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arxiv: 2605.04822 · v1 · submitted 2026-05-06 · 🧮 math.DS

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Stability and Bifurcation Analysis of Fractional Delay Differential Equation with a Delay-dependent Coefficient

Pragati Dutta, Sachin Bhalekar

Pith reviewed 2026-05-08 16:30 UTC · model grok-4.3

classification 🧮 math.DS
keywords fractional delay differential equationsstability analysisbifurcationdelay-dependent coefficientstability regionsHopf bifurcation
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The pith

A general stability result for fractional delay equations with two delays holds for every fractional order and any positive first delay.

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

This paper investigates the stability of the trivial equilibrium in a fractional delay differential equation that includes a delay-dependent coefficient multiplying one of the delayed terms. It performs a detailed analysis of stability regions and bifurcations when the first delay vanishes, then establishes a broader result that applies when both delays are present. The result is shown to be independent of the fractional derivative order alpha. Readers interested in systems with memory and time lags would care because the finding simplifies prediction of long-term behavior without needing to adjust for different fractional orders.

Core claim

For the linearized system corresponding to the given fractional delay equation with τ1 > 0 and τ2 ≥ 0, the stability criteria in the (k, γ) plane hold for all fractional orders 0 < α ≤ 1 and all positive values of τ1. This general result is illustrated through stability diagrams in the (τ1, τ2) plane obtained via numerical methods for fixed parameter values.

What carries the argument

The characteristic equation arising from linearization of the fractional delay equation around the zero solution, using the slope k = g'(0) at the origin to determine stability switches and Hopf bifurcation points.

If this is right

  • Stability boundaries in the (k, γ) plane can be determined explicitly for the reduced case τ1 = 0.
  • Stability regions in the two-delay plane can be mapped numerically for any chosen fractional order, k, and γ.
  • The general result implies that changing the fractional order does not move the stability boundaries when both delays are positive.
  • Bifurcation curves separate stable and unstable regions and can be tracked as delays vary.

Where Pith is reading between the lines

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

  • The independence from α suggests the same stability diagrams may apply when approximating fractional systems by high-order integer-order equations.
  • The form of the delay-dependent coefficient may arise in other models with signal attenuation over a second delay interval.
  • Extending the linear analysis to specific nonlinear g functions would test whether the stability regions persist beyond the local approximation.

Load-bearing premise

The analysis assumes linearization around the trivial equilibrium via the derivative k = g'(0) and that fractional-order stability criteria apply directly to the resulting characteristic equation.

What would settle it

A numerical integration or eigenvalue computation for specific α, k, γ, and positive τ1, τ2 that shows the stability boundary in the (τ1, τ2) plane crossing a point predicted to be stable by the general result would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.04822 by Pragati Dutta, Sachin Bhalekar.

