Fractional-Delay Mechanisms in Dynamic Systems

• Fractional-delay mechanisms are models that blend fractional-order operators with explicit delays to represent memory effects and nonlocal temporal dynamics.
• They enable accurate modeling in control systems, signal processing, and communications, capturing both finite-lag and distributed historical influences.
• Advanced analytical and numerical methods, including Mittag–Leffler functions and collocation schemes, facilitate precise stability and performance analysis.

Fractional-delay mechanisms are mathematical and engineering constructs in which the effect of time delay is coupled with memory via fractional-order operators. Unlike classical integer-order delay systems, fractional-delay models capture the combined influence of finite-lag (delay) and distributed, history-dependent (fractional) effects, leading to diverse and often richer dynamical regimes. These mechanisms pervade nonlinear and linear differential systems, control engineering, signal processing, communications, and scientific computation, enabling the modeling and analysis of phenomena with nonlocal temporal dependencies and asynchronous responses.

1. Mathematical Formulations of Fractional-Delay Systems

The core structure of a fractional-delay system incorporates a fractional-order differential (or difference) operator—typically in the Caputo, Riemann–Liouville, or Hilfer sense—in tandem with explicit time-lag terms. A canonical continuous-time prototype is

$$D_{t_0}^{\alpha} x(t) = f\left(x(t), x(t-\tau)\right),$$

where $D_{t_0}^{\alpha}$ is a Caputo derivative of order $0<\alpha<1$, and $\tau>0$ is the delay parameter. The Caputo derivative,

$${}_{t_0}^{C}D_t^{\alpha}f(t) = \frac{1}{\Gamma(1-\alpha)}\int_{t_0}^t (t-\sigma)^{-\alpha} f'(\sigma)\,d\sigma,$$

imbues the system with a power-law memory kernel, assigning algebraically decaying weights to past states. Discrete-time analogues leverage the fractional difference operator $\Delta^{\alpha} x(n)$, generating memory effects via binomial or convolutional kernels (Joshi et al., 2023).

The delay acts as a nonlocal, pointwise memory, introducing phase shifts and additional spectrum into the system. Multi-term fractional-delay systems generalize this further: $$\frac{d u}{dt} + \sum_{i=1}^m a_i {}^{C}D_t^{\alpha_i}u(t) + A u(t) = f\left(t, u(t-\nu)\right),$$ with $A$ an operator (possibly unbounded, accretive), and $\nu$ a fixed delay (Khatoon et al., 2024).

Engineering-oriented fractional-delay filters (in discrete time) are defined to delay signals by a non-integer multiple of the sampling interval, thereby manipulating intermediate, intersample behavior with high precision (Nagahara et al., 2013).

2. Dynamical Features and Stability Mechanisms

Fractional-delay systems exhibit a variety of dynamical behaviors due to the confluence of distributed memory and explicit temporal shift. Stability analysis reveals several key phenomena:

• Memory-induced regime shifts: The generalized kernel $(t-\sigma)^{-\alpha}$ renders the phase space non-Markovian, with stability boundaries and bifurcation loci deformed relative to their integer-order counterparts (Tuan et al., 2017, Yu et al., 2019, Bhalekar et al., 2024).
• Spectral conditions: Asymptotic stability for systems such as

$$-D^{\alpha}x(t) = A x(t-\tau) + g(x(t), x(t-\tau)),$$

requires the spectrum of $A$ to lie in a “fractional-delay stability sector” $S_{\alpha,\tau}$, determined by the roots of $s^{\alpha} - \lambda e^{-\tau s} = 0$ and $|\arg \lambda| > \tfrac12\alpha\pi$ (Tuan et al., 2017).

• Stability switches and SSR/SS regions: For multiple delays, e.g., $$D^{\alpha} x(t) = a x(t) + b x(t-\tau) - b x(t-2\tau)$$, the (a,b)-plane partitions into regions where stability holds for all delays (S), is lost for all (U), or where a critical delay bifurcation threshold induces a switch between regimes (SSR, SS). The location of these boundaries is modulated continuously by the fractional order α (Bhalekar et al., 2024).

Fractional delay also leads to phenomena such as Hopf bifurcation at non-integer delays, multi-window chaos, and quasi-periodic orbits, as demonstrated in epidemiological and immunological models (Yu et al., 2019).

3. Analytical and Computational Methods

Analytical solutions for fractional-delay systems involve a range of specialized techniques:

• Mittag–Leffler and Delayed Fractional Functions: The fundamental solution to homogeneous systems with Hilfer, Caputo, or Riemann–Liouville derivatives and delays can be expressed via delayed Mittag–Leffler type functions and their generalizations (cosine, sine analogues). These kernels allow explicit representation and stability estimates for initial value problems (Mahmudov, 2022).
• Semi-discretization and Numerical Schemes: Rothe's time semi-discretization, L1 formulae, and Galerkin finite element methods are used to approximate strong or mild solutions, with discrete Grönwall inequalities adapted for the nonlocal, delayed setting (Khatoon et al., 2024, Li et al., 2021).
• Mixed Steps–Collocation Approaches: For nonlinear fractional-delay DDEs, the method of steps eliminates the delay on each subinterval, and collocation techniques using shifted Legendre or Chebyshev polynomial bases (with explicit Caputo derivative expansions) produce high-accuracy spectral schemes (Mousa-Abadian et al., 2019).

