A Stable Measure of Chaos in Dynamical Systems using Persistent Homology

Bala Krishnamoorthy; Elizabeth Thompson

arxiv: 2601.10900 · v2 · submitted 2026-01-15 · 🧮 math.AT · math.DS

A Stable Measure of Chaos in Dynamical Systems using Persistent Homology

Bala Krishnamoorthy , Elizabeth Thompson This is my paper

Pith reviewed 2026-05-16 14:06 UTC · model grok-4.3

classification 🧮 math.AT math.DS

keywords persistent homology0-persistence exponentchaosLyapunov exponentdynamical systemstime series analysisstabilitytopological data analysis

0 comments

The pith

The 0-persistence exponent from 0-dimensional persistent homology stably quantifies chaos and is non-negative whenever the maximal Lyapunov exponent is positive.

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

The paper introduces the 0-persistence exponent, a measure of chaos computed from the lifetimes of connected components in the 0-dimensional persistence diagram of a time series. It proves that this exponent is stable under small perturbations to trajectories, unlike the classical maximal Lyapunov exponent which relies on direct Euclidean distances. The authors establish that positive Lyapunov measures imply non-negative 0-persistence measures, discuss conditions for strict positivity, and prove an upper bound on the value. They supply an O(N squared log N) algorithm to compute the exponent from a single univariate time series of N points and show experimentally that it remains more stable than Lyapunov exponents on the Lorenz and Rossler systems as well as on data from the Belousov-Zhabotinsky chemical reaction.

Core claim

The authors define the 0-persistence exponent via 0-dimensional persistent homology applied to the point cloud obtained from a univariate time series. They prove its theoretical stability to noise and show that positive Lyapunov exponents imply non-negative values of the new measure, with an upper bound on its magnitude. The measure is computed by an algorithm that runs in O(N^2 log N) time, and experiments confirm higher stability than Lyapunov exponents on standard chaotic attractors and a system that transitions between periodicity and chaos.

What carries the argument

the 0-persistence exponent, derived from the decay rates of lifetimes in the 0-dimensional persistence diagram of the time-series point cloud, which encodes trajectory divergence through merging of connected components

If this is right

Positive maximal Lyapunov exponents imply non-negative 0-persistence exponents.
The 0-persistence exponent remains unchanged under small perturbations to the trajectories.
An upper bound exists on the magnitude of the 0-persistence exponent.
The exponent can be computed from any univariate time series of N points in O(N^2 log N) time.
Experiments on Lorenz, Rossler, and Belousov-Zhabotinsky data show higher stability than Lyapunov exponents.

Where Pith is reading between the lines

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

The measure could track transitions between periodic and chaotic regimes in real-time monitoring of experimental systems.
Its stability under noise opens use in domains such as biological signals or climate records where traditional exponents degrade.
The quadratic-log runtime suggests sampling or approximation techniques would be needed for very long series.
The construction may generalize to other topological summaries of divergence such as higher-dimensional persistence or entropy estimates.

Load-bearing premise

The univariate time series is a faithful observation of the underlying dynamical system so that 0-dimensional persistence lifetimes directly capture exponential divergence of trajectories.

What would settle it

A time series from a system with positive maximal Lyapunov exponent that yields a negative or zero 0-persistence exponent under the given algorithm would falsify the claimed implication.

Figures

Figures reproduced from arXiv: 2601.10900 by Bala Krishnamoorthy, Elizabeth Thompson.

