{"id":"214d787f-9f29-44a8-a569-e4ffbee89ff4","arxiv_id":"1908.11195","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A Caputo-delta fractional variant of the Gompertz-like map is introduced, and an impulsive control algorithm is numerically shown to suppress chaos into numerically stable periodic orbits.","lead":"This paper defines a fractional-order version of a Gompertz-like discrete map from tumor growth modeling and shows numerically that periodic impulses can calm its chaos into apparent periodic orbits. A generalist might read it for an example of how discrete fractional calculus plus a simple control rule can regularize a biological model.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Long-horizon persistence of the claimed NSPOs is untested, and the LE evidence is explicitly left open; the suppression claim may be a finite-horizon artifact.","rationale":"The reader's CONDITIONAL verdict is reasonable. I do not object to the novelty or to the numerical nature of the study; the time series and the independent 0-1 test are genuine evidence. My concern is narrower: the examples select gamma from the same bifurcation diagrams used for verification, and the verification horizon is 1000 iterations. The paper explicitly concedes two limitations that bear on this: no true periodic solutions exist for fractional-order discrete systems (Remark 2), and the LE sign is open (Conclusion). These concessions make the finite-horizon persistence of the NSPOs the most load-bearing unverified condition. A single long-horizon rerun with released code would settle whether the reported periodic windows are intrinsic to the controlled dynamics or are transient features of a short simulation. Since this is a missing check rather than a demonstrated error, the reader's CONDITIONAL verdict should stand unchanged; the revision condition should be the long-horizon persistence check and an independent recomputation of the LE from the correct variational equation.","tokens_in":7360,"tokens_out":12453,"duration_ms":129701,"concrete_test":"Run the controlled algorithm for N=10^6 iterations at the three reported parameter triples (gamma=-0.0132, delta=1; gamma=-0.04, delta=3; gamma=-0.0722, delta=5; q=0.8, r=1), using the intended reading of (12): compute the fractional-sum update and then multiply by 1+gamma when n is a multiple of delta. In sliding windows, measure the NSPO closing error against the claimed periods (10, 5, and 19) and compute the 0-1 test K averaged over at least 100 random c values. If the closing error stays below the stated 1E-n bound and K remains below about 0.1 through 10^6 steps, the suppression claim is supported; if the trajectory drifts out of the periodic window or K rises, the reported suppression is a finite-horizon artifact.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the impulsive algorithm (12) suppresses the chaotic behavior of the fractional-order system (8) for q=0.8, r=1 and suitable gamma, producing numerically stable periodic orbits. For this claim to hold, the regular behavior must persist beyond the reported 1000-iteration horizon and must not be an artifact of the finite-horizon numerical integration. The paper itself supplies two warnings. First, Remark 2 states that fractional-order discrete systems cannot have nonconstant periodic solutions, so what is claimed is only an NSPO, defined by a closing error within a bound of 1E-n; but no closing errors are reported for the three example orbits (10-period for delta=1, gamma=-0.0132; 5-period for delta=3, gamma=-0.04; 19-period for delta=5, gamma=-0.0722). Second, the Conclusion states that the sign of the LE remains an open problem; since the LE is computed on the same short, parameter-selected windows, it cannot rule out a slow drift. The 0-1 test K values are also finite-sample estimates (the paper quotes errors of order 1e-2 for K near 0), so K values around 0.003, -0.004, and 0.032 do not by themselves establish asymptotic regularity. Because the convolution kernel in (9) carries the full pre-control chaotic history into every future state, the apparent periodicity could be a slowly decaying transient. If the orbit leaves the periodic window after, say, 10^4 or 10^5 steps, the central claim would reduce to a statement about a transient, not about suppression of chaos.