explores entropy-based diagnostics as a complementary tool to Lyapunov exponents for detecting chaos in gravitational systems, demonstrating their effectiveness in various dynamical regimes and system sizes.

Contribution

It introduces entropy-based diagnostics for chaos detection in gravitational dynamics and compares their effectiveness to Lyapunov exponents across different systems and parameters.

Findings

Entropy follows chaos transition in Henon-Heiles system.

Entropy decreases with increasing particle number in N-body simulations.

Lyapunov exponent remains nearly constant with N, unlike entropy.

Abstract

The largest Lyapunov exponent is widely used to diagnose chaos in gravitational dynamics, but in mixed phase spaces and finite-N systems it does not always provide a complete description of orbital complexity and phase-space transport. Entropy-based diagnostics may offer a complementary perspective. We investigate whether trajectory-based information entropy can provide a useful diagnostic of chaos in gravitational systems and how it relates to the largest Lyapunov exponent as a function of orbital energy and of the number of degrees of freedom. We computed the largest Lyapunov exponent and a coarse-grained Shannon entropy for ensembles of trajectories in the Henon-Heiles potential and for test-particle orbits in live N-body realizations of a Plummer model. We then compared the dependence of both quantities on orbital energy and, for the N-body case, on particle number. In the Henon-Heiles system, the Shannon entropy follows the transition from weak to widespread chaos and exhibits an energy dependence that closely mirrors that of the largest Lyapunov exponent. For test-particle orbits in live N-body potentials, both diagnostics indicate stronger chaos for more tightly bound trajectories. However, their dependence on N differs: the largest Lyapunov exponent remains nearly constant over the explored range of particle numbers, whereas the Shannon entropy decreases monotonically as N increases. These results show that the information entropy can complement the largest Lyapunov exponent and may better capture changes in global phase-space mixing, especially in systems where the leading Lyapunov exponent alone is not sufficiently informative. It therefore provides a promising alternative for diagnosing chaos when tangent-space dynamics is unavailable or computationally expensive, and it is naturally suited to systems with densely sampled trajectories, such as minor bodies in the Solar System.