A Stability Testing Algorithm for Incommensurate Fractional Differential Equation Systems

Kai Diethelm; Safoura Hashemishahraki

arxiv: 2603.02269 · v2 · pith:RTEGPXIXnew · submitted 2026-02-28 · 🧮 math.DS · cs.NA· math.NA

A Stability Testing Algorithm for Incommensurate Fractional Differential Equation Systems

Kai Diethelm , Safoura Hashemishahraki This is my paper

Pith reviewed 2026-05-22 10:40 UTC · model grok-4.3

classification 🧮 math.DS cs.NAmath.NA

keywords fractional differential equationsstability analysisincommensurate ordersnumerical algorithmlinear systemsasymptotic stabilityrational order ratios

0 comments

The pith

A simpler algorithm determines asymptotic stability for incommensurate fractional differential systems.

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

This paper presents an algorithm to decide whether a system of fractional differential equations with possibly different orders for each equation is asymptotically stable. Such systems arise in modeling complex phenomena where memory effects or anomalous processes play a role, and determining stability without full simulation is valuable for analysis and design. The method exploits numerical linear algebra to handle the linear case when the ratios of the fractional orders are rational numbers, converting the stability question into checking properties of a derived matrix. It indicates extensions to nonlinear cases using other established approaches. A MATLAB implementation is included to make the method immediately usable.

Core claim

For linear incommensurate fractional-order systems where the ratios of the orders are rational, the stability can be tested by an algorithm that reduces the problem to a standard linear algebra task, which is simpler than existing methods. The approach also suggests how to apply similar ideas to general nonlinear problems with arbitrary orders.

What carries the argument

Reduction via rational order ratios to a matrix eigenvalue problem from numerical linear algebra.

If this is right

For linear systems with rational order ratios, stability testing becomes a direct linear algebra computation.
The algorithm applies to both commensurate and incommensurate cases under the rationality condition.
Known techniques allow extension to nonlinear fractional systems with arbitrary orders.
Practical implementation in MATLAB enables direct computation of stability.

Where Pith is reading between the lines

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

Engineers could use this to quickly validate stability in fractional-order control systems.
Further research might explore numerical stability and efficiency of the reduction for high-dimensional systems.
Testing on real-world models like fractional-order neural networks could demonstrate broader applicability.

Load-bearing premise

The ratios of the fractional orders are rational numbers, allowing the linear case to reduce to a standard linear algebra problem.

What would settle it

A specific incommensurate linear fractional system with known rational order ratios whose stability is misclassified by the algorithm would disprove the method's correctness.

read the original abstract

We consider the question of determining whether or not a given system of fractional-order differential equations is (asymptotically) stable. In particular, we admit systems where each constituent equation may have its own order, independent of the order of the other equations in the system, i.e. we discuss the so-called incommensurate case. Exploiting ideas based in numerical linear algebra, we present an algorithm that can be used to answer this question that is much simpler than known methods. We discuss in detail the case of linear problems where the ratios of orders are rational and indicate how known techniques can be used to apply our findings also to general nonlinear problems with arbitrary orders. A MATLAB implementation of the code is provided.

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 paper gives a practical reduction of linear stability checks to standard eigenvalue problems when fractional order ratios are rational, plus MATLAB code, but the broader claim of a much simpler method for fully general incommensurate systems rests on existing approximation techniques rather than a new construction.

read the letter

The main point is that for linear incommensurate fractional systems where the orders have rational ratios, the authors turn the stability test into a standard linear algebra task such as eigenvalue computation or a matrix pencil. They supply MATLAB code to do it. This looks like a straightforward and usable contribution for that restricted setting, and the code makes it immediately testable by others working on models in control or biology that can live with rational ratios. That part of the work is concrete and could save people from more cumbersome custom fractional methods. The reduction itself draws on familiar numerical linear algebra ideas applied to the commensurate reformulation, which is a reasonable move and explains why it might feel simpler than some prior approaches for the rational case. The paper also sketches how to handle nonlinear problems and arbitrary orders by falling back to known approximation or continuation methods. On the soft side, the headline promise of a much simpler algorithm for incommensurate systems in general is not fully delivered for the irrational-ratio case that defines the hardest part of incommensurate behavior. There the manuscript indicates using existing techniques without showing a new, explicitly simpler procedure or any complexity comparison. That leaves the central simplicity claim resting more on the rational subcase than on a demonstrated advance for arbitrary orders. No obvious circularity or invented entities appear, and the citation pattern seems standard. The work is aimed at applied researchers who need a quick numerical stability check for fractional models rather than pure theorists. A reader who already works with fractional DEs and values reproducible code would get direct value from the rational-ratio algorithm and the implementation. It shows clear engagement with the practical side of the problem and deserves a serious referee because it supplies a usable tool even if the general-case discussion could be tightened. I would send it out for review and ask the authors to add a few concrete examples plus a short note on how much simpler the method remains once approximations enter for irrational ratios.

