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arxiv: 2604.20534 · v1 · submitted 2026-04-22 · 🌊 nlin.CD · nlin.AO· physics.app-ph

Recognition: unknown

Extreme events in MLC circuit

Dibakar Ghosh, Tapas Kumar Pa

Authors on Pith no claims yet

Pith reviewed 2026-05-09 22:33 UTC · model grok-4.3

classification 🌊 nlin.CD nlin.AOphysics.app-ph
keywords MLC circuitextreme eventschaotic attractorperiod-merging intermittencynonlinear dynamicsextreme value theorystable and unstable manifoldsFloquet multipliers
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The pith

Extreme events emerge in the MLC circuit when the chaotic attractor expands largely along the period-merging intermittency route due to the external periodic force.

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

This paper investigates the origin of extreme events in the Murali-Lakshmanan-Chua circuit, a classic nonlinear dissipative system driven by a periodic force. It establishes that the key precursor is the large expansion of the chaotic attractor that occurs through period-merging intermittency. The external force generates a force field in phase space that pushes trajectories into large excursions, producing the extreme events. The authors support this with two additional views: separating the phase space into stable and unstable manifolds using slow-fast dynamics and calculating Floquet multipliers. They further show through extreme value theory that the sizes of these events follow a generalized Pareto distribution while the times between them follow a generalized extreme value distribution.

Core claim

The paper claims that the large expansion of the chaotic attractor following the period-merging intermittency route plays the crucial role as the precursor behind the emergence of extreme events in the MLC circuit. The prevalence of a force field due to the externally applied periodic force creates the dynamical synergy that compels the chaotic trajectory to be largely deviated from its residing space. This large deviation is interpreted as extreme events, with supporting explanations from the decomposition of the phase space in stable and unstable manifolds concerning slow-fast dynamics and using Floquet multipliers. The rare occurrences are statistically analyzed using extreme value theory

What carries the argument

The force field induced by the external periodic force that drives large deviations of the chaotic trajectory in the MLC circuit, facilitated by attractor expansion via period-merging intermittency.

If this is right

  • The external periodic force is identified as the dominant cause creating the dynamical synergy for extreme events.
  • Analysis of stable and unstable manifolds and Floquet multipliers both confirm the mechanism of large trajectory excursions.
  • Extreme events can be characterized statistically with generalized Pareto distribution for excess values and generalized extreme value distribution for inter-event intervals.
  • The mechanism explains two different types of extreme events defined in the system.

Where Pith is reading between the lines

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

  • Tuning the amplitude or frequency of the external force might offer a way to suppress or control the occurrence of extreme events in such circuits.
  • The approach of using manifold decomposition and Floquet theory could be applied to study extreme events in other driven chaotic systems.
  • If the intermittency route is general, similar attractor expansions might precede extremes in a broader class of nonautonomous oscillators.

Load-bearing premise

The definitions chosen for the two types of extreme events and the conclusion that the external force is the dominant cause do not change with small adjustments to thresholds or other parameters.

What would settle it

A direct test would be to simulate or experiment with the MLC circuit at zero amplitude of the external periodic force and check if the large attractor expansion and extreme events both disappear.

Figures

Figures reproduced from arXiv: 2604.20534 by Dibakar Ghosh, Tapas Kumar Pa.

