Dual-Delay Framework: Dynamics, Sensing & Learning

• Dual-delay framework is a modeling approach that uses two distinct delayed channels to compensate for limitations inherent in single-delay systems.
• It is applied across domains such as delayed-feedback oscillators, consensus control, wireless sensing, and online learning to improve stability and identifiability.
• Leveraging complementary temporal scales, this framework enhances performance through improved feedback, robust synchronization, and effective delay compensation.

Searching arXiv for papers on “dual-delay framework” and closely related uses across sensing, control, learning, and dynamical systems. A dual-delay framework is a modeling and algorithmic pattern in which two distinct delays, two delayed information channels, or two temporally separated update mechanisms are used to represent dynamics, recover latent parameters, or stabilize learning and control. In the recent arXiv literature, this pattern appears in delayed-feedback oscillators, consensus and congestion control, integrated sensing and communication, multi-agent reinforcement learning, and online learning with delayed supervision; the common theme is that a single delayed observable is often insufficient, whereas two complementary delayed structures can restore identifiability, enlarge stability regions, or improve robustness (Semenov, 11 Dec 2025, Atay, 27 Dec 2025, Kim et al., 13 Mar 2026, Wang et al., 22 Jun 2026).

1. Conceptual scope

The term does not denote a single canonical formalism. In one class of works, it means a system with two explicit delay arguments, such as a bistable delayed-feedback oscillator with feedback loops at $\tau_1$ and $\tau_2$, or a multi-agent consensus protocol with one delay in the position channel and another in the velocity channel (Semenov, 11 Dec 2025, Cepeda-Gomez et al., 2014). In another class, the same structural role is played by two complementary measurements or two timescales: dual-chirp AFDM uses two distinct post-chirp parameters to turn an underdetermined delay-Doppler estimation problem into a full-rank system, and DDAM-based ISAC separates the fast coherence-time scale from the slower path-invariant-time scale (Kim et al., 13 Mar 2026, Xiao et al., 2023). In learning systems, the duality may be architectural rather than physical: CoDe combines intent alignment and timeliness alignment for asynchronous communication, while DT-GOL uses a persistent branch updated only from delayed labels and a transient branch updated from soft pseudo-labels inside the delay window (Song et al., 9 Jan 2025, Wang et al., 22 Jun 2026).

Domain	Dual-delay structure	Representative paper
Delayed dynamics	Two delayed feedback loops $\tau_1,\tau_2$	(Semenov, 11 Dec 2025)
Consensus/control	Two information channels or multiple time-delays	(Atay, 27 Dec 2025, Cepeda-Gomez et al., 2014, Peet, 2016)
Sensing/ISAC	Two chirps or two timescales	(Kim et al., 13 Mar 2026, Xiao et al., 2023)
Learning	Delayed ground truth plus fast surrogate updates	(Wang et al., 22 Jun 2026, Song et al., 9 Jan 2025, Hsieh et al., 2020)

This suggests that “dual-delay framework” is best understood as a structural principle: two delayed views of the same latent process are arranged so that each compensates for the limitations of the other.

2. Canonical mathematical structures

A direct dynamical realization is the bistable oscillator with two delayed-feedback loops,

$$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$

with $a,b>0$, $\gamma$ the feedback strength, $\tau_1=1000$ fixed in simulation, and $\tau_2\le \tau_1$ varied. The same form is implemented experimentally as

$$RC\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$

with $R=10\,\text{k}\Omega$, $\tau_2$0, $\tau_2$1, and $\tau_2$2. The cubic term creates bistability, while the dual delayed-feedback term acts as a centered sum of two self-feedbacks and is interpreted as a discrete second-difference operator in a temporal lattice (Semenov, 11 Dec 2025).

A distinct but related control-theoretic form is the two-delay second-order consensus protocol over a directed graph,

$\tau_2$3

where $\tau_2$4 affects the position channel and $\tau_2$5 the velocity channel. In state space this becomes a linear time-invariant multiple time-delay system,

$\tau_2$6

with two constant, uniform, rationally independent delays (Cepeda-Gomez et al., 2014).

More abstract multi-delay systems are written as

$\tau_2$7

and analyzed through a dual form of Lyapunov-Krasovskii functional. There, the central object is a bounded, positive, coercive, self-adjoint operator $\tau_2$8 satisfying a dual inequality of the form

$\tau_2$9

which is then translated into positivity and negativity conditions on multiplier-kernel operators and expressed as LMIs through SOS methodology (Peet, 2016).

