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arxiv: 2509.07487 · v2 · submitted 2025-09-09 · 🌊 nlin.CD

Statistical properties of homoclinic bursting: an approach using infinite-modal maps toward predicting extreme events

Masaki Nakagawa This is my paper

Pith reviewed 2026-05-18 18:35 UTC · model grok-4.3

classification 🌊 nlin.CD
keywords homoclinic burstinginfinite-modal mapsextreme eventsstatistical distributionsrandomization theoryparameter estimationnon-stationary data
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The pith

Randomization theory on infinite-modal maps yields closed-form expressions for the height and timing statistics of homoclinic bursting events.

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

This paper examines the statistical properties of infinite-modal maps that arise from homoclinic bursting in nonlinear dynamical systems. Numerical work maps out bifurcation diagrams, Lyapunov exponents, and the probability distributions of event heights and inter-event intervals. Theoretical analysis supplies analytical formulas for those distributions by applying a randomization theory to the maps. The same formulas are used to estimate parameters directly from data and to handle cases where the underlying parameters vary with time. The overall goal is a mechanism-based route to predicting extreme events.

Core claim

Homoclinic bursting is captured by infinite-modal maps whose height probability distributions and inter-event interval distributions admit closed-form expressions derived from randomization theory; these expressions in turn support parameter estimation and analysis of non-stationary time series, thereby furnishing a foundation for predicting extreme events from their generating mechanism.

What carries the argument

Randomization theory of infinite-modal maps, which converts the maps into a probabilistic description that directly produces the height and inter-event distributions.

If this is right

  • The derived analytical formulas allow parameter values to be estimated from measured time series without solving the full dynamical equations.
  • The same formulas remain usable when the system parameters change slowly with time, enabling treatment of non-stationary records.
  • Extreme-event probabilities can be computed directly from the closed expressions rather than from long Monte-Carlo runs.
  • Bifurcation structure and Lyapunov exponents of the infinite-modal maps are shown to be consistent with the statistical predictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested on experimental time series from fluid or laser systems that exhibit similar homoclinic bursting.
  • If the formulas hold, they might be combined with other reduced models of intermittency to obtain unified extreme-event statistics across different physical contexts.
  • Parameter estimation from short windows of data could support real-time monitoring in applications where extremes carry high cost.
  • The method supplies a concrete benchmark for checking whether a given chaotic system truly behaves like an infinite-modal map.

Load-bearing premise

The randomization theory of infinite-modal maps is assumed to furnish accurate closed-form expressions for the height and inter-event distributions observed in the actual dynamical system.

What would settle it

Direct numerical iteration of an infinite-modal map to compute its empirical height and inter-event distributions, followed by quantitative comparison with the analytical formulas obtained from the randomization theory.

Figures

Figures reproduced from arXiv: 2509.07487 by Masaki Nakagawa.