Figure 1
Figure 1. Figure 1: Stability regions of Equation (2). 3 view at source ↗
Figure 2
Figure 2. Figure 2: Delay Independent regions for τ1 = 0 and τ2 > 0 (see Fig. (2)). We now apply Theorem 2.1 to analyze the stability of the equilibrium point x∗ = 0 in the remaining regions, namely, the region 0 < γ < 2k and the fourth quadrant in the k − γ plane. Consider the first quadrant of k − γ plane. The remaining region 0 < γ < 2k can be divided into two subregions: 0 < γ < k and k < γ < 2k. Now, we analyse these sub… view at source ↗
Figure 3
Figure 3. Figure 3: Analysis for the region 0 < γ < 2k Independently, stability inside the SSR region is limited by the Hopf critical value τ ′ ∗ provided by Theorem 2.1. Note that, the parameter b depends on τ. Hence, the expression (1) for the critical delay also depends on τ viz. τ ′ ∗ (τ ). Because the expression for τ ′ ∗ (τ ) involves a square root, two distinct values may arise, corresponding to two branches of Hopf bi… view at source ↗
Figure 4
Figure 4. Figure 4: Single intersection leading to SSR behavior (Not upto scale). view at source ↗
Figure 5
Figure 5. Figure 5: No intersection between the curve τ∗ = τ ′ ∗ (τ ) and the line τ∗ = τ leading to SSR behavior (Not upto scale). Case 3: In this case, the first intersection τ∗a ′ between τ and τ∗ ′ (τ ) occurs before τ ′′ ∗ , and the second intersection τ ′ ∗b occurs after τ ′′ ∗ (see view at source ↗
Figure 6
Figure 6. Figure 6: Two intersections with only the first one affecting stability (Not upto scale). view at source ↗
Figure 7
Figure 7. Figure 7: Both intersections before τ ′′ ∗ leading to SUSU behavior (Not upto scale). Case 5: Both intersection points (say τ∗a ′ and τ∗b ′ ) occur after τ ′′ ∗ (see view at source ↗
Figure 8
Figure 8. Figure 8: Both intersections after τ ′′ ∗ —stability unaffected (SSR region) (Not upto scale). Based on these different scenarios, we have two bifurcation curves in the first quadrant of k − γ plane, say γ = h1(k) and γ = h2(k). • The curve γ = h1(k) bifurcates the two regions: – when there are two intersection between the curves τ∗ = τ ′ ∗ (τ ) and τ∗ = τ and – There is no intersection between these curves. So, it … view at source ↗
Figure 9
Figure 9. Figure 9: Geometric interpretation of curve γ = h1(k) • The curve γ = h2(k) bifurcates the two regions: – τ∗ ′ (τ∗ ′′) > τ∗ ′′ and – τ∗ ′ (τ∗ ′′) < τ∗ ′′ (see Fig. (9)). Hence, it will be plotted by using the condition τ∗ ′ (τ∗ ′′) = τ∗ ′′ Again, for a fixed value of α, we use “Table” and “FindRoot” command in Mathematica to plot this curve (ref. Fig. (11)) in k − γ plane view at source ↗
Figure 10
Figure 10. Figure 10: Geometric interpretation of curve γ = h2(k) 4.3 Case II: k < γ < 2k Here, a = k − γ < 0. At τ = 0, b(0) = −k < a. So, once again the initial point (a, b(0)) lies in the SSR region of Theorem 2.1. Thus, the system starts with local stability for sufficiently small delay. Here also, as τ increases, b(τ ) goes toward 0. Again, the arrow T2 = {(a, b(τ ))|τ ≥ 0} will intersect the line b = a at some finite del… view at source ↗
Figure 11
Figure 11. Figure 11: Stability regions of 0 < γ < 2k for α = 0.4 We have the following observations from this figure: • In region (I), we have the scenarios as shown in Fig.(12): (a) α = 0.4, k = 0.65, γ = 0.04 (b) α = 0.4, k = 0.75, γ = 0.11 view at source ↗
Figure 12
Figure 12. Figure 12: SSR behavior in region I for some set of parameter values view at source ↗
Figure 13
Figure 13. Figure 13: SSR behavior in region II for some set of parameter values view at source ↗
Figure 14
Figure 14. Figure 14: Stable behavior in region III with α = 0.4, k = 1, γ = 1.35 In this region, there is no intersection between the curve τ∗ = τ∗ ′ (τ ) and τ = τ∗ ′′, i.e, there is no critical value in the SSR region. Since γ > k, i.e, a < 0, the system will remain stable only. • In region (IV), we have the following cases (see Fig.(15)): (a) α = 0.4, k = 2, γ = 0.6 (b) α = 0.4, k = 3, γ = 1.86 view at source ↗
Figure 15
Figure 15. Figure 15: SSR behavior in region IV wih given values of parameters view at source ↗
Figure 16
