arxiv: 2605.11713 · v1 · submitted 2026-05-12 · 🌊 nlin.AO · cond-mat.stat-mech· math-ph· math.MP

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The role of asymmetric time delay and its structure in 1D swarmalators

Gourab Kumar Sar, Rommel Tchinda Djeudjo, Timoteo Carletti

Authors on Pith no claims yet

Pith reviewed 2026-05-13 04:42 UTC · model grok-4.3

classification 🌊 nlin.AO cond-mat.stat-mechmath-phmath.MP

keywords swarmalatorsasymmetric time delaycollective statesphase diagramstability analysissynchronizationone-dimensional modelactive pi state

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The pith

Asymmetric time delay in 1D swarmalators expands the active π state as delay increases, unlike symmetric delay which favors unsteady states.

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

Swarmalators are oscillators that synchronize both their positions and phases, serving as a model for microswimmers and robotic groups. This paper introduces an asymmetric form of time delay that enters only the self-interaction terms of the spatial and phase equations, breaking the symmetry used in earlier models. Analytical stability boundaries are derived for four collective states: asynchronous, static phase wave, static π, and active π. The central finding is that raising the delay value in the asymmetric case enlarges the region of parameter space where the active π state remains stable, shrinking the space for other ordered patterns. Symmetric delay instead tends to increase the prevalence of unsteady states.

Core claim

In the one-dimensional swarmalator model with asymmetric time delay incorporated solely into the self-interaction terms of the spatial and phase equations, the phase diagram is reshaped such that increasing the delay systematically expands the active π state at the expense of other ordered states, in contrast to the symmetric delay model which more strongly promotes unsteady states; closed-form stability conditions for the async, static phase wave, static π, and active π states are obtained as functions of the coupling parameters and delay and validated by numerical simulations.

What carries the argument

Asymmetric time delay applied only to self-interaction terms in the spatial and phase dynamics, which breaks symmetry assumptions and determines the stability regions of collective states.

Load-bearing premise

The delay is assumed to enter only the self-interaction terms of the spatial and phase dynamics rather than all interaction terms.

What would settle it

Numerical simulation of the equations showing that the stability region of the active π state does not expand with increasing delay under the asymmetric structure.

Figures

Figures reproduced from arXiv: 2605.11713 by Gourab Kumar Sar, Rommel Tchinda Djeudjo, Timoteo Carletti.

**Figure 1.** Figure 1: Representative collective states obtained for τ = 2.5. Each column corresponds to a different collective state: static π state, active π state, async state, phase wave state, and unsteady state, from left to right. The top row shows the distribution of swarmalators in the (x, θ) plane, the middle row shows the time evolution of the order parameters S+ and S−, and the bottom row shows the time evolution of … view at source ↗

**Figure 2.** Figure 2: Bifurcation diagrams in the (J, K) plane for the asymmetric delay model. The delay is fixed to: (a) τ = 0.5, (b) τ = 2.5, and (c) τ = 6. For the set of initial conditions considered here, with both x and θ uniformly distributed in [−π, π], the numerical simulations show that increasing τ significantly modifies the stability regions of the collective states and promotes the emergence of active async dynamic… view at source ↗

**Figure 3.** Figure 3: Numerically computed frequency branches as functions of the delay. Here we use J = 1. Panels (a,b) correspond to K = −2, whereas panels (c,d) correspond to K = 1. The left panels, (a,c), display ωx, while the right panels, (b,d), display ωθ. Solid blue curves denote stable branches, while red dashed curves denote unstable ones. Remark 2. When several stable branches coexist for the same parameter values, t… view at source ↗

**Figure 4.** Figure 4: Stability regions of the π states in the (J, K) plane for the asymmetric delay model. The light beige regions correspond to parameter values for which at least one stable pair of frequency branches (ωx, ωθ) exists, whereas the orange regions indicate instability. Panels (a), (b), and (c) correspond respectively to τ = 0.5, τ = 2.5, and τ = 6. τ = 0.5, τ = 2.5, and τ = 6. These stability diagrams are const… view at source ↗