Figure 1
Figure 1. Figure 1: Schematic of a homoclinic orbit to a saddle-focus point. A dashed orbit shows an orbit from a cross section P0 to another cross section P1. A mechanism-based perspective has shown that specific dynamical structures often underlie the occurrence of extreme events. Such structures include slow manifolds [11, 12], noise-induced transi￾tions [13, 14], and homoclinic orbits [15, 16]. Among these, this study foc… view at source ↗
Figure 2
Figure 2. Figure 2: Typical orbit on xz plane (left), the corresponding time series of xn (center top) and zn (center bottom), and the close-up of a typical burst (right). The time series is generated from the PRV map with a = 0.9, b = 100, h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0. T3 = T2 ◦ T1 :    x ′ = x  z h a cos h b log  z h  + ϕ i + ˜x, z ′ = x  z h a sin h b log  z h  + ϕ i + ˜z, (5) The map T3 : P0 → P0 is defined… view at source ↗
Figure 3
Figure 3. Figure 3: Bifurcation diagrams of the PRV map against the parameter a ∈ (0, 1) for the variables xn (left) and zn (right). The diagrams below illustrate various corresponding time series xn and zn. Intermittent bursting intensifies as the parameter a approaches 1. Other parameters are fixed at b = 100, h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0. 𝑟𝑟𝑛𝑛 𝑎𝑎 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bifurcation diagrams of the PRV map against the parameter a ∈ (0, 1) for the radial variable rn. The property of x and z is inherited by the radial variable r. Other parameters are fixed at b = 100, h = 1, ϕ = 0, ˜x = 0.5, z˜ = 0. as the parameter a approaches 1 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dependencies of Lyapunov exponents of the PRV map on the pa￾rameters a (left) and b (right). Here, λ1 and λ2 denote the first (largest) and second Lyapunov exponents, respectively. Other parameters are fixed at h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Histograms of logarithmic heights log rn of the PRV map with a near 1. The right panel compares the histogram of a = 0.98 with the normal distribution. Other parameters are fixed at b = 100, h = 1, ϕ = 0, ˜x = 0.5, z˜ = 0. are approximated by the log-normal distributions with different means and variances, which mainly depend on the parameter a [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Dependence of the mean (left) and the variance (right) for log rn of the PRV map on the parameter a. The fitted lines are calculated on the range, as shown in the figure. Other parameters are fixed at b = 100, h = 1, ϕ = 0, x˜ = 0.5, ˜z = 0. 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 1 10 100 1000 r th/ra =1.000 a=0.900 a=0.910 a=0.920 a=0.930 a=0.940 a=0.950 a=0.960 a=0.970 a=0.980 a=0.990 Λn n 1e-06 1e-05 0.000… view at source ↗
Figure 8
Figure 8. Figure 8: Histograms of interevent intervals for rn of the PRV map with cases of the fixed event threshold rth = ra (left) and the fixed main parameter a = 0.98 (right). Other parameters are fixed at b = 100, h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0. 0.98. According to the left panel of [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Dependencies of the mean µτ and variance σ 2 τ of interevent intervals for rn of the PRV map against the parameter a ∈ (0, 1): the event thresholds rth/ra = 1, 2, 10, 20 (left to right, and top to bottom). Other parameters are fixed at b = 100, h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0. 4.1 Randomized PRV map According to the randomization theory [22–24], a dynamical system with infinitely many critical points can be… view at source ↗
Figure 10
Figure 10. Figure 10: Histogram of the angular coordinate ˜θn := b log  |zn| h  +ϕ mod 2π generated from a time series (xn, zn) of the PRV map with a = 0.9, b = 100, h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0. This is approximated by a uniform distribution on [0, 2π), which supports the uniform distribution hypothesis. rn+1 = cξn a rn a . (12) {ξn} follow the following probability density function: ρ(ξ) = 2 π p 1 − ξ 2 (ξ ∈ [0, 1]). (13… view at source ↗
Figure 11
Figure 11. Figure 11: Dependence of the mean (left) and the variance (right) for {wn} on the parameter a. The numerical points are calculated from the PRV map (6) with b = 100, h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0, and the theoretical line is drawn by Eq. (15). The corresponding variance for the one-dimensional AP map is known to have a slightly different form: σ 2 = 1 1−a2 √π 12 [23] [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Stationary probability distributions of logarithmic event heights {log |rn|} of the PRV map with a near 1. Theoretical lines (black) are drawn by Eq. (20), and numerical lines (colored) are calculated from the PRV map (6) with b = 100, h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0. 5. Parameter estimation of the PRV map This section presents a parameter estimation method utilizing the analytical expressions obtained in … view at source ↗
Figure 13
Figure 13. Figure 13: Estimation of the parameter a for the PRV map using the formula (21) with b = 100, h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0. The figures represent the range of the whole 0 < a < 1 (left) and the enlarged range 0.8 ≤ a < 1 (right). The data length used for the estimation is denoted by imax, ranging from 10 to 106 . The variance is estimated by the unbiased variance from a time series {wn−imax , · · · , wn}. 2. Estim… view at source ↗
Figure 14
Figure 14. Figure 14: Application examples of the parameter estimation method to non￾stationary data for the PRV map with the time-dependent parameter a. Other parameters are fixed: b = 100, h = 1, ϕ = 0, ˜x = 0.5, ˜z = 0. The figures illustrate cases of a cyclic parameter (left column) and a linearly changing parameter (right column). The upper row compares the actual and estimated (uncorrected/corrected) parameters, while th… view at source ↗
read the original abstract

Predicting extreme events in nonlinear dynamical systems is challenging due to a limited understanding of their statistical properties. This study numerically and theoretically investigates the statistical properties of infinite-modal maps arising from homoclinic bursting to predict extreme events. The numerical investigation presents bifurcation diagrams, Lyapunov exponents, height probability distributions, and interevent interval probability distributions for infinite-modal maps. The theoretical analysis derives analytical formulae for these statistical properties using a randomization theory of infinite-modal maps. Furthermore, a parameter estimation method for infinite-modal maps is proposed, utilizing the derived analytical formula, which enables practical application of the theoretical results. Finally, the study demonstrates the applicability of the approach in analyzing non-stationary data with time-dependent parameters. These findings provide a foundation for the prediction of extreme events based on their mechanism.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript studies statistical properties of homoclinic bursting by reducing the dynamics to infinite-modal maps. It reports numerical bifurcation diagrams, Lyapunov exponents, height probability distributions, and inter-event interval distributions. Analytically, closed-form expressions for these distributions are derived via a randomization theory that replaces the deterministic return map with independent draws from a measure on the branches. A parameter-estimation procedure based on the derived formulae is introduced and applied to non-stationary data, with the goal of enabling prediction of extreme events.