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a fractional-order discrete Gompertz-like system, obtained by replacing the integer-order map with a Caputo delta fractional difference, and proposes an impulsive control algorithm (12) that periodically perturbs the state by a factor (1+gamma). The central claim is that for q=0.8, r=1, and suitable negative values of gamma, the chaotic dynamics of (8) is suppressed and replaced by numerically stable periodic orbits (NSPOs), with examples for delta=1,3,5 giving periods 10, 5, and 19. The numerical evidence consists of bifurcation diagrams, finite-time Lyapunov exponents from Eq. (11), the 0-1 test values K, p-and-q plots, and mean-square displacement M. The paper explicitly acknowledges that fractional-order discrete systems cannot have nonconstant periodic solutions, so only NSPOs are claimed, and it also states that the sign of the Lyapunov exponent remains an open problem.","tokens_in":7639,"tokens_out":7090,"duration_ms":71545,"significance":"If the numerical claims are correct, this is a modest but useful case study in chaos suppression for a fractional-order discrete biological model, extending earlier work by the same authors on impulsive control in continuous and discrete systems. The paper has the strength of being explicit about the nonexistence of true periodic solutions and about the NSPO interpretation, and it uses multiple diagnostics rather than relying on a single indicator. However, the central claim is under-supported as presented: the primary diagnostic, the finite-time Lyapunov exponent, is openly questioned by the authors themselves; no closing errors are reported for the claimed NSPOs; the simulations are limited to 1000 iterations; and the control parameters are selected from the controlled system's own bifurcation diagrams, making the verification partly in-sample. Since the paper releases no code or data, the reproducibility of the numerical results cannot be independently checked.","major_comments":[{"comment":"The central diagnostic used to certify regular dynamics is the finite-time Lyapunov exponent obtained from Eq. (11), but the Conclusion states that whether this LE is really negative or zero remains an open problem. Since the reported K values (0.0033, -0.0035, -0.0048, 0.0317) are all of the same order as the paper's own stated 1e-2 error margin for K close to zero, and since all runs use only 1000 iterations, the presented evidence does not distinguish regular motion from a slowly decaying transient that inherits the pre-control chaotic history through the memory kernel in (9). Please provide a reliable LE estimate with a corrected linearization and confidence bounds, or replace the LE with a direct quantitative regularity measure such as long-horizon closing errors and drift.","section":"§3 and Conclusion"},{"comment":"The definition of an NSPO requires a closing error within a bound of the form 1E-n, but no closing error is reported for any of the three claimed NSPOs (10-period for delta=1, 5-period for delta=3, 19-period for delta=5). Without these numbers, the statements '10-period,' '5-period,' and '19-period' are only visual classifications of the time series. Please report the maximal closing errors, the chosen bound, and the segment over which they are measured.","section":"§3, Remark 2"},{"comment":"The suppression claim is based on 1000-iteration runs, and the control parameter gamma is selected from the bifurcation diagram of the very same controlled system on which the diagnostics are then computed. Because the fractional kernel in (9) carries the full history, the apparent periodicity could be a transient that decays on a longer time scale. Please extend the runs to at least 10^4-10^5 iterations, report the diagnostics on a hold-out segment not used for parameter selection, and test several initial conditions. This would also reduce the in-sample character of the current confirmation.","section":"§3, '1000 iterations' and Eq. (9)"},{"comment":"As typeset, the variational equation (11) contains the factor (2/3)x(j) - 1/3 - 1, which is not the derivative 6.75r((2/3)x(j)^(-1/3) - 1) of the map f_r used in (9). If the exponent -1/3 is missing from the printed formula, then all reported LE values are computed from the wrong linearization and should be recomputed; if the formula is intentional, a derivation should be provided. This matters because the LE is the paper's main quantitative chaos indicator.","section":"Eq. (11)"}],"minor_comments":[{"comment":"The text states that for delta=3 and gamma=-0.04 a 'numerically 5-period orbit' is obtained, while the caption of Fig. 5 says the zoomed area reveals 'the six elements of the NSPO'; please reconcile this discrepancy.","section":"§3, Fig. 5"},{"comment":"The symbol p is used both for the power exponent in f_{r,p} and for the translation variable in the 0-1 test; this notation collision is confusing and should be resolved, for instance by using a different letter for one of them.","section":"§2 and Appendix"},{"comment":"In Remark 1, 'divergency' should be 'divergence', and the notation '1E-n' in Remark 2 should be replaced with a clearly defined bound such as 1 times 10^{-n}; the required value of n should also be quantified in relation to the