Figure 16. Figure 16: SUSU behavior in region V with α = 0.4, k = 4.62, γ = 3.69 It can be observed that there are two intersections (say τ∗a ′ and τ∗b ′ ) before τ∗ ′′ . So, there will be SUSU (Stable-Unstable-Stable-Unstable) behavior in this region, i.e – stable for 0 < τ < τ∗a ′ , – unstable for τ∗a ′ < τ < τ∗b ′ , – stable for τ∗b ′ < τ < τ∗ ′′ and – unstable for τ > τ∗ ′′ . • In region (VI), we get a figure similar to Fig.(16) view at source ↗
Figure 17
Figure 17. Figure 17: SUS behavior in region VI with α = 0.4, k = 9.8, γ = 10.56 However, a < 0 and hence τ > τ ′′ ∗ gives stable solutions. Since, γ > k in this region and there are two intersections (say τ ′ ∗a and τ ′ ∗b ) before τ ′′ ∗ . So, there will be SUS (Stable-Unstable-Stable) behavior in this region i.e – stable for 0 < τ < τ∗a ′ , – unstable for τ∗a ′ < τ < τ∗b ′ and – stable for τ > τ∗b ′ 4.4 Fourth Quadrant Anal… view at source ↗
Figure 18
Figure 18. Figure 18: Geometric interpretation of the curve γ = h2(k) in the fourth quadrant view at source ↗
Figure 19
Figure 19. Figure 19: Stability behaviour in the fourth quadrant view at source ↗
Figure 20
Figure 20. Figure 20: SSR behavior for region IV Example 4.2: Consider the parameter values k = 4.62 and γ = 3.69. These values lie in subregion V (see view at source ↗
Figure 21
Figure 21. Figure 21: a). When τ = 0.25 (see Fig. 21b), the system becomes unstable. Increasing τ further to 0.38 (see Fig. 21c) restores stability. However, for τ = 0.56 (see Fig. 21d), the equilibrium again loses stability and remains unstable thereafter. 30 35 40 45 50 t 0.000046 0.000048 0.000050 0.000052 0.000054 x (a) τ = 0.04 30 35 40 45 50 t -200 -100 100 200 x (b) τ = 0.25 30 35 40 45 50 t 0.00044 0.00045 0.00046 0.00… view at source ↗
Figure 22
Figure 22. Figure 22: SUS switching behavior in subregion VI Example 4.4: Let k = 0.23 and γ = −0.12. These parameter values lie in the fourth quadrant of the k–γ plane (see view at source ↗
Figure 23
Figure 23. Figure 23: USU switching behavior in the fourth quadrant view at source ↗
Figure 24
Figure 24. Figure 24: Unstable behavior for Example 1 Example 5.2: α = 0.8, k = 3.4, γ = −1.6 Since γ < 0 < k, the system is unstable for τ2 < − 1 γ log  k − γ k  = 0.2410, ∀ τ1 ≥ 0. Let τ2 = 0.15 < 0.2410. Then the system is unstable. Choose τ1 = 3.4 and τ1 = 6.1 (see view at source ↗
Figure 25
Figure 25. Figure 25: Unstable behavior for Example 2 17 view at source ↗
Figure 26
Figure 26. Figure 26: Stability diagram in the τ1–τ2 plane for α = 0.4, k = 1.02, and γ = 0.3. Ideally, the stable region should extend from the origin up to the boundary of the diagram. However, in this case, that does not occur. The boundary corresponds to Re(λ) = 0, but there exists a critical value of τ2 > 0 where λ = 0. Substituting λ = 0 into the characteristic equation (5) gives γ − k + ke−γτ2a∗ = 0, which simplifies to… view at source ↗
Figure 27
Figure 27. Figure 27: Critical points on the stability boundary in the view at source ↗
Figure 28
Figure 28. Figure 28: Stable behavior for τ2 = 1.04 2. Let τ2 = 1.1, so that τ2b∗ < τ2 < τ2a∗. In this regime, the equilibrium exhibits an S-U-S type stability switching (see view at source ↗
Figure 29
Figure 29. Figure 29: Stable cases corresponding to the S-U-S stability switching for view at source ↗
Figure 30
Figure 30. Figure 30: Stability switching behavior for τ2 = 1.1 Note: In Example 1, two branches arise, namely λ = iv (first branch) and λ = 0 (second branch). We seek a condition under which the second branch becomes irrelevant. Observe that if the critical value τ2∗ corresponding to λ = 0 is negative, then this branch does not influence the stability. Therefore, we determine the condition under which the expression for τ2∗ g… view at source ↗
Figure 31
Figure 31. Figure 31: Stability diagram in the τ1–τ2 plane for α = 0.4, k = −1.02, and γ = 0.3(Not upto scale). Here, λ = 0 gives τ2∗ = −0.85943, which cannot be a critical value. So, the stability boundary will be given by λ = iv. Hence, in this example, we can observe both the SSR region and the S-U-S-U region, i.e, say the local minima point A and the local maxima point correspond to the critical values τ2a∗ and τ2b∗ so we … view at source ↗
read the original abstract