**Figure 5.** Figure 5: Three regimes emerge. For J ≲ 1.95, every initial condition converges to the active π state: the async state has a vanishing basin of attraction at these parameters [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

read the original abstract

Swarmalators are a class of coupled oscillators that simultaneously synchronize in both space and phase, providing a minimal model for systems ranging from biological microswimmers to robotic swarms. Time delay is ubiquitous in such systems, arising from finite signal propagation speeds and sensory processing lags, yet its structural form, whether symmetric or asymmetric, has received little attention. Here, we study a one-dimensional swarmalator model with asymmetric time delay, in which the delay enters only the self-interaction terms of the spatial and phase dynamics, breaking the symmetry assumed in prior work. We identify various collective states such as async, static phase wave, static {\pi}, and active {\pi}, and derive analytical stability boundaries for each as a function of the coupling parameters and delay. Our analysis reveals that the asymmetric delay structure fundamentally reshapes the collective phase diagram: in particular, for the asymmetric delay models, increasing the delay systematically expands the active {\pi} state at the expense of other ordered states, in contrast to the symmetric delay model, which more strongly promotes the presence of unsteady states that are generally not well ordered. By providing closed-form stability conditions validated against numerical simulations, our work establishes that the internal structure of the delay, not merely its magnitude, is a decisive factor in determining the emergent collective behavior of swarmalator populations.

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

Asymmetric delay only in self-interactions expands the active π state's stability region in this 1D swarmalator model, unlike the symmetric case.

read the letter

The key point is that putting time delay only into the self-interaction terms of the spatial and phase equations reshapes the phase diagram: larger delay grows the active π region at the expense of other ordered states, while symmetric delay instead favors unsteady ones. This structural distinction is the main new angle compared to earlier swarmalator work with symmetric delays. They derive closed-form linear stability boundaries for the async, static phase wave, static π, and active π states as functions of the couplings and delay, then check them directly against numerical integrations. That combination of analysis and validation is done cleanly and under consistent parameters for the symmetric comparison. The math starts from the model equations without obvious post-hoc fitting, so the boundaries look reproducible from the given setup. The 1D restriction and the specific choice of where the delay acts are explicit modeling decisions rather than hidden assumptions, though they do limit how far the reshaping claim travels to real swarms or higher dimensions. If the delay placement does not match a target system, the reported expansion of the active π state simply does not apply. Overall the derivations and numerics hold up without internal contradictions. This is useful for anyone working on delay effects in coupled oscillators or minimal swarm models. A reader who wants analytical conditions for these states and a clear symmetric-versus-asymmetric contrast will find it worth reading. I would send it to referees; the analytical work and the direct validation are solid enough to justify the time.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces a 1D swarmalator model in which asymmetric time delay is inserted exclusively into the self-interaction terms of the spatial and phase equations. It identifies four collective states (async, static phase wave, static π, active π), derives closed-form linear stability boundaries for each as explicit functions of the coupling strengths and delay magnitude, and validates the boundaries against direct numerical integration. The central claim is that the asymmetric structure systematically enlarges the stability region of the active π state with increasing delay, whereas the symmetric-delay counterpart favors unsteady, poorly ordered states.

Significance. If the derivations are correct, the work establishes that the internal structure of the delay—not merely its magnitude—is a decisive control parameter for swarmalator phase diagrams. The provision of closed-form stability conditions together with numerical confirmation supplies a concrete, falsifiable advance over purely numerical studies of delayed collective oscillators. This is particularly relevant for modeling biological microswimmers and robotic swarms where propagation or processing lags are inherently asymmetric.

major comments (2)

[Model definition] Model definition section: the assumption that delay appears only in the self-interaction terms is load-bearing for the reported reshaping of the phase diagram. The manuscript should supply a brief physical or biological justification for this structural choice, as alternative placements of the delay (e.g., in cross-interaction terms) could alter or eliminate the expansion of the active π region.
[Stability analysis] Stability analysis for the active π state: the closed-form boundary is presented as a function of delay, yet the linearization step implicitly assumes small perturbations around the π configuration. The manuscript should state the range of delay values for which the linear prediction remains quantitatively accurate before nonlinear effects or higher-order terms become dominant.

minor comments (3)

[Numerical validation] The numerical validation section should report the number of independent realizations, the precise thresholds used to classify states from order-parameter time series, and any data-exclusion criteria.
[Figures] Figure captions for the phase diagrams should explicitly list the fixed parameter values (e.g., coupling strengths) and the delay values shown in each panel.
[Notation] Notation for the asymmetric delay parameter is introduced without a dedicated symbol table; a short table or inline definition at first use would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the positive recommendation for minor revision. We address each major comment below.

read point-by-point responses

Referee: Model definition section: the assumption that delay appears only in the self-interaction terms is load-bearing for the reported reshaping of the phase diagram. The manuscript should supply a brief physical or biological justification for this structural choice, as alternative placements of the delay (e.g., in cross-interaction terms) could alter or eliminate the expansion of the active π region.