In the context of signal processing and communications, Farrow filters enable efficient, parameterized fractional-delay filtering via polynomial approximation of sub-sample shifts. For sparse wireless channels or MIMO, fractional delay alignment (fDAM) combines upsampling and Farrow-based pre-compensation to eliminate ISI due to non-integer multipath (Zhou et al., 2024).

4. Applications Across Scientific and Engineering Domains

Fractional-delay mechanisms are utilized in diverse contexts:

• Biomodeling: Fractional-order delay models provide a natural framework for immune response and viral dynamics, as the nonlocal memory mirrors biological history dependence. In the hepatitis B virus model,

$${}_{t_0}^{C}D_t^{\alpha}x(t) = \lambda - d x(t) - \beta x(t) y(t),$$

with a CTL response term delayed by τ, the resulting dynamics display period-doubling and multiple chaotic windows, reflecting clinical phenomena (Yu et al., 2019).

• Control and Viscoelasticity: In finite- or infinite-dimensional systems on Hilbert spaces, fractional delays model viscoelastic aftereffects, distributed transport, and processing lags. Approximate controllability results leverage semigroup theory and Schauder's fixed-point theorem (Debbouche et al., 2013).
• Communications and Signal Processing: Fractional delay filters enable accurate sub-sample shifting indispensable for fractional symbol alignment, beamforming, or delay alignment modulation in communications, outperforming standard OFDM in ISI suppression and spectral efficiency (Nagahara et al., 2013, Zhou et al., 2024).
• System Identification and Model Reduction: The First-Order Plus Fractional Diffusive Delay (FOPFDD) model approximates complex RC-ladder networks, capturing both delay and “smeared” diffusive tails with a minimal parameter set (Juchem et al., 2021).

Typical applications include epidemiological modeling with memory and incubation lags, neural networks with both power-law synaptic plasticity and transmission delays, control of viscoelastic/dielectric materials, and discrete-time nonlinear maps with fractional memory and delay (Joshi et al., 2023).

5. Numerical and Algorithmic Realizations

Efficient and stable numerical solution is essential for practical deployment:

• Time-discretization and Finite Element Methods: For parabolic PDEs with nonlinear fractional delay, discrete convolution quadrature along with spatial FE leads to fully discrete schemes with proven convergence rates $O(\Delta t^2 + h^{r+1})$, supported by new discrete fractional Grönwall inequalities with delay dependencies (Li et al., 2021).
• Spectral Collocation: The Caputo derivatives of shifted Legendre/Chebyshev polynomials are leveraged to construct highly accurate collocation schemes, enabling “step-elimination” of delay and rapid convergence in applications with smooth solutions (Mousa-Abadian et al., 2019).
• Prony-Based Estimation: In high-mobility ISAC systems, two-stage Prony stacking and DFT-based decomposition jointly estimate continuous fractional delay and Doppler shifts with sub-sample accuracy, outperforming classical AIC/BIC model selection (Jitsumatsu et al., 21 Jun 2025).
• Farrow Filtering and fDAM: For sub-sample delay in digital communication, polynomial Farrow filters serve as a universal mechanism for fractional-delay realization with fixed coefficients, obviating real-time optimization and facilitating deployment in large-scale MIMO and beamforming applications (Zhou et al., 2024).

6. Theoretical Implications and Open Directions

Fractional-delay mechanisms radically expand the descriptive and analytic capacity of dynamical system models:

• Memory–Delay Interplay: The coaction of memory kernels and explicit delays admits “reentrant” stability, multiple Hopf bifurcations, and rich bifurcation scenarios absent in pure delay or pure fractional systems (Bhalekar et al., 2024).
• Spectral and Control Insights: The parameter dependence of critical delay/α values offers levers for control intervention. For example, therapeutic modulation in immune models or PAPR/efficiency trade-offs in communications can be optimized by tuning α and τ (Yu et al., 2019, Zhou et al., 2024).
• Numerical and Model Reduction Advances: Fractional delay structures allow accurate low-order representations of inherently high-dimensional, complex-memory systems—e.g., diffusive-resistor networks—while preserving both short- and long-term kinetic features (Juchem et al., 2021).

Future research is poised to address provable performance bounds (CRLB in estimation), integration of machine learning for structure and parameter selection, extension to distributed multi-delay/fractional systems, and generalizations to stochastic fractional-delay equations. The evidence from diverse fields indicates that fractional-delay mechanisms are indispensable for modeling and analyzing systems governed by both hereditary and delayed feedback effects.