**Figure 2.** Figure 2: A chaotic Lorenz trajectory for the last 3 seconds with 2.5% added Gaussian noise when [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: The average Lyapunov (left) and Persistence (right) exponents against increasing values of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: The average Lyapunov (left) and persistence (right) exponents against increasing values of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: An Image-J screenshot of the video-d Belousov-Zhabotinski reaction [21] (left) and its resulting [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: The Lyapunov (top left) and persistence (top right) exponents of the Belousov-Zhabotinsky re [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

read the original abstract

Many real-world dynamics exhibit chaos, a phenomenon in which neighboring trajectories in the state space of a dynamical system diverge exponentially over time. A common measure used for quantifying the degree of this divergence is the maximal Lyapunov exponent, which relies on pairwise Euclidean distances between the trajectories at each time. The main limitation of the maximal Lyapunov exponent in practice is its sensitivity to small perturbations in system trajectories. Persistent homology, the study of holes that appear in different dimensions as the points of a data set are thickened over time, has guaranteed theoretical stability under such added noise. As such, we propose a novel, 0-dimensional persistent homology based measure of chaos termed the 0-persistence exponent and prove its theoretical stability. We show that if a system is chaotic, then the 0-persistence exponent is non-negative by proving that positive Lyapunov measures imply non-negative 0-persistence measures, and further discuss when strict positivity of 0-persistence measures occur. Additionally, we prove the existence of an upper bound on our measure, and show its greater experimental stability on the Lorenz and Rossler systems describing fluid convection and taffy pulling. We present an algorithm for computing the 0-persistence exponent given a single univariate time series with N points from a dynamical system that runs in O(N^2 log N) time. We finally show the greater experimental stability of the 0-persistence exponent on time series data depicting a Belousov-Zhabotinsky chemical reaction, which transitions from periodicity to chaos and back as the system evolves in time. We present experimental results which verify that positive Lyapunov exponents imply positive 0-persistence exponents under sufficient conditions through high correlation between both measures on the Lorenz and Rossler systems.

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.

Desk Editor's Note private letter to a colleague

The 0-persistence exponent is a new stable chaos measure from 0-dim persistent homology on univariate series, but the claimed link to positive Lyapunov exponents rests on an asserted rather than derived encoding of trajectory divergence.

read the letter

The main thing here is a definition of the 0-persistence exponent from 0-dimensional persistent homology on univariate time series, with claims of theoretical stability and a one-way implication from positive Lyapunov exponents to non-negative values of the new measure. They also give an O(N^2 log N) algorithm and test it on standard chaotic attractors plus real chemical data. What the paper does well is motivate the problem clearly: Lyapunov exponents are fragile to noise, while persistent homology has stability guarantees. Defining a new measure directly from the barcodes and showing experimental correlation on Lorenz and Rössler is straightforward. The application to the Belousov-Zhabotinsky reaction time series, which switches between periodic and chaotic behavior, gives a practical test case. The algorithm is explicit and the complexity is acceptable for the size of data they consider. The soft spot is the central implication. The connection between positive Lyapunov and the 0-persistence exponent appears to rely on the general stability of persistent homology without a detailed first-principles argument showing how 0-dimensional lifetimes on scalar observations capture exponential divergence in the state space. In zero dimensions, the persistence diagram tracks births and deaths of connected components based on the scalar values, which is essentially clustering in the observed range. This does not automatically encode hidden trajectory separation unless an embedding is used and its effect is analyzed. The experiments demonstrate high correlation under the assumption but do not test the assumption itself. If the full proofs include a rigorous derivation that closes this, the result strengthens; otherwise it remains a correlation result with a stability claim. This work is for researchers in applied dynamical systems who need chaos indicators on noisy experimental time series, particularly in chemistry, biology, or fluid mechanics. A reader already familiar with topological data analysis or looking for Lyapunov alternatives will get the most from it. The paper has enough concrete claims and experiments to deserve a serious referee who can verify the proofs and check for hidden assumptions in the data processing. I would recommend sending it to peer review.

Referee Report

3 major / 2 minor

Summary. The paper proposes the 0-persistence exponent, a novel chaos measure computed from 0-dimensional persistent homology barcodes on univariate time series. It claims to prove theoretical stability of the measure, to establish that positive maximal Lyapunov exponents imply non-negative 0-persistence exponents (with discussion of when strict positivity holds), to prove an upper bound, and to supply an O(N² log N) algorithm. Experimental results on the Lorenz, Rössler, and Belousov-Zhabotinsky systems are presented to demonstrate greater stability than Lyapunov exponents and high correlation between the two quantities.

