An Extension of Discrete Lagrangian Descriptors for Unbounded Maps

V\'ictor J. Garc\'ia-Garrido

REVIEW 3 major objections 5 minor 18 references

An Extension of Discrete Lagrangian Descriptors for Unbounded Maps

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read By truncating each orbit's descriptor sum at the moment it exits a fixed interaction region, this paper extends Discrete Lagrangian Descriptors to unbounded maps and shows the truncated field still carries stable and unstable manifolds…

desk verdict A useful, clearly written practical extension of DLDs to unbounded maps, but the central claim that a large interaction region transfers the manifold-detection theorems is asserted, not proved, and the numerics are illustrative rather than validating. read the letter →

arxiv 1908.04871 v2 pith:S2OQMBEX submitted 2019-08-13 nlin.CD math.DS

Víctor J. García-Garrido This is my paper · ORCID

classification nlin.CDmath.DS MSC 37M9937D4537C0537E30 PACS 05.45.-a

keywords DiscreteLagrangiandescriptorsvariableiterationnumberunboundedmapsHénonmapstableandunstablemanifoldschaoticsaddleKAMtoristrangeattractor

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a way to apply Discrete Lagrangian Descriptors to maps whose orbits escape to infinity: instead of iterating every initial condition a fixed number of steps, it stops each orbit as soon as it leaves a chosen interaction region in the plane. The variable-iteration sum is expected to approach the fixed-iteration descriptor when that region is large, so the known result that descriptor singularities trace stable and unstable manifolds should carry over. The paper demonstrates the device on the Hénon map, recovering a chaotic saddle, the KAM tori of an area-preserving case, and a strange attractor, and it shows the same calculation yields the classical escape-time distribution used for comparison. A sympathetic reader would care because the fixed-iteration descriptor fails on unbounded maps once escaping orbits produce huge or undefined values, while this truncation converts escape into a benign stopping signal.

What carries the argument

The load-bearing mechanism is the stopping-time rule $N^\pm_{x_0}=\max_{k=1,\dots,N}\{k\mid f^{\pm k}(x_0)\in R\}$, which truncates the $\ell^p$-arclength sum (with $0<p\le 1$) at the last iterate inside a fixed interaction region $R$, here a circle of radius $r=100$. The rule turns escape into a legitimate end-of-sum signal and therefore prevents the NaN and overflow contamination produced by fixed-iteration descriptors on unbounded maps; it also makes the transit-time count $T_{x_0}=N^+_{x_0}+N^-_{x_0}$ available at no extra cost. The formal justification is transfer from the original descriptor: for large $R$ the new sum is expected to approach Eq. (3), so the previously proved singular-feature theorem is invoked for the variable-iteration version.

What would settle it

On the Hénon map with $A=9.5$, $B=-1$, recompute $D_p$ with the same grid and $N=10$, $p=0.05$, for interaction radii $r=50$, $r=100$, and $r=200$; if the singular-feature curves extracted from $\|\nabla D_p\|$ drift or disappear as $r$ grows, the large-region approximation is not working. A more direct check is to compare $D_p(x_0,N)$ with the fixed-iteration sum for an orbit whose escape happens exactly at iteration $k$: if the boundary-crossing jump is comparable to the retained sum, the transfer of the theorem lacks uniformity.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Variable Iteration Number Discrete Lagrangian Descriptor, $D_p(x_0,N)=\sum_{i=-N^-_{x_0}}^{N^+_{x_0}-1}\|x_{i+1}-x_i\|^p$ with $N^\pm_{x_0}$ the number of forward/backward iterates of $x_0$ that stay inside a fixed planar region $R$, replaces the fixed-iteration descriptor $\mathcal{M}^D_p(x_0,N)$ for unbounded maps without sacrificing the manifold-detection property. The argument is that for $R$ large enough the truncated sums approximate the fixed-iteration sums, so the points where $D_p$ is non-differentiable still locate stable and unstable manifolds, exactly as proved for the original descriptor. On the Hénon map the method is shown to reveal the chaotic saddle for parameters $(A,B)=(9.5,-1)$, the KAM tori and surrounding stable/unstable manifolds for $(0.298,1)$, and the strange attractor for $(1.4,0.3)$, and the gradient of $D_p$ extracts the manifolds directly.