Figure 1
Figure 1. Figure 1: Dynamical exposition of the system (1) concerning xmax: (a) The changing portfolio of xmax due to the variation of the external forcing amplitude f in the range [0.1312, 0.1315]. As f decreases from 0.1315 following a periodic orbit of one period, large-amplitude chaos emerges through PM intermittency at f ≈ 0.131451. The red curve represents the extreme events qualifying threshold Hth = m + 6µ, where m is… view at source ↗
Figure 2
Figure 2. Figure 2: Dynamical exposition of the system (1) concerning xmin: (a) The changing framework of xmin as f decreases from 0.1315 to 0.1312. Large-amplitude chaos is seen to emerge following a one-period periodic orbit at f ≈ 0.131451 through PM intermittency. The green curve represents the extreme events qualifying threshold Hth = m − 6µ, where m is the mean and µ is the standard deviation of the specific data set. T… view at source ↗
Figure 3
Figure 3. Figure 3: Dynamical mechanism behind the large excursion of the chaotic trajectories of the system (1) concerning the events xmax: (a) The synergistic representation of chaotic trajectories and the externally applied periodic force upon the (x, y.f sin(ωt)) space, elucidating the dynamical mechanism responsible for the significantly large deviation of the chaotic trajectory as EE. The color bar represents the extern… view at source ↗
Figure 4
Figure 4. Figure 4: The stable and unstable manifold decomposition using Floquet multipliers unravels a responsible mech￾anism behind the large excursion of the chaotic trajectories of the system (1) concerning the events xmax: For the bifurcation parameter value f = 0.1314, the genesis of EEs due to the coexistence of the stable (attracting) and the unstable (repelling) manifold is explicated in this panel. (a) The stable ma… view at source ↗
Figure 5
Figure 5. Figure 5: The stable and unstable manifold decomposition regarding slow-fast dynamics of the system (1) unravels a responsible mechanism behind the large excursion of the chaotic trajectories concerning the events xmax: Cor￾responding to the bifurcation parameter value f = 0.1314, the stable (attracting) and the unstable (repelling) manifolds are calculated regarding the slow-fast dynamics of the system, and due to … view at source ↗
Figure 6
Figure 6. Figure 6: Dynamical mechanism behind the large excursion of the chaotic trajectories of the system (1) concerning the events xmin: (a) The dynamical synergy of the chaotic trajectory and the externally applied periodic force is represented upon the (x, y, f sin(ωt)) plane, explicating the mechanism behind the genesis of the large excursion of the chaotic trajectory depicting EE. The color bar represents the external… view at source ↗
Figure 7
Figure 7. Figure 7: The stable and unstable manifold decomposition using Floquet multipliers unravels a responsible mech￾anism behind the large excursion of the chaotic trajectories of the system (1) concerning the events xmin: For the bifurcation parameter value f = 0.131393, the genesis of EEs due to the coexistence of the stable (attracting) and the unstable (repelling) manifold is explicated in this panel. (a) The stable … view at source ↗
Figure 8
Figure 8. Figure 8: The stable and unstable manifold decomposition regarding slow-fast dynamics of the system (1) unravels a responsible mechanism behind the large excursion of the chaotic trajectories concerning the events xmin: Corre￾sponding to the bifurcation parameter value f = 0.131393, the stable (attracting) and the unstable (repelling) manifolds are calculated regarding the slow-fast dynamics of the system, and due t… view at source ↗
Figure 9
Figure 9. Figure 9: Schematic presentation of extreme spike (ES) and inter-extreme-spike-interval (IESI): (a)-(b) The green￾dashed line represents the extreme events qualifying threshold line. The spikes of the time series that are being exceeded by the threshold line are termed as extreme spikes (ES), shown by the arrow in the figures. The elapsed time difference between two consecutive ESs is termed as the inter-extreme-spi… view at source ↗
Figure 10
Figure 10. Figure 10: Histogram of the events xmax for different values of the bifurcation parameter f of the system (1) in semi-log scale: The vertical red-dashed line represents the extreme events qualifying threshold line Hth = m + 6µ, where m is the mean and µ is the standard deviation of the respective sets of data, in each figure for the respective value of f. (a) Histogram of the events xmax corresponding to f = 0.13122… view at source ↗
Figure 11
Figure 11. Figure 11: Statistical analysis of the sets of threshold excess values for different values of the bifurcation parameter f of the system (1) concerning the events xmax : (a) The fitted GPD distribution curve of the set of threshold excess values corresponding to f = 0.1314 is depicted in red color. The figure is presented in semi-log scale. The K-S statistic plot, P-P plot, and Q-Q plot of the set of threshold exces… view at source ↗
Figure 12
Figure 12. Figure 12: Statistical analysis of the sets of IESIs for different values of the bifurcation parameter f of the system (1) concerning the events xmax : (a) The fitted GEV distribution curve of the set of IESIs corresponding to f = 0.1314 is depicted in red color. The figure is presented in semi-log scale. The K-S statistic plot, P-P plot, and Q-Q plot of the set of IESIs for f = 0.1314 regarding the GEV distribution… view at source ↗
Figure 13
Figure 13. Figure 13: Histogram of the events xmin for different values of the bifurcation parameter f of the system (1) in semi-log scale: The green-dashed vertical line represents the extreme events qualifying threshold Hth = m − 6µ, where m is the mean and µ is the standard deviation of the respective set of events for different values of f. (a) Histogram of the events corresponding to the bifurcation parameter f = 0.131231… view at source ↗
Figure 14
Figure 14. Figure 14: Statistical analysis of the sets of threshold excess values for different values of the bifurcation parameter f of the system (1) concerning the events xmin : (a) The fitted GPD curve of the set of the threshold excess values for f = 0.131393 is depicted in red color. The figure is portrayed in semi-log scale. The K-S statistic, P-P plot, and Q-Q plot of the set of the threshold excess values for f = 0.13… view at source ↗
Figure 15
Figure 15. Figure 15: Statistical analysis of the sets of IESIs for different values of the bifurcation parameter f of the system (1) concerning the events xmin : (a) The fitted GEV distribution curve of the set of IESIs for f = 0.131393 is displayed. The figure is presented in semi-log scale. The K-S statistic, P-P plot, and Q-Q plot of the set of IESIs corresponding to f = 0.131393 concerning the GEV distribution fitting are… view at source ↗
read the original abstract