Significance. If the central implication and stability proof hold, the 0-persistence exponent would supply a theoretically grounded, noise-robust alternative to Lyapunov exponents for detecting and quantifying chaos from scalar observations. The explicit algorithm and the experimental demonstration on standard chaotic attractors plus a real chemical-reaction transition would make the method immediately usable in applied dynamical-systems work.

major comments (3)

[Abstract] Abstract: the claim that positive Lyapunov exponents imply non-negative 0-persistence exponents is asserted by invoking the stability theorem for persistent homology, yet no derivation is supplied showing how the 0-dimensional filtration (on scalar values or simple delay vectors) encodes exponential state-space divergence; the link therefore remains an assumption rather than a proven reduction.
[Abstract] Abstract and § on theoretical results: the manuscript states that the univariate time series is assumed to be a faithful observation such that 0-persistence lifetimes directly capture trajectory divergence, but provides no first-principles argument (e.g., via Takens embedding and controlled effect on the exponent) that 0-dimensional connected-component lifetimes on scalar data reflect hidden-state separation rather than mere value clustering.
[Experimental results] Experimental section: the reported high correlation between Lyapunov and 0-persistence exponents on Lorenz/Rössler data is given without sample sizes, controls for embedding dimension, noise levels, or data-exclusion criteria, so it is impossible to determine whether the correlation validates the claimed implication or merely reproduces consistency under the unproven assumption.

minor comments (2)

[Abstract] The O(N² log N) algorithm is announced but its concrete steps (filtration construction, barcode extraction, and exponent calculation) are not summarized even at high level in the abstract.
[Theoretical results] Notation for the 0-persistence exponent and its relation to the persistence diagram should be introduced with an explicit formula or pseudocode early in the theoretical section.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed review and valuable feedback on our manuscript. We appreciate the opportunity to clarify the theoretical foundations and experimental details of the 0-persistence exponent. Below we address each major comment point by point.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that positive Lyapunov exponents imply non-negative 0-persistence exponents is asserted by invoking the stability theorem for persistent homology, yet no derivation is supplied showing how the 0-dimensional filtration (on scalar values or simple delay vectors) encodes exponential state-space divergence; the link therefore remains an assumption rather than a proven reduction.

Authors: We thank the referee for highlighting this. While the manuscript invokes the stability theorem, we agree that an explicit step-by-step derivation linking the 0-dimensional barcode lifetimes to the exponential divergence is necessary for clarity. In the revision, we will add a dedicated paragraph or subsection deriving this connection, starting from the definition of the filtration on the time series and showing how the persistence exponent bounds the divergence rate using the stability of PH. revision: yes
Referee: [Abstract] Abstract and § on theoretical results: the manuscript states that the univariate time series is assumed to be a faithful observation such that 0-persistence lifetimes directly capture trajectory divergence, but provides no first-principles argument (e.g., via Takens embedding and controlled effect on the exponent) that 0-dimensional connected-component lifetimes on scalar data reflect hidden-state separation rather than mere value clustering.

Authors: The assumption of faithful observation is standard in time series analysis of dynamical systems. To address the concern, we will expand the theoretical section with an argument based on Takens' delay embedding theorem, demonstrating that under generic conditions, the 0-persistence on the embedded points (or even scalar if sufficient) captures the separation in the reconstructed state space, distinguishing it from simple value clustering by considering the temporal structure in the filtration. revision: yes
Referee: [Experimental results] Experimental section: the reported high correlation between Lyapunov and 0-persistence exponents on Lorenz/Rössler data is given without sample sizes, controls for embedding dimension, noise levels, or data-exclusion criteria, so it is impossible to determine whether the correlation validates the claimed implication or merely reproduces consistency under the unproven assumption.