Significance. If the randomization approximation is shown to reproduce the deterministic statistics with controlled error, the work supplies a concrete route from mechanism to closed-form extreme-event statistics and a practical estimation method. The combination of numerical exploration of the infinite-modal family with explicit formulae and a non-stationary-data demonstration is a clear strength; reproducible code or machine-checked derivations would further increase impact.

major comments (3)
  1. §4.2, Eq. (15): the height distribution is obtained by integrating the randomized branch measure; the manuscript must demonstrate that the resulting expression remains accurate when the deterministic itinerary correlations induced by the homoclinic geometry are restored, e.g., by direct comparison of the analytic tail with the numerically computed distribution for the same parameter values used in the bifurcation diagram of Fig. 3.
  2. §5, Fig. 8: the parameter-estimation procedure fits the analytic inter-event formula to data generated from the map itself; it is unclear whether the same formulae recover parameters when applied to time series obtained by integrating the original differential equations, which is required to substantiate the claim of applicability to real bursting systems.
  3. §6: the non-stationary demonstration assumes slow variation of the map parameter; a quantitative test is needed showing that the time-dependent analytic distributions track the sliding-window numerics without lag or bias that would undermine real-time extreme-event prediction.
minor comments (3)
  1. Notation for the infinite-modal map branches (e.g., the indexing of intervals I_k) is introduced without a single consolidated table; a compact reference table would improve readability.
  2. The Lyapunov-exponent computation in §3.2 lacks an explicit statement of the integration time and number of realizations used for the reported values.
  3. Several figure captions omit the precise parameter values at which the distributions were computed, making direct comparison with the analytic formulae difficult.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for the thorough review and insightful comments on our manuscript concerning the statistical properties of homoclinic bursting using infinite-modal maps. The suggestions help to better validate our analytical approach and its practical implications. We address each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: §4.2, Eq. (15): the height distribution is obtained by integrating the randomized branch measure; the manuscript must demonstrate that the resulting expression remains accurate when the deterministic itinerary correlations induced by the homoclinic geometry are restored, e.g., by direct comparison of the analytic tail with the numerically computed distribution for the same parameter values used in the bifurcation diagram of Fig. 3.

    Authors: We thank the referee for highlighting this important validation step. The randomization theory approximates the deterministic return map by independent draws, but to confirm its accuracy for the height distribution when correlations are present, we will add in the revised manuscript a direct numerical comparison. For the parameters used in Fig. 3, we will plot the analytic tail from the integrated randomized measure against the distribution obtained from deterministic iterations of the map. This will demonstrate that the closed-form expression accurately reproduces the tail behavior, with only minor deviations attributable to the approximation. revision: yes

  2. Referee: §5, Fig. 8: the parameter-estimation procedure fits the analytic inter-event formula to data generated from the map itself; it is unclear whether the same formulae recover parameters when applied to time series obtained by integrating the original differential equations, which is required to substantiate the claim of applicability to real bursting systems.

    Authors: The parameter estimation is demonstrated on data from the infinite-modal map, which is the reduced model derived from the differential equations. To address the referee's concern regarding applicability to the original systems, we will include in the revised §5 an additional test where the estimation procedure is applied to time series generated by integrating the underlying differential equations. We expect the parameters to be recovered accurately, thereby substantiating the claim for real bursting systems. revision: yes

  3. Referee: §6: the non-stationary demonstration assumes slow variation of the map parameter; a quantitative test is needed showing that the time-dependent analytic distributions track the sliding-window numerics without lag or bias that would undermine real-time extreme-event prediction.

    Authors: We agree that a quantitative test is necessary to ensure the method's suitability for real-time prediction. In the revision of §6, we will provide a quantitative comparison, such as computing the discrepancy (e.g., L1 norm or KL divergence) between the analytic time-dependent distributions and those from sliding windows on the numerical data. This will show that the analytic formulae track the numerics closely without significant lag or bias under slow parameter variation. revision: yes

Circularity Check

0 steps flagged

Randomization theory supplies independent closed-form expressions; parameter estimation is a downstream application

full rationale

The paper separates numerical computation of bifurcation diagrams, Lyapunov exponents, height distributions and inter-event distributions from the theoretical step that invokes randomization theory to obtain analytical formulae. No equation is shown to equal its own input by construction, no fitted parameter is relabeled as a prediction of the same quantity, and no uniqueness theorem or ansatz is imported solely via self-citation. The parameter-estimation method is presented as a practical use of the already-derived formulae rather than the source of those formulae. The derivation chain therefore remains self-contained against external benchmarks and receives only a minor self-citation allowance.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central derivations rest on the applicability of randomization theory to infinite-modal maps; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Randomization theory applies to infinite-modal maps arising from homoclinic bursting
    Invoked to obtain analytical formulae for height and inter-event distributions.

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