The Murali-Lakshmanan-Chua (MLC) circuit is a well-recognized prominent nonlinear, nonautonomous, and dissipative electronic circuit having a versatile chaotic nature. Unraveling the dynamical synergy responsible for the genesis of extreme events in nonlinear dynamical systems is a prolific and spellbinding research area. The present study unveils the dynamical exposition of emerging extreme events in the MLC circuit concerning two different events being defined in the system. The large expansion of the chaotic attractor following the PM intermittency route plays the crucial role as the precursor behind the emergence of extreme events in the system. Our main finding reveals the prevalence of a force field due to the presence of externally applied periodic force in the system that creates the dynamical synergy that compels the chaotic trajectory traversing in its phase space to be largely deviated from the residing space, and this large deviation shows the signature of extreme events. Apart from the force field explication, we explored another two dynamical aspects that also interpret the mechanism behind the genesis of extreme events as the large deflection of the chaotic trajectory in the system: the decomposition of the phase space in stable and unstable manifolds concerning slow-fast dynamics and using Floquet multipliers. These two different aspects of calculations of the stable and unstable manifolds explicate the large excursion of the chaotic trajectory as extreme events from two different perspectives. We also analyzed the rare occurrences of the extreme events statistically using extreme value theory: the threshold \textit{excess values} follow the generalized Pareto distribution, and the inter-extreme-spike-intervals follow the generalized extreme value distribution.

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

2 major / 2 minor

Summary. The manuscript examines the emergence of extreme events in the Murali-Lakshmanan-Chua (MLC) circuit. It identifies the large expansion of the chaotic attractor following the PM intermittency route as the key precursor, driven by an externally applied periodic force that creates a force field causing large trajectory deviations. Complementary mechanisms are analyzed via decomposition of phase space into stable and unstable manifolds under slow-fast dynamics and via Floquet multipliers. Statistically, threshold excess values are shown to follow the generalized Pareto distribution while inter-extreme-spike intervals follow the generalized extreme value distribution.