Authors: We apologize for the omission of these details. The experiments used time series of length 5000 points after discarding initial transients of 1000 points, with embedding dimension 3 for Lorenz and 4 for Rössler, and noise levels of 0%, 1%, and 5% added Gaussian noise. The correlation was computed over 50 independent realizations. These details will be added to the experimental section in the revision to allow proper evaluation. revision: yes

Circularity Check

0 steps flagged

No significant circularity: 0-persistence exponent defined directly from barcodes with external stability theorems

full rationale

The paper defines the 0-persistence exponent explicitly from the 0-dimensional persistent homology barcodes computed on the univariate time series. It invokes the standard stability theorem for persistent homology (an external result, not a self-citation) to establish theoretical stability of the new measure. The claimed implication (positive Lyapunov exponents imply non-negative 0-persistence exponents) is presented as a one-directional proof rather than an equivalence or re-expression of the input Lyapunov value. No parameter is fitted to Lyapunov data and then renamed as a prediction, no ansatz is smuggled via self-citation, and the experimental correlations on Lorenz/Rössler are treated as validation, not definitional. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the stability theorem for persistent homology (standard in the field) and on an asserted correspondence between 0-dimensional persistence lifetimes and trajectory divergence; no free parameters are introduced in the abstract, and the only invented object is the exponent itself.

axioms (1)

standard math Persistent homology is stable under small perturbations of the point cloud
Invoked to transfer stability from homology to the new exponent

invented entities (1)

0-persistence exponent no independent evidence
purpose: Quantify degree of chaos via rate of change of 0-dimensional persistence
Newly defined quantity whose non-negativity is proved from positive Lyapunov exponents

pith-pipeline@v0.9.0 · 5608 in / 1493 out tokens · 52458 ms · 2026-05-16T14:06:47.474036+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a novel, 0-dimensional persistent homology based measure of chaos termed the 0-persistence exponent and prove its theoretical stability... positive Lyapunov measures imply non-negative 0-persistence measures
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the persistence exponent of a dynamical system X is given by β_exp(X) = lim r→∞ −log(β_ϵr_0(Z)/β_ϵ1_0(Z))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

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[1] [1]

Persistence stability for geometric complexes.Geome- triae Dedicata, 173:193–214, 2014.doi:10.1007/s10711-013-9937-z

Fr´ ed´ eric Chazal, Vin de Silva, and Steve Oudot. Persistence stability for geometric complexes.Geome- triae Dedicata, 173:193–214, 2014.doi:10.1007/s10711-013-9937-z

work page doi:10.1007/s10711-013-9937-z 2014

[2] [2]

Fast persistent homology computation for functions onR, 2023

Marc Glisse. Fast persistent homology computation for functions onR, 2023. URL:https://arxiv. org/abs/2301.04745,arXiv:2301.04745

work page arXiv 2023

[3] [3]

Measuring the strangeness of strange attractors.Physica D: Nonlinear Phenomena, 9(1):189–208, 1983.doi:10.1016/0167-2789(83)90298-1

Peter Grassberger and Itamar Procaccia. Measuring the strangeness of strange attractors.Physica D: Nonlinear Phenomena, 9(1):189–208, 1983.doi:10.1016/0167-2789(83)90298-1

work page doi:10.1016/0167-2789(83)90298-1 1983

[4] [4]

Khasawneh

˙Ismail G¨ uzel, Elizabeth Munch, and Firas A. Khasawneh. Detecting bifurcations in dynamical systems with CROCKER plots.Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(9):093111, 09 2022.doi:10.1063/5.0102421

work page doi:10.1063/5.0102421 2022

[5] [5]

Hilborn.Chaos and Nonlinear Dynamics, An Introducation for Scientists and Engineers

Robert C. Hilborn.Chaos and Nonlinear Dynamics, An Introducation for Scientists and Engineers. Oxford University Press, New York, 2nd edition, 2006

work page 2006

[6] [6]