This paper investigates the stability of different regions in the $(k,\gamma)$-plane for a class of fractional delay differential equations given by \begin{equation} D^{\alpha} x(t) = -\gamma x(t) + g\big(x(t - \tau_1)\big) - e^{-\gamma \tau_2}\, g\big(x(t - \tau_1 - \tau_2)\big), \qquad 0 < \alpha \le 1, \end{equation} where $k = g'(0)$. The primary focus is on the stability of the trivial equilibrium of the corresponding linearized system. A detailed stability and bifurcation analysis is carried out for the particular case $\tau_1 = 0$ and $\tau_2 \ge 0$. Furthermore, a general result is established for the case $\tau_1 > 0$, $\tau_2 \ge 0$, which holds for all values of $\alpha$ and $\tau_1$. In addition, illustrative examples are provided in the form of stability diagrams in the $(\tau_1,\tau_2)$-plane for fixed values of $\alpha$, $k$, and $\gamma$. These diagrams are generated using appropriate numerical methods to visualize the stability regions and to support the theoretical results.

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 analyzes stability and bifurcations of the trivial equilibrium for the fractional DDE D^α x(t) = -γ x(t) + g(x(t-τ1)) - e^{-γ τ2} g(x(t-τ1-τ2)) with 0<α≤1. Detailed stability regions in the (k,γ)-plane are derived for the special case τ1=0, τ2≥0. A general stability result is asserted for τ1>0, τ2≥0 that is claimed to hold for every α in (0,1] and every τ1. Numerical stability diagrams in the (τ1,τ2)-plane are generated for fixed α, k, γ to illustrate the regions.

Significance. If the general result is rigorously established, the work would be significant for fractional-order delay systems: it would identify a structural feature (the specific combination of delayed terms with the exponential coefficient) that renders stability regions independent of α, which is atypical and could simplify analysis in applications such as control or population models. The numerical diagrams provide concrete support for fixed-α cases, though their value is reduced by the lack of α-variation testing.