Referee Report

1 major / 0 minor

Summary. The paper presents an algorithm for testing asymptotic stability of incommensurate fractional differential equation systems, exploiting numerical linear algebra ideas. It provides a detailed treatment for linear problems when fractional order ratios are rational (reducing to a standard linear algebra problem such as eigenvalue computation), and indicates that known techniques can extend the findings to general nonlinear problems with arbitrary orders. A MATLAB implementation is supplied.

Significance. If the algorithm delivers a demonstrably simpler and correct stability test for general incommensurate FDEs, it would offer a practical tool for dynamical systems analysis in fields relying on fractional-order models. The explicit provision of reproducible MATLAB code is a strength that supports verification and adoption.

major comments (1)

[Abstract] Abstract: the central claim that the algorithm 'is much simpler than known methods' for incommensurate systems rests on the detailed reduction only for rational order ratios (which converts the system to commensurate form via a common multiple). For irrational ratios—the defining incommensurate case—the manuscript refers to 'known techniques' without supplying an explicit, simpler construction, complexity comparison, or error analysis, leaving the headline claim unsupported by the presented material.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful review and for identifying an area where the abstract's phrasing could be tightened to better align with the manuscript's detailed contributions. We address the major comment below and will revise accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the algorithm 'is much simpler than known methods' for incommensurate systems rests on the detailed reduction only for rational order ratios (which converts the system to commensurate form via a common multiple). For irrational ratios—the defining incommensurate case—the manuscript refers to 'known techniques' without supplying an explicit, simpler construction, complexity comparison, or error analysis, leaving the headline claim unsupported by the presented material.

Authors: We appreciate this observation and agree that the abstract should more carefully distinguish the scope of the new algorithm. The manuscript's core contribution is a stability-testing procedure for linear incommensurate FDE systems that exploits numerical linear algebra (primarily eigenvalue computations after a common-multiple reduction). This procedure is fully detailed and, we maintain, demonstrably simpler than prior approaches precisely when the order ratios are rational, because the reduction yields a standard linear system whose stability is settled by a single matrix eigenvalue test rather than more involved frequency-domain or Lyapunov constructions common in the literature. For the broader class that includes irrational ratios, the paper indicates how established extension techniques (e.g., approximation or embedding methods already present in the fractional-order literature) can be combined with the same linear-algebra core; we do not supply a new explicit construction or complexity analysis for that fully general setting. We will revise the abstract to state explicitly that the simpler algorithm is developed in detail for rational-ratio linear systems and that known techniques are invoked for the remaining cases. We will also add a short paragraph in the introduction clarifying this boundary so that the headline claim is supported exactly by the material that is worked out in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithm reduces commensurate case to standard linear algebra without self-referential derivation

full rationale

The paper's core contribution is an algorithm that, for linear systems with rational order ratios, reformulates the stability test as a standard eigenvalue or matrix pencil problem from numerical linear algebra. This is a direct application of existing techniques to the fractional-order stability criterion rather than a self-definitional loop or fitted prediction. For irrational ratios and nonlinear cases, the manuscript explicitly defers to known approximation or continuation methods without claiming a new closed-form derivation that would require self-citation or ansatz smuggling. No load-bearing self-citations, uniqueness theorems from prior author work, or renaming of empirical patterns appear in the derivation chain. The approach is therefore self-contained against external benchmarks in linear algebra.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard theory of fractional derivatives and stability for FDEs, plus the assumption that numerical linear algebra techniques apply directly to the characteristic matrix when order ratios are rational.

axioms (2)

domain assumption Standard definitions and properties of fractional derivatives and asymptotic stability for systems of differential equations hold.
Invoked implicitly when discussing stability of the incommensurate system.
domain assumption When order ratios are rational, the system can be analyzed via a commensurate embedding or equivalent matrix formulation.
Central to the detailed linear case treatment described in the abstract.

pith-pipeline@v0.9.0 · 5650 in / 1180 out tokens · 72977 ms · 2026-05-22T10:40:54.230041+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/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

We discuss in detail the case of linear problems where the ratios of orders are rational and indicate how known techniques can be used to apply our findings also to general nonlinear problems with arbitrary orders.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

p(μ) = sum A_j μ^{q_j} ... polynomial eigenvalue problem ... companion form ... generalized linear eigenvalue problem μXv = Yv

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

24 extracted references · 24 canonical work pages

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(eds.): Handbook of fractional calculus with applications, Vol