A different canonical structure appears in delayed consensus-as-Markov formulations. Information processing delays lead to

$\tau_1,\tau_2$0

whereas information propagation delays lead to

$\tau_1,\tau_2$1

The two delay types are not interchangeable: they preserve different invariants and produce different consensus conditions (Atay, 27 Dec 2025).

3. Identifiability and sensing

A dual-delay framework is especially useful when a single delayed observable collapses multiple latent parameters into one scalar. In the Rydberg atomic quantum receiver setting, the RAQR self-heterodyne output maps the target delay $\tau_1,\tau_2$2 and Doppler $\tau_1,\tau_2$3 to a fluctuation frequency

$\tau_1,\tau_2$4

For a single AFDM chirp rate, all subcarriers produce the same $\tau_1,\tau_2$5, so the system is rank-1 and the joint delay-Doppler estimation problem is underdetermined. The proposed dual-chirp AFDM framework changes the post-chirp parameter between two consecutive frames and yields

$\tau_1,\tau_2$6

which stack into the $\tau_1,\tau_2$7 system

$\tau_1,\tau_2$8

If $\tau_1,\tau_2$9, the matrix is full rank and $$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$0 is uniquely recoverable. The paper explicitly characterizes this as a “dual-delay / dual-structure framework,” emphasizing that the ambiguity is not a conventional radar ambiguity-function issue but a structural ambiguity induced by the RAQR optical mapping and self-heterodyne readout (Kim et al., 13 Mar 2026).

An allied but not identical construction appears in DDAM-based ISAC through dual timescales. The channel coherence time is

$$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$1

while the path invariant time is defined as

$$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$2

Under the paper’s geometric bounds,

$$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$3

and under typical conditions $$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$4, $$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$5, and $$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$6, the paper derives $$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$7. In the numerical example at $$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$8 GHz with $$\frac{dx}{dt}=-x(x-a)(x+b)+F(t)+\frac{\gamma}{2}\big(x(t-\tau_1)+x(t-\tau_2)-2x(t)\big),$$9 MHz, $a,b>0$0, $a,b>0$1 m/s, and $a,b>0$2 m, it reports $a,b>0$3 ms and $a,b>0$4 ms, about a factor $a,b>0$5 apart (Xiao et al., 2023).

These two sensing examples instantiate the same structural idea in different ways. In the RAQR case, two distinct chirps create two independent affine projections of $a,b>0$6; in DDAM, two timescales separate fast CSI variation from slower path-state variation. A plausible implication is that dual-delay design becomes valuable whenever the observable generated by one timescale or one waveform is intrinsically one-dimensional.

4. Learning and optimization with delayed information

In online optimization, delays are incorporated directly into the dual variable rather than treated as exogenous timestamps. Delayed Dual Averaging uses the available gradient index set $a,b>0$7 and updates

$a,b>0$8

The paper’s framework is built around information sets rather than explicit delay values, and in the single-agent case it extends adaptive dual averaging to potentially unbounded delays. In the multi-agent case it addresses full asynchronicity by using only the information available for producing each prediction rather than a global clock. It derives adaptive strategies with optimal regret bounds and an optimistic variant that exploits predictability of slowly varying problems (Hsieh et al., 2020).

CoDe addresses communication delay in MARL through two distinct alignments. First, it learns an intent representation by future action inference, so the communicated message reflects stable future behavioral trends rather than only instantaneous state. Second, it applies a dual alignment mechanism of intent and timeliness: intent alignment weights messages according to relevance in intent space, and timeliness alignment discounts messages according to age. The paper defines both a fixed delay setting $a,b>0$9, where $\gamma$0, and a time-varying delay setting $\gamma$1, where $\gamma$2, and reports that CoDe outperforms baseline algorithms in three MARL benchmarks without delay and exhibits robustness under fixed and time-varying delays (Song et al., 9 Jan 2025).

DT-GOL reformulates fixed label delay in nonstationary online classification as a semi-supervised problem over the delay window. Its core dual-track architecture consists of a persistent learner $\gamma$3, updated only on delayed ground truth $\gamma$4, and a transient learner $\gamma$5, re-initialized from the master at each step and updated on geometry-derived soft pseudo-labels in the current delay window. The paper states that this decoupling resolves the stability-plasticity dilemma; empirically, removing the dual-track component causes the largest average ACC drop, $\gamma$6, among the reported ablations. It also reports that DT-GOL achieves the best CER and AUC on the majority of datasets, specifically $\gamma$7 real datasets, and shows the highest average ACC in the blind adaptation zone on SEA, about $\gamma$8 (Wang et al., 22 Jun 2026).

