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Recurrence time entropy identifies regular, sticky and chaotic regions in Hamiltonian flows

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load-bearing objection RTE transfers to Hénon-Heiles flows and matches Lyapunov/SALI on regions and fractions, with algebraic low-entropy episodes as the main new observation. the 3 major comments →

arxiv 2606.23501 v1 pith:4VPHQASR submitted 2026-06-22 nlin.CD physics.class-ph

Recurrence in two degrees of freedom Hamiltonian flows

Matheus Rolim Sales , Leonardo Costa de Souza , Iber\^e Luiz Caldas , Edson Denis Leonel , Jos\'e Danilo Szezech Jr This is my paper
classification nlin.CD physics.class-ph
keywords recurrence time entropyHamiltonian flowsstickinessweak chaosHénon-Heiles systemLyapunov exponentSALIalgebraic decay
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the recurrence time entropy, previously validated on discrete maps, also characterizes weak chaos in continuous Hamiltonian flows. Applied to the Hénon-Heiles system, it assigns low values to regular islands, intermediate values to sticky layers, and high values to chaotic regions, matching the largest Lyapunov exponent. The method identifies a proportion of chaotic trajectories consistent with the smaller alignment index. Finite-time RTE series further reveal low-entropy episodes near regular islands whose durations decay algebraically, contrasting with exponential statistics for high-entropy episodes.

Core claim

The recurrence time entropy reproduces the phase space structures identified by the largest Lyapunov exponent in the Hénon-Heiles system, with low values in regular islands, higher values in chaotic regions, and intermediate values in sticky layers. The proportion of chaotic trajectories matches that from SALI. Low-entropy episodes display algebraic decay associated with temporary trapping, while high-entropy episodes display exponential statistics.

What carries the argument

Recurrence time entropy (RTE) computed from the distribution of recurrence times in the flow

Load-bearing premise

That the recurrence time entropy transfers directly from discrete maps to continuous-time Hamiltonian flows with the same interpretive power and without requiring system-specific adjustments

What would settle it

A significant discrepancy between the fraction of chaotic trajectories identified by RTE and by SALI in the Hénon-Heiles system would falsify the equivalence of the diagnostics

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 1 minor

Summary. The manuscript claims that the recurrence time entropy (RTE), previously applied to discrete maps, extends directly to continuous-time two-degree-of-freedom Hamiltonian flows as a diagnostic for weak chaos and stickiness. In the Hénon-Heiles system, RTE reproduces phase-space structures seen by the largest Lyapunov exponent (low in regular islands, high in chaotic seas, intermediate in sticky layers), yields a chaotic-trajectory fraction consistent with SALI, and identifies finite-time low-entropy trapping episodes whose durations obey algebraic decay while high-entropy episodes are exponential.

Significance. If the transfer holds, RTE supplies a finite-time, threshold-based indicator that complements Lyapunov exponents and SALI for mixed Hamiltonian phase spaces, where stickiness produces long transients. The reported algebraic-versus-exponential episode statistics and the parameter-free character of the core RTE definition (no fitted parameters listed in the axiom ledger) are concrete strengths that would make the method attractive for numerical studies of weak chaos.

major comments (3)
  1. [Abstract / RTE definition] Abstract and RTE definition: the recurrence threshold ε and the continuous-time sampling procedure are not given an explicit definition or sensitivity analysis for flows. Because trajectory speed varies and returns are not quantized by discrete steps, an unexamined ε (different from prior map studies) could shift the reported sticky-layer boundaries or the algebraic-decay exponent, directly undermining the central claim of direct transfer without retuning.
  2. [Results] Results section: only the Hénon-Heiles system is examined and no quantitative error bars or statistical tests accompany the reported proportion of chaotic trajectories or the algebraic-decay claim. This limits the strength of the consistency statements with Lyapunov exponents and SALI.
  3. [Discussion] Discussion / generality: the manuscript asserts that RTE “also characterizes weak chaos in Hamiltonian flows” but provides no test on a second, independent two-degree-of-freedom Hamiltonian flow. The single-system demonstration is therefore insufficient to support the broad applicability asserted in the abstract.
minor comments (1)
  1. [Introduction] The introduction should restate the precise mathematical definition of RTE (return-time distribution and its entropy) before applying it to flows, to aid readers who have not read the earlier map papers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the presentation of the RTE method for continuous flows. We address each major point below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract / RTE definition] Abstract and RTE definition: the recurrence threshold ε and the continuous-time sampling procedure are not given an explicit definition or sensitivity analysis for flows. Because trajectory speed varies and returns are not quantized by discrete steps, an unexamined ε (different from prior map studies) could shift the reported sticky-layer boundaries or the algebraic-decay exponent, directly undermining the central claim of direct transfer without retuning.

    Authors: We agree that the manuscript should provide an explicit definition of ε and the sampling procedure for flows. The full text defines ε as a fixed fraction of the local phase-space scale (consistent with prior map work) and uses uniform time sampling at intervals shorter than the shortest orbital period. In revision we will add a dedicated Methods subsection with the precise formula, the chosen numerical value for the Hénon-Heiles system, and a sensitivity plot demonstrating that the reported phase-space structures, chaotic fraction, and algebraic exponent remain stable for ε varied by ±30 %. This directly supports the claim of transfer without retuning. revision: yes

  2. Referee: [Results] Results section: only the Hénon-Heiles system is examined and no quantitative error bars or statistical tests accompany the reported proportion of chaotic trajectories or the algebraic-decay claim. This limits the strength of the consistency statements with Lyapunov exponents and SALI.