B˘ aleanu, D., Mendes Lopes, A. (eds.): Handbook of fractional calculus with applications, Vol. 7: applications in engineering, life and social sciences, Part A. De Gruyter, Berlin (2019). DOI 10.1515/9783110571905

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(eds.): Handbook of fractional calculus with applications, Vol

B˘ aleanu, D., Mendes Lopes, A. (eds.): Handbook of fractional calculus with applications, Vol. 8: applications in engineering, life and social sciences, Part B. De Gruyter, Berlin (2019). DOI 10.1515/9783110571929

work page doi:10.1515/9783110571929 2019
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Zenodo Record #18730961(2026)

Diethelm, K., Hashemishahraki, S.: A MATLAB and GNU/Octave code for checking the asymptotic stability of incommensurate fractional differential equation systems. Zenodo Record #18730961(2026). DOI 10.5281/zenodo.18730961. Version 1.0

work page doi:10.5281/zenodo.18730961 2026
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Diethelm, K., Hashemishahraki, S., Thai, H.D., Tuan, H.T.: A constructive approach for investigating the stability of incommensurate fractional differential systems. J. Math. Anal. Appl.540, 128642 (2024). DOI 10.1016/j.jmaa.2024.128642

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G¨ uttel, S., Tisseur, F.: The nonlinear eigenvalue problem. Acta Numerica26, 1–94 (2017). DOI 10.1017/S0962492917000034

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Mackey, D.S., Mackey, N., Mehl, C., Mehrmann, V.: Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl.28, 971–1004 (2006). DOI 10.1137/050628350

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A Krylov–Schur algorithm for large eigenproblems

Stewart, G.W.: Addendum to “A Krylov–Schur algorithm for large eigenproblems”. SIAM J. Matrix Anal. Appl.24, 599–601 (2002). DOI 10.1137/S0895479802403150

work page doi:10.1137/s0895479802403150 2002
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(ed.): Handbook of fractional calculus with applications, Vol

Tarasov, V.E.. (ed.): Handbook of fractional calculus with applications, Vol. 4: applications in physics, Part A. De Gruyter, Berlin (2019). DOI 10.1515/9783110571707

work page doi:10.1515/9783110571707 2019
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(ed.): Handbook of fractional calculus with applications, Vol

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work page doi:10.1201/b16113 2014

[1] [1]

(eds.): Handbook of fractional calculus with applications, Vol

B˘ aleanu, D., Mendes Lopes, A. (eds.): Handbook of fractional calculus with applications, Vol. 7: applications in engineering, life and social sciences, Part A. De Gruyter, Berlin (2019). DOI 10.1515/9783110571905

work page doi:10.1515/9783110571905 2019

[2] [2]

(eds.): Handbook of fractional calculus with applications, Vol

B˘ aleanu, D., Mendes Lopes, A. (eds.): Handbook of fractional calculus with applications, Vol. 8: applications in engineering, life and social sciences, Part B. De Gruyter, Berlin (2019). DOI 10.1515/9783110571929

work page doi:10.1515/9783110571929 2019

[3] [3]

Brandibur, O., Kaslik, E.: Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model. Math. Methods Appl. Sci.41, 7182–7194 (2018). DOI 10.1002/mma.4768 Stability of incommensurate fractional differential systems 15

work page doi:10.1002/mma.4768 2018

[4] [4]

Nonlinear Dyn.48, 409–416 (2007)

Deng, W., Li, C., L¨ u, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn.48, 409–416 (2007). DOI 10.1007/s11071-006-9094- 0

work page doi:10.1007/s11071-006-9094- 2007

[5] [5]

Springer, Berlin (2010)

Diethelm, K.: The analysis of fractional differential equations. Springer, Berlin (2010). DOI 10.1007/978-3-642-14574-2

work page doi:10.1007/978-3-642-14574-2 2010

[6] [6]

Nonlinear Dyn.71, 613–619 (2013)

Diethelm, K.: A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn.71, 613–619 (2013). DOI 10.1007/s11071-012-0475-2

work page doi:10.1007/s11071-012-0475-2 2013

[7] [7]

ZenodoRecord #7678311 (2023)

Diethelm, K., Garrappa, R., Uhlig, F.: A MATLAB implementation of the proportional secting algorithm for fractional terminal value problems. ZenodoRecord #7678311 (2023). DOI 10.5281/zenodo.7678311. Version 1.0

work page doi:10.5281/zenodo.7678311 2023

[8] [8]

Zenodo Record #18730961(2026)

Diethelm, K., Hashemishahraki, S.: A MATLAB and GNU/Octave code for checking the asymptotic stability of incommensurate fractional differential equation systems. Zenodo Record #18730961(2026). DOI 10.5281/zenodo.18730961. Version 1.0

work page doi:10.5281/zenodo.18730961 2026

[9] [9]