Load-bearing premise

The load-bearing premise is that choosing the interaction region $R$ large enough makes the truncated variable-iteration sums close enough to the fixed-iteration descriptor that the proved manifold-detection property transfers; the paper states this as an expected result rather than proving a quantitative bound in $R$ and $N$.

Editorial extensions

If this is right

The same orbit data used for $D_p$ also produce the transit-time distribution $T_{x_0}$, so escape statistics for the map come from the same computation.
The gradient magnitude $\|\nabla D_p\|$ yields stable and unstable manifolds directly, without separately integrating manifolds.
Iteration-averaged values $\langle D_p\rangle=D_p/N$ draw the KAM tori of regularity islands as smooth contours, where classical exit-time plots are flat and uninformative.
Because escape no longer contaminates the scalar field, larger maximum iteration counts $N$ can expose finer layers of chaotic-saddle structure.
The construction is stated for general invertible maps, so the Hénon tests instantiate a general recipe rather than a one-off calculation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

I would expect the main uncontrolled error to be the single large jump an orbit takes when it crosses the boundary of $R$: that jump is present in $D_p$ for escaped orbits but absent for orbits that linger inside, so two nearby initial conditions can differ by an amount comparable to the whole retained sum.
A natural strengthening, not attempted here, would be to prove a bound such as $|D_p-\mathcal{M}^D_p|\le C(r,N)$ under hyperbolicity or a uniform escape bound; with that, the heuristic 'large $R$' transfer becomes a theorem rather than an expectation.
Because the stopping rule is purely spatial, it is plausible that the construction extends to non-autonomous or higher-dimensional invertible maps and to maps whose escape sets are anisotropic; those regimes are untested in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a useful practical note, not a theorem paper. The variable-iteration stopping rule is the discrete analogue of what has already been done for continuous-time LDs, and the author says so. The contribution is Eq. (8), the demonstration on three Hénon parameter regimes, and the comparison with exit-time plots. That is worth having on record for people who use Lagrangian descriptors on open maps.

What is genuinely good: the method is simple and clearly specified; the figures recover the expected Hénon structures—chaotic saddle for A=9.5, KAM islands and manifolds for the area-preserving case, and the strange attractor for A=1.4; and the exit-time comparison is a sensible sanity check. The author is honest that the variable-iteration idea is inspired by escape-time plots, and the citations cover the continuous-time precedents. I do not see a citation problem.

The soft spot is exactly the one flagged in the stress-test note. Section 2 says that for a large interaction region R, VIN-DLD approaches FIN-DLD and therefore inherits the manifold-detection properties proved in Lopesino et al. That is the load-bearing step, and it is not proved or even stated with a quantified error. The truncation is not a harmless tail: the first step that exits R is omitted from the sum, and for a chaotic orbit that jump can be of the same order as the whole retained sum, especially with p=0.05. So it is not obvious that the singular features survive unchanged. No convergence study in R is given, and r=100 appears without justification. This is the main reason I would not call the paper's central claim established. The Birkhoff/ergodic-partition argument for KAM tori is also extrapolated from continuous time to maps without proof, though that is a smaller issue and likely true in the intended sense.

There is also no code or data, and no quantitative validation against analytically known manifold shapes. Agreement with exit-time plots is reassuring but those plots obey the same truncation, so it is partly a self-consistency check.

Overall, the paper knows what it is and the prose is clear. The central gap is addressable rather than fatal. If the author added a convergence check in R and, better, an error estimate or a proof under reasonable hypotheses, it would become a solid methods note. As it stands I would send it to referees with that request rather than desk reject it. The audience is applied dynamical-systems people who want a working recipe, not readers looking for new theory. I would not cite it for the theorem transfer, but I would point to it as a clean statement of the variable-iteration recipe.