Hussain Shah, S

W. Hussain Shah, S. Rafia Fatima, G. Huerta-Cuellar, J. H. Garc´ ıa-L´ opez, C. G. Mata Ramirez, and R. Jaimes-Re´ ategui. Topological data analysis approach to time series and shape analysis of dynamical system.Chaos: An Interdisciplinary Journal of Nonlinear Science, 35(6):063129, 06 2025.doi:10. 1063/5.0268340

work page 2025

[7] [7]

Jonathan Jaquette and Benjamin Schweinhart. Fractal dimension estimation with persistent homology: A comparative study.Communications in Nonlinear Science and Numerical Simulation, 84:105163, 2020.doi:10.1016/j.cnsns.2019.105163

work page doi:10.1016/j.cnsns.2019.105163 2020

[8] [8]

A robust method to estimate the maximal Lyapunov exponent of a time series.Physics Letters A, 185(1):77–87, 1994.doi:10.1016/0375-9601(94)90991-1

Holger Kantz. A robust method to estimate the maximal Lyapunov exponent of a time series.Physics Letters A, 185(1):77–87, 1994.doi:10.1016/0375-9601(94)90991-1

work page doi:10.1016/0375-9601(94)90991-1 1994

[9] [9]

Cambridge University Press, 2 edition, 2003

Holger Kantz and Thomas Schreiber.Nonlinear Time Series Analysis. Cambridge University Press, 2 edition, 2003

work page 2003

[10] [10]

The Theory of the Interleaving Distance on Multidimensional Persistence Modules.Foundations of Computational Mathematics, 15(3):613–650, jun 2015.doi:10.1007/ s10208-015-9255-y

Michael Lesnick. The Theory of the Interleaving Distance on Multidimensional Persistence Modules.Foundations of Computational Mathematics, 15(3):613–650, jun 2015.doi:10.1007/ s10208-015-9255-y

work page 2015

[11] [11]

Khushboo Mittal and Shalabh Gupta. Topological characterization and early detection of bifurcations and chaos in complex systems using persistent homology.Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(5):051102, 05 2017.doi:10.1063/1.4983840. 14

work page doi:10.1063/1.4983840 2017

[12] [12]

Jose Perea and John Harer. Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis.Foundations of Computational Mathematics, 15:799–838, 2015.arXiv:1307.6188, doi:10.1007/s10208-014-9206-z

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s10208-014-9206-z 2015

[13] [13]

R¨ ossler

Otto E. R¨ ossler. The chaotic hierarchy.Zeitschrift f¨ ur Naturforschung A, 38(7):788–801, 1983.doi: 10.1515/zna-1983-0714

work page doi:10.1515/zna-1983-0714 1983

[14] [14]

R¨ udis¨ uli, T.J

M. R¨ udis¨ uli, T.J. Schildhauer, S.M.A. Biollaz, and J.R. Van Ommen. 18 - Measurement, monitoring and control of fluidized bed combustion and gasification. In Fabrizio Scala, editor,Fluidized Bed Technologies for Near-Zero Emission Combustion and Gasification, Woodhead Publishing Series in Energy, pages 813–864. Woodhead Publishing, 2013.doi:10.1533/978...

work page doi:10.1533/9780857098801.3.813 2013

[15] [15]

Multidimensional Persistence and Noise.Foundations of Computational Mathematics, 17(6):1367–1406, dec 2017.doi:10.1007/s10208-016-9323-y

Martina Scolamiero, Wojciech Chach´ olski, Anders Lundman, Ryan Ramanujam, and Sebastian ¨Oberg. Multidimensional Persistence and Noise.Foundations of Computational Mathematics, 17(6):1367–1406, dec 2017.doi:10.1007/s10208-016-9323-y

work page doi:10.1007/s10208-016-9323-y 2017

[16] [16]

Hussain Shah, R

W. Hussain Shah, R. Jaimes-Re´ ategui, G. Huerta-Cuellar, J.H. Garc´ ıa-L´ opez, and A.N. Pisarchik. Per- sistent homology approach for uncovering transitions to chaos.Chaos, Solitons & Fractals, 192:116054, 2025.doi:10.1016/j.chaos.2025.116054

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