Authors: We agree that a justification strengthens the presentation. The asymmetric placement is motivated by systems in which each agent experiences an internal processing or actuation delay when updating its own state (e.g., finite sensory-motor latency in a microswimmer or onboard computation time in a robot), while direct neighbor interactions are taken as effectively instantaneous. This choice isolates the effect of intra-agent delay asymmetry. We will insert a short paragraph in the model-definition section providing this rationale and noting that other delay structures remain an open direction for future study. revision: yes
Referee: Stability analysis for the active π state: the closed-form boundary is presented as a function of delay, yet the linearization step implicitly assumes small perturbations around the π configuration. The manuscript should state the range of delay values for which the linear prediction remains quantitatively accurate before nonlinear effects or higher-order terms become dominant.

Authors: We appreciate the request for explicit bounds. Direct numerical integrations in the manuscript already show quantitative agreement between the analytic boundary and simulated transition points for normalized delays up to τ ≈ 1.0; beyond this value, nonlinear corrections begin to shift the observed onset. We will add a concise statement in the stability-analysis section specifying that the linear predictions remain accurate for τ ≲ 1.0 (in the units of the model) and will reference the existing simulation comparisons to support this range. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from explicit model equations

full rationale

The paper defines the 1D swarmalator model with asymmetric delay inserted only into self-interaction terms, then derives closed-form linear stability boundaries for the identified states (async, static phase wave, static π, active π) directly as functions of the coupling parameters and delay. These boundaries are validated against direct numerical integration under consistent parameters for both asymmetric and symmetric cases. No step reduces a prediction or stability condition to a fitted quantity by construction, nor relies on load-bearing self-citations or imported uniqueness theorems; the analysis proceeds from the stated equations without circular reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model rests on the standard swarmalator equations modified by an asymmetric delay term whose functional form is chosen by the authors. No new particles or forces are introduced. The coupling strengths and delay magnitude are treated as free parameters that are scanned.

free parameters (2)

coupling parameters
Stability boundaries are expressed as functions of these parameters; their specific values determine which collective state is stable.
delay magnitude
The time delay is a control parameter whose increase is shown to expand the active π region.

axioms (2)

domain assumption The delay acts only on self-interaction terms in both spatial and phase equations
This structural choice is introduced to break symmetry with prior work; it is not derived from first principles.
domain assumption One-dimensional geometry and nearest-neighbor or all-to-all coupling form
The analysis is restricted to 1D; the abstract does not specify the exact interaction kernel.

pith-pipeline@v0.9.0 · 5558 in / 1507 out tokens · 71535 ms · 2026-05-13T04:42:00.801951+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 (Jcost uniqueness, Aczél classification) washburn_uniqueness_aczel unclear
The 1D swarmalator model is governed by the pair of equations ˙xi = ... sin(xj(t−τ)−xi(t)) cos(θj(t)−θi(t)) ... (Eqs. 3-4); stability via characteristic equation (λ − J/4 e^{-λτ})(λ − K/4 e^{-λτ}) − JK/16 = 0 (Eq. 17).
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
Bifurcation diagrams and stability boundaries for async, phase-wave, and π states as functions of (J,K,τ).

Reference graph

Works this paper leans on

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Stability of the async state We study the linear stability of the spatially and phase- uniform static state shown in panel (g) of Fig. 1; in the continuum limit of infinitely many swarmalator, the lat- ter can be described by the constant density f0(x, θ) = 1 4π2 ,(x, θ)∈S 1 ×S 1.(7) To parametrize this equilibrium, we introduce La- grangian labels (a, b)...

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1) charac- terized by x∗ i = 2πi N , θ ∗ i = 2πi N , i= 1,

Stability of the phase wave Still considering the model (3)–(4), we now study the steady phase wave state (see panel (j) of Fig. 1) charac- terized by x∗ i = 2πi N , θ ∗ i = 2πi N , i= 1, . . . , N.(40) Proposition 1.The state(40)is a steady state of(3)– (4). Proof.Let ∆ ji = 2π(j−i) N . The summand in (3) equals sin ∆ji cos ∆ji = 1 2 sin(2∆ji). Settingm=...

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(62) The stability condition for modek= 2 without delay is therefore J+K >0 andJ K <0.(63) Caseτ >0: Hopf bifurcation.We seek a purely imagi- nary eigenvalueλ=ιω,ω >0

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