examples.","section":"§2, Remark 1"},{"comment":"The description of the 0-1 test omits implementation details essential for reproducibility: the number of iterations N, the number of randomly chosen c values, and the regression procedure used to estimate K. Please add these details.","section":"Appendix"},{"comment":"No code or data are made available, and no data-availability statement is provided; for a purely numerical paper this strongly limits reproducibility and verification of the reported bifurcation diagrams and diagnostics.","section":"General"},{"comment":"There are several editorial errors, including 'Institute od Science and Technology' in the affiliation and a reference list that begins with Nicol et al. rather than following the chronological order of citations; the reference formatting should be made consistent with the journal style.","section":"Front matter and references"}],"recommendation":"major_revision","confidential_remarks":"The paper is a numerical case study whose main claim is not supported by released code or data, and the authors themselves state that the sign of the Lyapunov exponent is an open problem. If the journal is willing to accept a purely numerical demonstration of chaos suppression, the revised version needs substantially stronger evidence: quantified NSPO closing errors, much longer horizons, and a corrected or explicitly qualified LE computation. I would also encourage the editor to ask for an independent implementation check, since no code is provided and the numerical integration of the fractional kernel is described as delicate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a modest, real contribution. The authors introduce a Caputo delta fractional-order version of a Gompertz-like map (system (8)) and apply an old periodic-impulse control idea from Guemez-Matias/Codreanu-Danca. The fractional variant is new, and the first numerical study of impulse control in that setting. The numerical presentation is clear: bifurcation diagrams, time series, 0-1 test K-values, (p,q) plots, and mean-square displacement. The citation pattern looks clean; earlier control and fractional-difference work is credited.\n\nThe paper is also honest about its own limits. Remark 2 correctly notes that discrete fractional-order systems cannot have nonconstant periodic solutions, so the claim is only about NSPOs. The Conclusion explicitly says whether the LE is really negative or zero is open. That candor counts for something.\n\nThe soft spots are genuine and mostly acknowledged. Gamma is chosen from the controlled system's own bifurcation diagram, and the same trajectory then supplies LE and K, so the verification is partly confirmation on the fitting set. The finite-time LE from (11) is a natural linearization, but the paper does not prove it is the correct variational equation for the fractional integral. The 0-1 test K-values carry errors of order 1e-2, so values like 0.003 or -0.004 do not by themselves establish asymptotic regularity. And all runs are only 1000 iterations. Because the convolution kernel in (9) carries the full pre-control history into every future state, the apparent NSPO could in principle drift after longer horizons. The stress-test worry about finite-horizon artifact is legitimate, but it is not fatal; the authors already flag the main caveats. What is missing is the evidence that would resolve them: code/data release, longer runs, reported closing errors, and a clearer statement on the LE.\n\nProportionately: the claimed phenomenon likely exists in the numerical sense the authors define, but the evidence is not conclusive. The paper serves the niche of fractional discrete dynamics; it will not change practice elsewhere.\n\nRecommendation: this deserves peer review, not a desk reject. A serious referee should ask for code/data, longer horizons, closing-error magnitudes, and a sharper treatment of the LE. I would not cite it in my own work in the next year, but I would bring it to a reading group focused on fractional discrete systems.","headline":"A modest but real niche contribution: new fractional-order Gompertz map with honest numerical evidence of impulse-induced chaos suppression, yet the case rests on short parameter-selected runs and an LE whose sign the authors themselves leave open.","tokens_in":8193,"tokens_out":2104,"would_cite":false,"duration_ms":23926,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D45","39A12"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that periodic impulses can suppress chaos in a fractional-order Gompertz-like discrete map.","keywords":["fractional-order discrete system","Gompertz-like map","chaos suppression","impulsive control","Lyapunov exponent","0-1 test","numerically stable periodic orbits","Caputo delta fractional