Diethelm, K., Hashemishahraki, S., Thai, H.D., Tuan, H.T.: A constructive approach for investigating the stability of incommensurate fractional differential systems. J. Math. Anal. Appl.540, 128642 (2024). DOI 10.1016/j.jmaa.2024.128642

work page doi:10.1016/j.jmaa.2024.128642 2024

[10] [10]

Mathematics6, 16 (2018)

Garrappa, R.:, Numerical solution of fractional differential equations: a survey and a soft- ware tutorial. Mathematics6, 16 (2018). DOI 10.3390/math6020016

work page doi:10.3390/math6020016 2018

[11] [11]

Johns Hopkins Univ

Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins Univ. Press, Baltimore (1996)

work page 1996

[12] [12]

Acta Numerica26, 1–94 (2017)

G¨ uttel, S., Tisseur, F.: The nonlinear eigenvalue problem. Acta Numerica26, 1–94 (2017). DOI 10.1017/S0962492917000034

work page doi:10.1017/s0962492917000034 2017

[13] [13]

G¨ uttel, S., Van Beeumen, R., Meerbergen, K., Michiels, W.: NLEIGS: A class of fully rational Krylov methods for nonlinear eigenvalue problems. SIAM J. Sci. Comput.36, A2842–A2864 (2014). DOI 10.1137/130935045

work page doi:10.1137/130935045 2014

[14] [14]

Mackey, D.S., Mackey, N., Mehl, C., Mehrmann, V.: Vector spaces of linearizations for matrix polynomials. SIAM J. Matrix Anal. Appl.28, 971–1004 (2006). DOI 10.1137/050628350

work page doi:10.1137/050628350 2006

[15] [15]

ESAIM, Proc.5, 145–158 (1998)

Matignon, D.: Stability properties for generalized fractional differential systems. ESAIM, Proc.5, 145–158 (1998). DOI 10.1051/proc:1998004

work page doi:10.1051/proc:1998004 1998

[16] [16]

Stewart, G.W.: A Krylov–Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl.23, 601–614 (2002). DOI 10.1137/S0895479800371529

work page doi:10.1137/s0895479800371529 2002

[17] [17]

A Krylov–Schur algorithm for large eigenproblems

Stewart, G.W.: Addendum to “A Krylov–Schur algorithm for large eigenproblems”. SIAM J. Matrix Anal. Appl.24, 599–601 (2002). DOI 10.1137/S0895479802403150

work page doi:10.1137/s0895479802403150 2002

[18] [18]

(ed.): Handbook of fractional calculus with applications, Vol

Tarasov, V.E.. (ed.): Handbook of fractional calculus with applications, Vol. 4: applications in physics, Part A. De Gruyter, Berlin (2019). DOI 10.1515/9783110571707

work page doi:10.1515/9783110571707 2019

[19] [19]

(ed.): Handbook of fractional calculus with applications, Vol

Tarasov, V.E.. (ed.): Handbook of fractional calculus with applications, Vol. 5: applications in physics, Part B. De Gruyter, Berlin (2019). DOI 10.1515/9783110571721

work page doi:10.1515/9783110571721 2019

[20] [20]

com/help/matlab/ref/eig.html

The MathWorks, Inc.: MATLAB R2025b Documentation: eig.https://www.mathworks. com/help/matlab/ref/eig.html. Accessed: 2026-02-21

work page 2026

[21] [21]

com/help/matlab/ref/eigs.html

The MathWorks, Inc.: MATLAB R2025b Documentation: eigs.https://www.mathworks. com/help/matlab/ref/eigs.html. Accessed: 2026-02-21

work page 2026

[22] [22]

Discrete Contin

Tuan, H.T., Siegmund, S., Son, D.T., Cong, N.: An instability theorem for nonlinear fractional differential systems. Discrete Contin. Dyn. Syst., Ser. B22, 3079–3090 (2017). DOI 10.3934/dcdsb.2017164

work page doi:10.3934/dcdsb.2017164 2017

[23] [23]

Van Beeumen, R., Jarlebring, E., Michiels, W.: A rank-exploiting infinite Arnoldi algo- rithm for nonlinear eigenvalue problems. Numer. Linear Algebra Appl.23, 607–628 (2016). DOI 10.1002/nla.2043

work page doi:10.1002/nla.2043 2016

[24] [24]

Voss, H.: Nonlinear eigenvalue problems. In: L. Hogben (ed.) Handbook of Linear Algebra, 2nd enlarged edn., pp. 60–1–60–24. Chapman & Hall/CRC, Boca Raton (2014). DOI 10.1201/b16113

work page doi:10.1201/b16113 2014