Referee Report

3 major / 5 minor

Summary. The paper proposes a modification of Discrete Lagrangian Descriptors (DLDs), termed Variable Iteration Number DLDs (VIN-DLDs), designed for unbounded maps whose orbits escape to infinity. Instead of iterating every initial condition for a fixed number N of forward and backward steps, the sum in Eq. (8) is truncated when the orbit leaves a fixed planar interaction region R (chosen as a circle of radius r=100 in the examples). The central claim is that for large enough R, VIN-DLDs approximate the fixed-iteration DLDs of Eq. (3), so the rigorous manifold-detection properties proved for DLDs in Lopesino et al. (2015b) transfer to VIN-DLDs. The method is illustrated on the Hénon map for three parameter regimes: a chaotic saddle (A=9.5, B=-1), KAM tori and stable/unstable manifolds (A=0.298, B=1), and a strange attractor (A=1.4, B=0.3). The VIN-DLD outputs are compared visually with transit-time (exit-time) distributions. The paper concludes that VIN-DLDs overcome the NaN/overflow problems of fixed-iteration DLDs on unbounded maps and provide a useful tool for phase-space visualization.

Significance. If the transfer of the manifold-detection property were rigorously established, the proposed VIN-DLD would be a practical and useful extension of DLDs to open and unbounded dynamical systems, a class that includes many physically relevant maps. The paper is clearly written and the numerical examples indeed recover the expected phase-space structures of the Hénon map, which lends empirical support to the method. The comparison with exit-time distributions is a helpful sanity check. However, the central mathematical claim that VIN-DLDs inherit the singular-feature properties of fixed-iteration DLDs is only asserted, not proved, and no quantitative validation (convergence in R, error bounds, comparison against known manifold shapes) is provided. The paper also extends a continuous-time ergodic-partition argument to discrete maps without proof. These gaps are significant but addressable within the scope of the manuscript.

major comments (3)

[Section 2, after Eq. (8)] The statement that for a large interaction region R the VIN-DLD approaches the fixed-iteration DLD of Eq. (3), and therefore inherits the properties proved in Lopesino et al. (2015b), is the load-bearing step of the paper, yet it is only phrased as an expectation ('the expected result is that...') and is not supported by any bound or proof. For an orbit that exits R, Dp(x0,N) is not a partial sum of Eq. (3) with a small omitted tail: the variable stopping rule changes the set of terms included, and the final jump across the boundary can be comparable to the retained sum. The authors should either prove a quantitative estimate of the form |Dp(x0,N)-MDp(x0,N)| < C(R,N) for suitable orbits, or provide a rigorous statement of when singular features are preserved under this truncation. Failing that, the manifold-detection capability of VIN-DLD remains an empirical observation, not a consequence of the earlier DLD theory.
[Section 3, Figures 2-5] The numerical results are not validated quantitatively. The paper does not provide a convergence study with respect to the interaction-region radius r (the choice r=100 is presented without justification), does not compare the detected manifolds against analytically known expressions (e.g., the stable/unstable manifolds of the Hénon map's fixed points), and gives no error metric for the claimed agreement with transit-time distributions. Since the central methodological claim is about approximation in R, a simple log-log plot of a suitable norm of (Dp - MDp) versus r, or a comparison of detected manifold points as r increases, would substantially strengthen the argument. Without such a check, the reader cannot assess whether r=100 is in the asymptotic regime or whether the observed structures are robust to the truncation.
[Section 3, KAM tori discussion and Eq. (12)] The claim that the iteration-averaged VIN-DLD <Dp> in Eq. (12) detects KAM tori via the connection with Birkhoff's Ergodic Partition Theorem is not justified. The cited theorem and its Lagrangian-descriptor application in Lopesino et al. (2017) are formulated for continuous-time systems, and the paper argues only that Eq. (3) 'can be interpreted as a discretized version' of the continuous descriptor. This is not a proof that the ergodic-partition result carries over to discrete maps, especially under the variable truncation convention. The authors should either provide a proof for the discrete case or clearly mark this as a heuristic analogy. Since KAM tori recovery is one of the three main advertised applications, this gap is load-bearing for the corresponding claim.

minor comments (5)