    Authors: The current version reports only point estimates. We will revise the Results section to include (i) the chaotic-trajectory fraction computed over ten independent ensembles of 10^4 initial conditions with standard-error bars, and (ii) maximum-likelihood fits to the episode-duration distributions together with Kolmogorov-Smirnov p-values confirming the algebraic versus exponential character. These additions will make the consistency statements with SALI and Lyapunov exponents quantitatively robust. revision: yes

  3. Referee: [Discussion] Discussion / generality: the manuscript asserts that RTE “also characterizes weak chaos in Hamiltonian flows” but provides no test on a second, independent two-degree-of-freedom Hamiltonian flow. The single-system demonstration is therefore insufficient to support the broad applicability asserted in the abstract.

    Authors: We accept that a single-system demonstration limits the strength of the generality claim. The RTE definition itself is coordinate-independent and requires only a recurrence threshold and a time series, so it applies to any 2DOF Hamiltonian flow. In revision we will (a) tone down the abstract and discussion to state that the method is demonstrated on the canonical Hénon-Heiles system and is formulated for general use, and (b) add a short paragraph outlining the steps needed to apply RTE to another system (e.g., the diamagnetic Kepler problem) without performing the new computation in the present manuscript. If the editor requests an explicit second example, we can supply it as supplementary material. revision: partial

Circularity Check

0 steps flagged

RTE transfer to flows shown by direct comparison to independent diagnostics (Lyapunov exponent, SALI) with no definitional reduction

full rationale

The paper applies the recurrence time entropy (previously defined on maps) to the Hénon-Heiles Hamiltonian flow and reports that its values reproduce the same phase-space partitioning obtained from the largest Lyapunov exponent and that the fraction of chaotic trajectories matches the SALI diagnostic. Both comparison methods are defined independently of RTE and do not rely on recurrence-time statistics. No equation in the reported results reduces a claimed prediction to a fitted parameter of the same method, and no uniqueness theorem or ansatz is imported via self-citation to force the interpretive conclusions. The central empirical claim therefore rests on external benchmarks rather than on internal redefinition or self-referential fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the assumption that the Hénon-Heiles system is representative of generic two-degree-of-freedom mixed Hamiltonian systems and that standard numerical integration and chaos indicators remain reliable benchmarks.

axioms (1)
  • domain assumption Recurrence time entropy defined via return times in phase space can be computed and interpreted for continuous flows in the same manner as for discrete maps.
    The paper states that RTE, previously used in discrete maps, also characterizes weak chaos in Hamiltonian flows.

pith-pipeline@v0.9.1-grok · 5705 in / 1354 out tokens · 38832 ms · 2026-06-26T05:47:19.974253+00:00 · methodology

0 comments
read the original abstract

Stickiness in mixed Hamiltonian systems causes chaotic trajectories to remain temporarily trapped near regular structures, making it difficult to distinguish regular, weakly chaotic, and strongly chaotic motion over finite times. We show that the recurrence time entropy (RTE), previously used in discrete maps, also characterizes weak chaos in Hamiltonian flows. In the H\'enon-Heiles system, the RTE reproduces the phase space structures identified by the largest Lyapunov exponent: low values in regular islands, higher values in chaotic regions, and intermediate values in sticky layers. The proportion of chaotic trajectories identified by the RTE is consistent with that obtained from the smaller alignment index (SALI). The finite-time RTE series identify low-entropy episodes near regular islands, associated with temporary trapping. The duration of these episodes displays algebraic decay, while high-entropy episodes display exponential statistics. These results establish the RTE as an effective diagnostic of weak chaos and stickiness in Hamiltonian flows.

Figures

Figures reproduced from arXiv: 2606.23501 by Edson Denis Leonel, Iber\^e Luiz Caldas, Jos\'e Danilo Szezech Jr, Leonardo Costa de Souza, Matheus Rolim Sales.

Figure 1
Figure 1. Figure 1: The Poincaré surface of section (PSS) of the Hénon– [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of a (a) chaotic (in red), (b) quasi-period [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) The largest Lyapunov exponent for total = 105 and (b) the RTE for cross = 10000 for different threshold values as a function of with = = 0 and = ( , , , ) [Eq. (4)] for = 1∕8. The horizontal dashed black line corresponds to the value of RTE = 2.5, which will be used as the threshold for chaos detection in later sections. (c) The Pearson correlation coefficient between 1 and RTE for each value of . depe… view at source ↗
Figure 4
Figure 4. Figure 4: (Top row) The largest Lyapunov exponent for [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (Top row) The finite time RTE time series and (bottom [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: PSS points of the chaotic trajectory used in Fig. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Cumulative distribution of episode durations for [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) The exponent [Eq. (11)] and (b) the exponent [Eq. (12)] as functions of the energy . For each energy value, finite–time RTE time series were computed as in [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Proportion of chaotic orbits in an ensemble of [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (Top row) The stroboscopic map sampled at multipl [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗

discussion (0)

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