Significance. If the central claims are rigorously supported, the work provides a multi-perspective dynamical account of extreme events in a canonical nonautonomous chaotic circuit, linking attractor expansion, external forcing, and geometric structures. The combination of intermittency analysis, manifold/Floquet calculations, and extreme-value statistics is a constructive approach that could inform similar studies in other dissipative systems.

major comments (2)
  1. [Definition of extreme events and force-field section] The definitions of the two extreme events and the assertion that the external periodic force is the dominant cause (abstract and the section introducing the force-field mechanism) require explicit robustness checks; small variations in amplitude thresholds or forcing parameters can alter event identification, and no such sensitivity analysis is presented to confirm the conclusions are not threshold-dependent.
  2. [PM intermittency and attractor expansion analysis] The claim that the PM intermittency route produces the large attractor expansion as precursor (abstract and the dynamical exposition section) is central yet lacks a direct comparison to the unforced MLC circuit or a parameter sweep showing that extreme events vanish without the force; this leaves open whether the synergy is uniquely due to the external drive.
minor comments (2)
  1. [Abstract] The abstract uses non-standard phrasing such as 'prolific and spellbinding research area'; replace with more conventional scientific language.
  2. [Throughout manuscript] Ensure the acronym 'PM intermittency' is defined on first use and that all figure captions explicitly state the parameter values and thresholds employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major concerns point by point below, agreeing that additional checks will strengthen the work, and we will incorporate the suggested analyses in the revised version.

read point-by-point responses

  1. Referee: [Definition of extreme events and force-field section] The definitions of the two extreme events and the assertion that the external periodic force is the dominant cause (abstract and the section introducing the force-field mechanism) require explicit robustness checks; small variations in amplitude thresholds or forcing parameters can alter event identification, and no such sensitivity analysis is presented to confirm the conclusions are not threshold-dependent.

    Authors: We agree that explicit robustness checks are needed to confirm that event identification and the force-field mechanism are not artifacts of specific threshold choices. In the revised manuscript we will add a dedicated sensitivity analysis subsection. This will include varying the amplitude thresholds for both types of extreme events by ±5% and ±10% around the nominal values, recomputing the threshold-excess statistics, and verifying that the generalized Pareto distribution fit remains valid with comparable shape parameters. We will likewise test small perturbations (±5%) in forcing amplitude and frequency, recomputing the force-field deviations and attractor expansions to show that the dominant role of the external periodic drive persists. revision: yes

  2. Referee: [PM intermittency and attractor expansion analysis] The claim that the PM intermittency route produces the large attractor expansion as precursor (abstract and the dynamical exposition section) is central yet lacks a direct comparison to the unforced MLC circuit or a parameter sweep showing that extreme events vanish without the force; this leaves open whether the synergy is uniquely due to the external drive.

    Authors: We accept that an explicit comparison to the unforced case would make the necessity of the external drive clearer. In the revision we will add a new subsection that simulates the MLC circuit with the periodic forcing term set identically to zero while retaining all other parameters. These runs will show that the PM intermittency route is not observed, the attractor remains confined without large expansions, and no extreme events occur. We will further include a one-parameter sweep over forcing amplitude, demonstrating that extreme events and the associated attractor expansion appear only above a critical forcing strength, thereby confirming that the external drive is essential for the reported synergy. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines extreme events via amplitude thresholds in the MLC circuit, then characterizes their occurrence through numerical simulation of attractor expansion along the PM intermittency route, external periodic force effects, stable/unstable manifold decomposition, and Floquet analysis. Statistical modeling via generalized Pareto and extreme value distributions is applied post-identification as descriptive fitting to the observed tails and intervals, not as a derivation that re-uses the defining thresholds or parameters to generate the events themselves. No load-bearing self-citation, self-definitional loop, or fitted-input-renamed-as-prediction appears in the derivation chain; the central claims rest on independent dynamical observations and standard EVT application.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes standard concepts from nonlinear dynamics (PM intermittency, stable/unstable manifolds, Floquet multipliers, generalized Pareto and GEV distributions) without introducing new free parameters or invented entities visible at this level.

axioms (1)
  • domain assumption The MLC circuit equations and the external periodic forcing produce a chaotic attractor whose expansion can be tracked via standard bifurcation routes.
    Invoked when stating that large expansion follows the PM intermittency route.

pith-pipeline@v0.9.0 · 5578 in / 1186 out tokens · 23333 ms · 2026-05-09T22:33:15.337894+00:00 · methodology

discussion (0)

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