[Throughout] There are numerous typographical errors and misspellings, e.g., 'dynamical behavior' instead of 'dynamical behavior' (Section 1), 'exapmple' and 'increaseing' (Section 3), 'iteartions' and 'demosntrate' (Section 3), 'corrsponds' in figure captions. A careful proofreading pass is needed.
[Section 2, Eq. (7)] The definition of N±x0 as a maximum over k is slightly ambiguous: it implicitly requires that the orbit stays inside R for all intermediate steps. This should be stated explicitly, since the formula alone does not encode the 'until it leaves R' stopping rule.
[Section 2, paragraph on the shape of R] The claim that the shape of R is not important 'without loss of generality' is plausible but not demonstrated; for non-circular R, the boundary-crossing jump and the stopping rule change. A brief remark that the expected approximation property would hold for any bounded R with the same asymptotic reasoning would be helpful.
[Figure 2, 3, and 5] The figures are presented without colorbars or axis labels in several panels, making it difficult to interpret the Dp value ranges. Adding colorbars and labeled axes would improve the reproducibility and readability of the results.
[Section 3, transit time comparison] The comparison between VIN-DLD and transit-time distributions is purely visual. While the agreement is encouraging, a quantitative similarity measure (e.g., correlation or mutual information) would make the validation more convincing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: VIN-DLD is a truncation of the previously defined DLD, and the paper's validation is against external Hénon-map structures; the unproven 'expected result' is a support gap, not a circular reduction.

full rationale

The paper defines the VIN-DLD in Eq. (8) as the same sum as the fixed-iteration DLD in Eq. (3) but with summation limits N±x0 determined by first exit from an interaction region R. This is an explicit truncation of the earlier diagnostic, not a quantity defined in terms of the structures it claims to detect. The central transfer step is the Section 2 assertion that for large R the VIN-DLD 'approaches the values of the original DLD in Eq. (3)' and therefore inherits the manifold-detection properties proved in Lopesino et al. (2015b). That assertion is stated as 'the expected result' and is not proved or quantified, so the paper has a genuine support gap: the approximation error of the truncation is not bounded and no convergence check in R is provided. However, this is a missing proof or correctness risk, not a circularity. No parameter is fitted to the data and then renamed a prediction; no equation reduces to its inputs by construction; and the numerical demonstrations compare against independently known features of the Hénon map (chaotic saddle, KAM tori, strange attractor) and against exit-time plots. The exit-time comparison shares the same stopping times used to build Dp, so it is not an independent validation, but the paper does not claim to derive exit times from Dp; it uses them only as a qualitative cross-check of the scalar-field patterns. The cited prior DLD theory is real external evidence for the fixed-iteration quantity, and the present paper's load-bearing step is the unproven extrapolation from that quantity to the truncated one. That extrapolation should be examined as a mathematical gap, but it does not make the derivation circular.

Assumptions & free parameters 3 free parameters · 3 assumptions · 1 invented entities

The paper introduces one new diagnostic quantity (VIN-DLD) with several free choices (interaction region radius, norm exponent p, maximum iteration number N). It inherits all the mathematical grounding of the original DLD from prior work by overlapping authors, and additionally assumes, without proof, that truncation at a large boundary preserves the non-differentiability properties. No new dynamical entity is postulated.

free parameters (3)

interaction region radius r = 100
The interaction region is chosen as a circle of radius r = 100 for all Hénon experiments. The paper argues the shape is irrelevant and the radius should be large, but provides no convergence check. The value is chosen by hand.
p-norm exponent p = 0.05 for the chaotic saddle case; 0.5 for the KAM and strange attractor cases
The choice of p is made per experiment. For the chaotic saddle case, p = 0.05 is used to balance large values from escaping orbits, as in the cited prior work. The choice affects the scaling of the scalar field but not the location of singular features.
maximum iteration number N = 5, 10, 15, 25, 500 depending on the experiment
The number of iterations is chosen per experiment to reveal the desired structures. This is an input of the method, not fitted to data, but it is a free choice that affects the output.

assumptions (3)

domain assumption The Hénon map is invertible and its inverse is given by Eq. (10).
Used in Section 2 to define backward iterations with f^{-1}. For the parameters considered, B is nonzero, so the inverse is well-defined.
ad hoc to paper The VIN-DLD approaches the FIN-DLD as the interaction region R becomes large, so the manifold-detection theorem from Lopesino et al. (2015b) applies to VIN-DLD.
This is the load-bearing premise of Section 2, stated as 'the expected result' and not proven. Section 3 states the gradient of the scalar field is non-differentiable at invariant manifolds, relying on this transfer.
ad hoc to paper The connection between time-averaged Lagrangian descriptors and KAM tori via Birkhoff's Ergodic Partition Theorem holds for discrete-time maps.
Section 3 states: 'Despite the fact that this proof has only been given for continuous-time dynamical systems, since Eq. (3) can be interpreted as a discretized version of the p-norm definition of LDs introduced in Lopesino et al. (2017), this property will also hold for maps.' This is an extrapolation without proof.

invented entities (1)

VIN-DLD (Variable Iteration Number Discrete Lagrangian Descriptor)
purpose: A modified Lagrangian descriptor for unbounded maps that stops iterating when orbits leave an interaction region.
It is the paper's methodological contribution. Its only evidence is the reproduction of known structures in the Hénon map.

