arxiv: 2604.07924 · v1 · submitted 2026-04-09 · 🧮 math.DS

Recognition: 2 theorem links

· Lean Theorem

Analysis of Chaos and Bifurcation in Nonlinear two-delay differential equation

Pragati Dutta , Sachin Bhalekar

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:05 UTC · model grok-4.3

classification 🧮 math.DS

keywords chaosbifurcationdelay differential equationsynchronizationfractional orderfeedback controlmulti-scroll attractor

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The pith

A nonlinear equation with two time delays produces chaos and multi-scroll attractors that linear feedback can suppress and synchronize, including in fractional-order versions.

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

The paper examines a delay differential equation whose present rate of change depends on the state at two earlier times through a sine nonlinearity. Numerical integration, Lyapunov exponents, and phase portraits establish the existence of chaotic regimes and multi-scroll attractors for ranges of the gain and delay parameters. A simple linear feedback term is added to drive the irregular motion to a stable equilibrium. The same feedback structure yields a delay-independent sufficient condition that guarantees synchronization between two copies of the system. The analysis is repeated for the fractional-order counterpart, where positive Lyapunov exponents continue to appear even when the derivative order falls below one.

Core claim

The two-delay system exhibits chaotic motion and multi-scroll attractors for suitable choices of the parameters k, gamma, tau1 and tau2. These irregular behaviors are suppressed by the addition of a linear state-feedback term. A delay-independent criterion derived from the feedback law ensures that a slave system converges to a master system. The same chaotic attractors and feedback control remain effective when the integer derivative is replaced by a fractional derivative of order alpha less than or equal to one.

What carries the argument

The two-delay differential equation with sinusoidal nonlinearity, together with linear feedback control laws that stabilize the origin and enforce master-slave synchronization.

If this is right

Multi-scroll chaos appears for ranges of the sine gain k and the two delays.
A single linear feedback term suffices to eliminate the irregular oscillations.
Synchronization occurs under a condition that does not depend on the size of either delay.
Positive Lyapunov exponents and multi-scroll patterns persist when the system order is lowered to fractional values.

Where Pith is reading between the lines

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

The delay-independent synchronization condition may allow controller design without precise knowledge of the delays.
Similar linear feedback might stabilize other multi-delay systems whose nonlinearity is bounded.
Fractional-order persistence of chaos suggests that memory effects alone do not destroy the irregular dynamics once they appear in the integer-order case.

Load-bearing premise

The chosen numerical parameter ranges, integration methods, and initial conditions are representative of the system's global dynamics and do not miss other attractors or instabilities.

What would settle it

Compute the largest Lyapunov exponent for a specific parameter tuple inside the reported chaotic region; if the exponent is negative while the phase portrait shows bounded irregular motion, or if synchronization fails for delays satisfying the reported inequality, the central claims are falsified.

Figures

Figures reproduced from arXiv: 2604.07924 by Pragati Dutta, Sachin Bhalekar.

**Figure 2.** Figure 2: Bifurcation diagram of system (3) with respect to [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Phase portraits showing convergence to equilibrium and the emergence of periodic oscillations for [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Transition from periodic motion to single-scroll chaotic oscillations. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: Periodic window followed by chaotic oscillations. [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Transition from periodic dynamics to double-scroll chaotic behavior as [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 7.** Figure 7: Reappearance of periodic oscillations within a chaotic regime followed by a return to double-scroll [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 8.** Figure 8: Bifurcation diagram with respect to τ2 for k = 1.5, γ = 0.3, and τ1 = 4.5. where µ > 0 denotes the control gain. Linearizing system (4) about the equilibrium point x = 0 yields the following linear delay differential equation: x˙(t) = −(γ + µ)x(t) + kx(t − τ1) − ke−γτ2 x(t − τ1 − τ2). (5) The corresponding characteristic equation is given by λ = −(γ + µ) + ke−λτ1 − ke−γτ2−λ(τ1+τ2) . (6) For the parameter … view at source ↗

**Figure 9.** Figure 9: Numerical verification of chaos suppression using linear state feedback control for [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗

**Figure 10.** Figure 10: Numerical verification of chaos suppression using linear state feedback control for [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗

**Figure 11.** Figure 11: Synchronization of master and slave states under linear state feedback control. The trajectories [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗

**Figure 12.** Figure 12: Decay of the synchronization error e(t) confirming asymptotic synchronization for parameter sets satisfying γ + δ > 2|k|. To better show how fast the synchronization error decreases, [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗

**Figure 13.** Figure 13: Numerical demonstration of synchronization for the parameter set [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗

**Figure 14.** Figure 14: Phase portraits of system (20) illustrating the effect of decreasing fractional order [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗

**Figure 15.** Figure 15: Evolution of the chaotic attractor as the fractional order [PITH_FULL_IMAGE:figures/full_fig_p013_15.png] view at source ↗

**Figure 16.** Figure 16: Phase portraits showing the emergence of chaos for smaller fractional orders with parameters [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗

**Figure 17.** Figure 17: Effect of the time delay τ2 on system dynamics for α = 0.4, k = 2.4, γ = 0.3, and τ1 = 2.5. The system exhibits periodic behavior at τ2 = 0.3, highly periodic oscillations at τ2 = 1.2, and transitions to chaotic dynamics at τ2 = 4.2, highlighting delay-induced complexity. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗

read the original abstract

This paper studies how complicated and irregular behavior, known as chaos, can arise in a simple mathematical model that includes time delays. The model is a delay differential equation in which the present rate of change depends not only on the current state but also on past states at two different delay times. The system is described by \begin{equation} \dot{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 $g(x)=k \sin{x}, k\in\mathbf{R}$. Here, the delays $\tau_1$ and $\tau_2$ represent memory effects in the system, while the sine terms introduce strong nonlinearity. Numerical simulations are used to study the system behavior for different parameter values. Chaotic motion is identified using Lyapunov exponents and phase portraits, which show irregular and unpredictable dynamics. For certain parameter ranges, the system exhibits multi-scroll chaotic attractors, in which the motion alternates among several complex patterns. Finally, chaos is controlled by adding a simple linear feedback term, which suppresses irregular oscillations and stabilizes the system. In addition, synchronization between master and slave systems is investigated using linear state feedback control, and a delay-independent sufficient condition for synchronization is derived and verified numerically. The results show that even complex delayed systems can be effectively controlled and synchronized using simple feedback techniques. The study is further extended to a fractional-order version of the system to examine the influence of memory effects, where it is observed that chaotic behavior can persist even for lower fractional orders.

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

Numerical study of chaos control in a two-delay sine DDE with a claimed delay-independent sync condition, but the work stays at the level of standard simulations.

read the letter

The key takeaway is that this is a numerical investigation of chaos and its control in one particular two-delay sine DDE, including a fractional version. They derive a delay-independent condition for synchronization and check it with simulations, but the overall contribution stays within established methods for delayed chaotic systems. The new part is applying these ideas to this specific model and getting the delay-independent sync condition for it. They do a decent job showing multi-scroll attractors via phase portraits and Lyapunov exponents, and demonstrating that linear feedback can stabilize the system and achieve synchronization in the cases they test. Extending to fractional orders and noting chaos at low alpha is also a plus for the applied side. The main weakness is the reliance on numerical evidence without enough transparency on the methods. For DDEs, choosing initial histories and integration schemes matters a lot, and without details on step-size control or exhaustive checks, it's hard to be sure the simulations capture the true dynamics. Bounded parameter grids might miss co-existing attractors or cases where the feedback fails. The derivation of the sync condition needs to be examined closely to confirm it's truly delay-independent and not just holding for the chosen parameters. This kind of work is useful for researchers in nonlinear dynamics who want concrete examples to build on or compare against. It won't change the field, but it adds to the catalog of controlled delayed systems. A serious referee should look at it to verify the math and numerics, as the claims are plausible but rest on simulations that could have gaps. I recommend putting it through peer review rather than desk rejecting it.

Referee Report

3 major / 3 minor

Summary. The manuscript presents a numerical study of chaos and bifurcations in a nonlinear two-delay differential equation (DDE) of the form dot x(t) = -γ x(t) + k sin(x(t-τ1)) - exp(-γ τ2) k sin(x(t-τ1-τ2)), extended to fractional orders 0 < α ≤ 1. Through phase portraits and Lyapunov exponent calculations, the authors identify chaotic regimes including multi-scroll attractors for various delay values and nonlinearity strengths k. They propose a linear feedback control to suppress chaos and derive a delay-independent sufficient condition for synchronization between master and slave systems, which is then verified through simulations. The fractional-order analysis suggests that chaotic behavior persists even at lower fractional orders.