difference"],"falsifier":"Recompute the finite-time Lyapunov exponent for the same three controlled parameter sets using a full numerical Jacobian of (9) or a perturbation-tracking method, and extend the 0-1 test beyond 1000 iterations. If the exponent becomes clearly positive, or if $K$ drifts toward 1 on longer time series, the reported chaos suppression would be shown to be an artifact of the chosen simulation window.","tokens_in":7120,"feed_emoji":"🌀","tokens_out":10601,"duration_ms":95873,"temperature":0.7,"pith_summary":"This paper introduces a fractional-order version of a Gompertz-like discrete growth map and argues that its chaotic motion can be suppressed by a simple periodic impulse: at every $\\delta$-th step, the state is multiplied by $1+\\gamma$. For fractional order $q=0.8$ and parameter $r=1$, the paper reports that suitable negative values of $\\gamma$ — $\\gamma=-0.0132$ with $\\delta=1$, $\\gamma=-0.04$ with $\\delta=3$, and $\\gamma=-0.0722$ with $\\delta=5$ — turn the chaotic orbit into what it calls a numerically stable periodic orbit. The supporting diagnostics are a non-positive finite-time Lyapunov exponent, a 0-1 test value $K$ close to zero, disc-like translation-variable plots, and bounded mean-square displacement. A sympathetic reader would care because the result extends a familiar integer-order control idea to systems with memory, where true periodic solutions are known not to exist and only numerically stable periodic orbits are available.","feed_headline":"Periodic kicks suppress chaos in a fractional Gompertz map","feed_subtitle":"At q=0.8, impulses every 1, 3, or 5 steps force numerically stable periodic orbits.","key_machinery":"The central mechanism is the discrete fractional integral (9), $$x(n)=x(0)+\\frac{6.75r}{\\Gamma(q)}\\sum_{j=1}^{n}\\frac{\\Gamma(n-j+q)}{\\Gamma(n-j+1)}\\left(x_{j-1}^{2/3}-x_{j-1}\\right),$$ which encodes the memory of the Caputo delta fractional difference (7)–(8). The control is the periodic impulse (12), which multiplies the next state by $1+\\gamma$ at every $\\delta$-th step. The verification uses the natural linearization (11) to define a finite-time Lyapunov exponent $\\lambda\\simeq(1/n)\\ln|a(n-1)|$, and the 0-1 test, whose value $K\\approx 0$, disc-like $(p,q)$ graphs, and bounded mean-square displacement $M$ are taken to indicate regular motion. Because of the memory term, the target states are called numerically stable periodic orbits (NSPOs), i.e., closed trajectories whose closing error stays within a prescribed bound.","core_discovery":"The paper claims that the fractional-order discrete Gompertz-like system (8), built from the Caputo delta fractional difference with $q=0.8$ and $r=1$, is chaotic, and that the impulsive algorithm (12) suppresses that chaos for properly chosen $\\gamma$. The algorithm acts as $x_{n+1}\\leftarrow (1+\\gamma)x_{n+1}$ whenever $n$ is a multiple of $\\delta$. For $\\delta=1$, $\\gamma=-0.0132$ yields a numerically stable period-10 orbit; for $\\delta=3$, $\\gamma=-0.04$ yields a period-5 orbit; and for $\\delta=5$, $\\gamma=-0.0722$ yields a period-19 orbit, with no controlled orbits found for $\\delta>5$. The paper emphasizes that these are numerically stable periodic orbits, not genuine periodic solutions, because fractional-order discrete systems admit no nonconstant periodic solutions. The numerical confirmation combines the finite-time Lyapunov exponent from the natural linearization (11) with the 0-1 test output: $K$ near zero, disc-like $(p,q)$, and bounded $M$.","pith_inferences":["An extension the paper does not pursue is that the same periodic-impulse scheme could be tested on other memory-carrying discrete maps, such as the fractional logistic map, to see whether the reported $\\delta\\le 5$ control window is a general feature of fractional discrete systems.","If the scalar linearization (11) is not the exact variational equation, the finite-time Lyapunov exponent may understate the true instability; a full Jacobian or perturbation-based exponent for (9) would settle whether the controlled orbits are genuinely stable or merely slow on the simulated horizon.","The near-zero plateau of the Lyapunov exponent across broad parameter ranges hints that the suppressed regime may be a slow nonchaotic dynamics inherited from the memory kernel rather than true periodicity; spectral analysis of the NSPO would distinguish these cases.","The observed sign asymmetry of algorithm (12) versus the symmetry of algorithm (13) suggests an energy-balance explanation for the control, which could in principle be derived analytically for unimodal maps with a power-law memory."],"forward_implications":["For $\\delta=1,3,5$, the controlled system settles onto numerically stable periodic orbits of periods 10, 5, and 19, respectively, and no controlled