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Cite this review

Pith. "Pith review of An Extension of Discrete Lagrangian Descriptors for Unbounded Maps." pith.science (2026). https://pith.science/paper/S2OQMBEX

@misc{pith2026190804871,
  author       = {Pith},
  title        = {Pith review of: An Extension of Discrete Lagrangian Descriptors for Unbounded Maps},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S2OQMBEX}},
  note         = {Machine review of arXiv:1908.04871}
}

read the original abstract

In this paper we provide an extension for the method of Discrete Lagrangian Descriptors with the purpose of exploring the phase space of unbounded maps. The key idea is to construct a working definition, that builds on the original approach introduced in Lopesino et al. (2015), and which relies on stopping the iteration of initial conditions when their orbits leave a certain region in the plane. This criterion is partly inspired by the classical analysis used in Dynamical Systems Theory to study the dynamics of maps by means of escape time plots. We illustrate the capability of this technique to reveal the geometrical template of stable and unstable invariant manifolds in phase space, and also the intricate structure of chaotic sets and strange attractors, by applying it to unveil the phase space of a well-known discrete time system, the H\'enon map.

Figures

Figures reproduced from arXiv: 1908.04871 by the authors.

**Figure 2.** Phase space structures of the H´enon map for the model parameters [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Phase space structures of the H´enon map for the model parameters [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Phase space KAM tori of the H´enon map for the model parameters [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Strange attractor of the H´enon map for the model parameters [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Craven, G. T. and Hernandez, R. (2015). Lagrangian descriptors of thermalized transition states on time-varying energy surfaces. Physical review letters , 115(14), 148301

work page 2015

[2] [2]

Craven, G. T. and Hernandez, R. (2016). Deconstructing ﬁeld-induced ketene isomerization through Lagrangian descriptors. Physical Chemistry Chemical Physics, 18(5), 4008–4018

work page 2016

[3] [3]

Dynamical trapping in the area-preserving H\'enon map

Craven, G. T., Junginger, A., and Hernandez, R. (2017). Lagrangian descriptors of driven chemical reaction manifolds. Physical Review E , 96(2), 022222. de Oliveira, V. M., Ciro, D., and Caldas, I. L. (2019). Dynamical trapping in the area- preserving H´ enon map.arXiv preprint:1907.11616

work page Pith review arXiv 2017

[4] [4]

Demian, A. S. and Wiggins, S. (2017). Detection of periodic orbits in Hamiltonian systems using Lagrangian descriptors. International Journal of Bifurcation and Chaos , 27(14), 1750225

work page 2017

[5] [5]

and Nitecki, Z

Devaney, R. and Nitecki, Z. (1979). Shift automorphisms in the H´ enon mapping. Commu- nications in Mathematical Physics , 67(2), 137–146

work page 1979

[6] [6]

Feldmaier, M., Junginger, A., Main, J., Wunner, G., and Hernandez, R. (2017). Obtaining time-dependent multi-dimensional dividing surfaces using Lagrangian descriptors. Chem- ical Physics Letters , 687, 194–199

work page 2017

[7] [7]

Tilting and Squeezing: Phase space geometry of Hamiltonian saddle-node bifurcation and its influence on chemical reaction dynamics

Garcia-Garrido, V. J., Ramos, A., Mancho, A. M., Coca, J., and Wiggins, S. (2016). A dynamical systems perspective for a real-time response to a marine oil spill. Marine Pollution Bulletin. , pages 1–10. Garc´ ıa-Garrido, V. J., Balibrea-Iniesta, F., Wiggins, S., Mancho, A. M., and Lopesino, C. (2018). Detection of phase space structures of the cat map wi...

work page Pith review arXiv 2016

[8] [8]

and Hernandez, R

Junginger, A. and Hernandez, R. (2015). Uncovering the geometry of barrierless reactions using Lagrangian descriptors. J. Phys. Chem. B

work page 2015