Taken together, these works use dual-delay structure to separate slow trustworthy information from fast approximate information: delayed gradients versus current decision sets, future-intent messages versus stale messages, and delayed labels versus geometry-derived soft supervision.

5. Stability, bifurcation, and synthesis

For delayed consensus, the Markov-duality formulation yields two sharply different regimes. With information propagation delays,

$\gamma$9

the paper proves that if zero is a simple eigenvalue of $\tau_1=1000$0, then $\tau_1=1000$1 is a simple characteristic root and all others have strictly negative real parts; consensus therefore holds without a delay-size restriction under the normalization condition on $\tau_1=1000$2. With information processing delays,

$\tau_1=1000$3

stability depends on the delay kernel. For symmetric coupling and discrete delay, the necessary and sufficient condition for consensus is

$\tau_1=1000$4

and a sufficient bound is

$\tau_1=1000$5

The consensus value also differs between the two cases: propagation delays modify it through the initial history and mean delay, whereas processing delays preserve $\tau_1=1000$6 whenever consensus remains stable (Atay, 27 Dec 2025).

In second-order directed consensus with two rationally independent delays, the characteristic quasi-polynomial is factorized via the eigenstructure of the row-stochastic matrix $\tau_1=1000$7, and stability boundaries in $\tau_1=1000$8 are computed exactly through CTCR and the Spectral Delay Space. The paper emphasizes two consequences. First, the overall stability region is the intersection of the stable regions of all modal factors. Second, relative stability need not vary monotonically with delay: the performance may be improved by increasing the delays, leading to the delay scheduling concept. In the reported five-agent example, point $\tau_1=1000$9 is unstable whereas point $\tau_2\le \tau_1$0, with both delays larger, lies in the stable region (Cepeda-Gomez et al., 2014).

The nonlinear consequences of dual delays are particularly sharp in congestion control. For the proportionally fair dual algorithm with two delays, the local stability condition can be written explicitly, and the Hopf bifurcation is shown to be always super-critical, leading to asymptotically orbitally stable small-amplitude limit cycles. By contrast, the TCP fair and Delay dual algorithms with two delays can undergo sub-critical Hopf bifurcation. The paper states that such a bifurcation can result in either large amplitude limit cycles or unstable limit cycles and should be avoided in engineering applications (Abuthahir et al., 2019).

For synthesis of general multi-delay systems, the dual Lyapunov-Krasovskii formulation supplies a convex route. The paper constructs positive, self-adjoint, structure-preserving Lyapunov operators $\tau_2\le \tau_1$1, rewrites the stability condition in dual form, and then expresses positivity and negativity of the associated quadratic forms as SOS constraints and LMIs. Its stated significance is that these results open the way for dynamic output $\tau_2\le \tau_1$2 optimal control of systems with multiple time-delays (Peet, 2016).

6. Applications, limitations, and open directions

The applications already span several distinct scientific programs. In delayed-feedback nonlinear dynamics, adding the second delayed-feedback loop reproduces the effects of unidirectional nonlocal coupling of radius two and can speed up deterministic and stochastic wavefront propagation, achieve stabilization of propagating fronts at lower noise intensity, and prevent fronts from noise-induced destruction that occurs with single delayed-feedback (Semenov, 11 Dec 2025). In wireless sensing, dual-chirp AFDM is presented as the first framework to explicitly design AFDM for RAQR-based joint delay-Doppler estimation, while DDAM exploits the disparity between path invariant time and channel coherence time to reduce pilot overhead and improve spectral efficiency and PAPR relative to OFDM (Kim et al., 13 Mar 2026, Xiao et al., 2023). In learning, dual-track or dual-alignment constructions are used to bridge delayed supervision or asynchronous communication, and in control they enable exact delay-domain stability maps or convex synthesis (Wang et al., 22 Jun 2026, Song et al., 9 Jan 2025, Cepeda-Gomez et al., 2014, Peet, 2016).

Several limitations are also explicit. The bistable oscillator work is scalar and produces a one-dimensional pseudo-space with unidirectional ring topology; higher-dimensional analogues would require more sophisticated multi-delay architectures (Semenov, 11 Dec 2025). The RAQR study focuses on a single target in a quasi-monostatic setup and assumes known reference delay $\tau_2\le \tau_1$3 (Kim et al., 13 Mar 2026). DT-GOL is presented for online binary classification with fixed label delay and incurs graph-construction overhead that can be quadratic in buffer size without approximation [(Wang et al., 22 Jun 2026)