Significance. If the numerical results prove robust, the work offers practical insights into controlling and synchronizing chaotic dynamics in delayed systems using straightforward linear feedback techniques. The derivation of a delay-independent synchronization condition, if rigorously established, would be a valuable contribution to the field of dynamical systems with memory effects. The observation of chaos in fractional-order versions adds to understanding how memory influences complex behavior. However, the significance is tempered by the purely numerical nature of the evidence without analytical proofs or exhaustive exploration of the infinite-dimensional state space.

major comments (3)

The delay-independent sufficient condition for synchronization is stated as derived, but without the explicit steps of the derivation (e.g., the Lyapunov function or inequality used), it is difficult to assess its validity independently of the numerical verification. The numerical tests use limited initial conditions and parameter ranges, which may not confirm the delay-independence across all possible histories in the DDE state space.
No information is provided on the specific numerical integrator used for the DDE (such as method of steps, Runge-Kutta with delay handling, or software like MATLAB's dde23), the step size, or error tolerances. This is particularly important for computing reliable Lyapunov exponents and identifying multi-scroll attractors, as numerical artifacts could mimic or obscure chaotic behavior.
The fractional-order extension mentions persistence of chaos for lower α, but lacks specification of the fractional derivative type and the numerical scheme employed (e.g., Adams-Bashforth-Moulton or Grünwald-Letnikov). Given the sensitivity of fractional systems to discretization, this omission undermines confidence in the claim that chaos persists at low orders.

minor comments (3)

The equation in the abstract has a comma before the final g term, which appears to be a typesetting error; it should read as multiplication by the exponential term.
The parameter γ is introduced without explanation of its physical or mathematical significance in the model.
Phase portraits and bifurcation diagrams would benefit from clearer labeling of parameter values used and axes scales for better reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which will help strengthen the presentation of our results. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The delay-independent sufficient condition for synchronization is stated as derived, but without the explicit steps of the derivation (e.g., the Lyapunov function or inequality used), it is difficult to assess its validity independently of the numerical verification. The numerical tests use limited initial conditions and parameter ranges, which may not confirm the delay-independence across all possible histories in the DDE state space.

Authors: We thank the referee for this observation. In the revised manuscript we will include the complete, explicit derivation of the delay-independent synchronization condition, showing the Lyapunov function employed and the sequence of inequalities that establish the sufficient condition. While the state space of a DDE is infinite-dimensional, our numerical tests already employed several distinct initial history functions over intervals determined by the maximum delay together with a range of parameter values; we will augment this section with additional examples using further varied initial histories to better illustrate the robustness of the result. revision: yes
Referee: No information is provided on the specific numerical integrator used for the DDE (such as method of steps, Runge-Kutta with delay handling, or software like MATLAB's dde23), the step size, or error tolerances. This is particularly important for computing reliable Lyapunov exponents and identifying multi-scroll attractors, as numerical artifacts could mimic or obscure chaotic behavior.

Authors: We agree that the absence of these details limits reproducibility. We will add a dedicated paragraph in the revised manuscript that specifies the numerical integrator, software, step size, and error tolerances used for the integer-order DDE simulations, including the computation of Lyapunov exponents. This addition will allow readers to assess the reliability of the reported chaotic regimes and multi-scroll attractors. revision: yes
Referee: The fractional-order extension mentions persistence of chaos for lower α, but lacks specification of the fractional derivative type and the numerical scheme employed (e.g., Adams-Bashforth-Moulton or Grünwald-Letnikov). Given the sensitivity of fractional systems to discretization, this omission undermines confidence in the claim that chaos persists at low orders.

Authors: We appreciate the referee drawing attention to this omission. We will revise the fractional-order section to state explicitly the derivative definition employed and the numerical scheme together with its discretization parameters. These additions will provide the necessary information to evaluate the persistence of chaos at lower fractional orders. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are independent of target results

full rationale

The paper presents an analytical derivation of a delay-independent sufficient condition for master-slave synchronization via linear feedback, followed by separate numerical verification through direct simulation. Chaos and multi-scroll attractors are identified via explicit computation of Lyapunov exponents and phase portraits on the given two-delay DDE (and its fractional extension). No parameters are fitted to the synchronization error or chaos metrics, no self-definitional loops appear in the model equations or control laws, and no load-bearing self-citations are used to justify uniqueness or ansatzes. The numerical sweeps, while necessarily finite, do not retroactively force the analytical steps by construction.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard existence theory for delay differential equations and the reliability of numerical chaos diagnostics. No new physical entities are postulated. All parameters are explored rather than fitted to external data.

free parameters (3)

delay values τ1 and τ2
Varied across ranges to produce different dynamical regimes including chaos
nonlinearity strength k
Varied to induce and control chaotic behavior
fractional order α
Varied below 1 to test persistence of chaos

axioms (2)

standard math Solutions to the delay differential equation exist and are unique for the chosen parameters
Required for the system to be well-posed before any simulation
domain assumption Numerical computation of Lyapunov exponents reliably detects chaos
Standard assumption in numerical studies of nonlinear dynamics

pith-pipeline@v0.9.0 · 5626 in / 1538 out tokens · 69265 ms · 2026-05-10T18:05:48.789865+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Theorem 5.1 (Delay-Independent Synchronization). If γ+δ>2|k| ...

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.

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