orbits are found for $\\delta>5$.","Under the multiplicative algorithm (12), only negative values of $\\gamma$ suppress chaos, suggesting the controlled system must shed energy every $\\delta$ steps.","The alternative additive algorithm (13) suppresses chaos for both negative and positive $\\gamma$, showing that either energy loss or energy gain can stabilize the system.","Over wide ranges of $\\gamma$, the finite-time Lyapunov exponent stays constant and close to zero, leaving open whether the true exponent is negative or zero.","In both integer- and fractional-order versions, increasing the power exponent $p$ produces reverse period-doubling and extinction of chaos, with exterior crises visible in the fractional-order bifurcation diagram."],"supporting_citations":[{"why":"Supplies the discrete fractional integral formula (9) used as the numerical integration of the fractional system.","marker":"[Wu & Baleanu, 2014]"},{"why":"Supplies the natural linearization (11) from which the finite-time Lyapunov exponent is computed.","marker":"[Wu & Baleanu, 2015]"},{"why":"Provides the 0-1 test whose K, (p,q), and M diagnostics are used to verify chaos suppression.","marker":"[Gottwald & Melbourne, 2004]"},{"why":"Introduces the periodic impulsive-perturbation control strategy that algorithm (12) adapts to the fractional system.","marker":"[Guemez & Matias, 1993]"},{"why":"Applies the control strategy to the integer-order Gompertz-like map (4), the chaotic baseline this paper extends to fractional order.","marker":"[Codreanu & Danca, 1997]"},{"why":"Introduces the Gompertz-like discrete model (2)-(3) from which the rescaled map (4) is obtained.","marker":"[Ahmed, 1992]"},{"why":"Supplies the notion of numerically stable periodic orbits used to describe the controlled regular motion.","marker":"[Danca et al., 2018]"},{"why":"Establishes that fractional difference systems have no nonconstant periodic solutions, which is why the controlled orbits are called numerically stable.","marker":"[Diblik et al., 2015]"}],"fun_headline_variants":["Impulsive kicks tame chaos in fractional Gompertz map","Periodic kicks quell chaos in fractional Gompertz system","Impulsive control stops fractional Gompertz chaos","Fractional Gompertz chaos suppressed by discrete kicks","Periodic impulses stabilize fractional Gompertz dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the finite-time Lyapunov exponent computed from the scalar linearization (11) is a valid measure of the stability of the fractional integral (9) along the controlled orbit; the paper itself leaves open whether the exponent is genuinely negative or merely zero, and if the linearization is not the true variational equation, the Lyapunov-based evidence for chaos suppression loses its force.","fun_headline_variants_meta":{"raw":{"variants":["Impulsive kicks tame chaos in fractional Gompertz map","Periodic kicks quell chaos in fractional Gompertz system","Impulsive control stops fractional Gompertz chaos","Fractional Gompertz chaos suppressed by discrete kicks","Periodic impulses stabilize fractional Gompertz dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00066,"raw_usage":{"total_tokens":2960,"prompt_tokens":831,"completion_tokens":2129,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":2052}},"tokens_in":447,"tokens_out":2129,"duration_ms":14524,"temperature":1.0,"reasoning_tokens":2052,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:37:33.429292+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the finite-time Lyapunov exponent for the same three controlled parameter sets using a full numerical Jacobian of (9) or a perturbation-tracking method, and extend the 0-1 test beyond 1000 iterations. If the exponent becomes clearly positive, or if $K$ drifts toward 1 on longer time series, the reported chaos suppression would be shown to be an artifact of the chosen simulation window.","supporting_citations":[{"cited_title":"[2004] ``A new test for chaos in deterministic systems'', Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 460(2042)603--611","cited_arxiv_id":null,"evidence_quote":"Provides the 0-1 test whose K, (p,q), and M diagnostics are used to verify chaos suppression."},{"cited_title":"and Matias, M.A","cited_arxiv_id":null,"evidence_quote":"Introduces the periodic impulsive-perturbation control strategy that algorithm (12) adapts to the fractional system."},{"cited_title":"[1992] ``Fractals and chaos in cancer models'', Int","cited_arxiv_id":null,"evidence_quote":"Introduces the Gompertz-like discrete model (2)-(3) from which the rescaled map (4) is obtained."},{"cited_title":"and Chen G","cited_arxiv_id":null,"evidence_quote":"Supplies the notion of numerically stable periodic orbits used to describe the controlled regular motion."}],"review_version":1}