major comments (2)
  1. [General result for τ1 > 0] The section establishing the general result for τ1>0, τ2≥0: the claim that the result holds for all 0<α≤1 rests on the characteristic equation s^α + γ - k e^{-s τ1} + k e^{-γ τ2} e^{-s(τ1+τ2)}=0. Standard DDE crossing criteria (pure imaginary roots, Routh-Hurwitz on the quasi-polynomial) are derived under Re(s)=0, but fractional-order stability requires |arg(s)| > α π/2. No explicit argument is given showing that the stability boundary or crossing conditions are invariant under changes in α, nor is it shown that roots cannot enter the unstable sector for α<1 while remaining stable for α=1. This directly undermines the universality assertion, which is load-bearing for the central claim.
  2. [Numerical examples] Numerical examples section: the stability diagrams are produced only for fixed values of α. While they illustrate the (τ1,τ2) regions for those specific α, they provide no test of the α-independence asserted in the general result; a single diagram varying α (or an analytic proof that the boundary curves do not move) is needed to support the claim.
minor comments (2)
  1. [Abstract] The abstract and introduction should explicitly state the precise form of the general result (e.g., the explicit stability condition in the (k,γ) or (τ1,τ2) plane) rather than referring only to its existence.
  2. [Linearization] Notation for the linearized coefficient k = g'(0) is introduced without a dedicated equation number; adding an equation label would improve traceability when the characteristic equation is later written.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. The comments have helped us identify areas where the presentation of the general stability result and its numerical support can be strengthened. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [General result for τ1 > 0] The section establishing the general result for τ1>0, τ2≥0: the claim that the result holds for all 0<α≤1 rests on the characteristic equation s^α + γ - k e^{-s τ1} + k e^{-γ τ2} e^{-s(τ1+τ2)}=0. Standard DDE crossing criteria (pure imaginary roots, Routh-Hurwitz on the quasi-polynomial) are derived under Re(s)=0, but fractional-order stability requires |arg(s)| > α π/2. No explicit argument is given showing that the stability boundary or crossing conditions are invariant under changes in α, nor is it shown that roots cannot enter the unstable sector for α<1 while remaining stable for α=1. This directly undermines the universality assertion, which is load-bearing for the central claim.

    Authors: We agree that an explicit demonstration of α-invariance is necessary to rigorously support the general result. The specific structure of the system, with the delay-dependent coefficient e^{-γ τ2} multiplying the second delayed term, leads to a characteristic equation whose stability boundaries in the (k, γ) and (τ1, τ2) planes are determined by magnitude and frequency conditions that coincide for all α ∈ (0,1]. In particular, substituting s = iω into the equation and separating real and imaginary parts yields crossing conditions independent of the fractional order because the exponential prefactor compensates for the argument shift introduced by s^α. We acknowledge that this reasoning was not spelled out in sufficient detail. In the revised manuscript we will insert a new subsection immediately following the derivation of the characteristic equation that explicitly derives the Hopf crossing conditions for general α, shows that the critical curves remain unchanged, and confirms that no additional unstable roots enter the sector |arg(s)| < απ/2 for α < 1. revision: yes

  2. Referee: [Numerical examples] Numerical examples section: the stability diagrams are produced only for fixed values of α. While they illustrate the (τ1,τ2) regions for those specific α, they provide no test of the α-independence asserted in the general result; a single diagram varying α (or an analytic proof that the boundary curves do not move) is needed to support the claim.

    Authors: We concur that numerical confirmation across α values would provide useful supporting evidence. In the revised version we will augment the numerical examples section with an additional figure that overlays stability boundaries in the (τ1, τ2)-plane for three representative fractional orders (α = 0.5, 0.8, and 1.0) at the same fixed k and γ. This will visually demonstrate that the curves coincide, thereby corroborating the analytic claim of α-independence. If space constraints arise, we will instead include a short table of critical (τ1, τ2) pairs computed for multiple α values to illustrate the invariance. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation from linearized fractional DDE characteristic equation is self-contained.

full rationale

The paper starts from the given fractional DDE, linearizes around the trivial equilibrium to obtain the characteristic equation involving s^α, and applies stability analysis for the cases τ1=0 and τ1>0. No parameters are fitted to produce the claimed stability regions, no self-citation chain justifies the core result, and the general claim for all α is presented as following from the model equations rather than being redefined by them. Numerical diagrams are illustrative only. The derivation chain does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on established mathematical frameworks for fractional calculus and delay equations without introducing new free parameters or entities.

axioms (2)
  • standard math The fractional derivative operator D^α satisfies standard properties for 0 < α ≤ 1.
    Invoked in the definition of the system equation and subsequent linearization.
  • domain assumption The function g is continuously differentiable at zero so that k = g'(0) is well-defined.
    Required for linearization around the trivial equilibrium.

pith-pipeline@v0.9.0 · 5540 in / 1317 out tokens · 32489 ms · 2026-05-08T16:30:32.222530+00:00 · methodology

discussion (0)

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