Retarded Differential Equations (RDE)

• RDEs are functional differential equations characterized by their dependence on past values, introducing memory effects into system dynamics.
• Key analytical results include exponential bounds and spectral theory, which are instrumental in establishing existence, stability, and control via multiplicity-induced dominancy.
• Advanced numerical methods such as recursive ODE embedding, collocation, and randomized schemes are developed to address the challenges posed by the infinite-dimensional history space.

A retarded differential equation (RDE) is a type of functional differential equation in which the evolution of a state variable at time $t$ depends not only on its current value and its derivatives but also on its values at previous times. This class of equations embodies memory effects and finite propagation speeds inherent in numerous real-world systems, such as control engineering, population biology, epidemiology, and physical sciences. The "retarded" terminology distinguishes these equations by the absence of delayed highest derivatives, in contrast to neutral and advanced types. Mathematically, RDEs bridge ordinary differential equations (ODEs) and delay differential equations (DDEs), presenting unique analytical and numerical challenges due to their infinite-dimensional phase space and history dependence.

1. Formulation and Core Properties

A general linear RDE of order $n$ with a single delay $\tau > 0$ takes the canonical form: $$y^{(n)}(t) + a_{n-1} y^{(n-1)}(t) + \cdots + a_0 y(t) + \alpha_{n-1} y^{(n-1)}(t - \tau) + \cdots + \alpha_0 y(t-\tau) = 0,$$ where delayed terms are present only in lower-order derivatives or the function itself (Mazanti et al., 2020). More generally, a nonlinear or time-varying RDE may be expressed as: $$x'(t) = f\left(t, x(t), x(t-\tau_1), \ldots, x(t-\tau_m)\right),$$ with $f$ continuous and suitable initial history prescribed for $t \leq t_0$.

Crucial features of RDEs:

• The phase space is infinite-dimensional, as the solution at $t_0$ depends on a segment $[t_0 - r, t_0]$ of the history.
• The solution $x(t)$ typically has points of non-smoothness (for example, discontinuities in $x'(t)$) that propagate due to the delay.
• The spectral theory for linear RDEs is governed by characteristic equations of quasipolynomial type, blending polynomial and exponential dynamics.

2. Analytical Bounds and Existence Theory

A central theoretical advance is the derivation of explicit exponential a priori bounds on the combined sum of the absolute values of the solution and its lower-order derivatives. For the $n$-th order retarded equation

$$L(y) = y^{(n)}(t) + M_1(t) y^{(n-1)}(t - A(t)) + \cdots + M_n(t) y(t - A(t)) = 0,$$

where $A(t)$ is a continuous delay function and $M_j(t)$ are bounded coefficient functions, if we define the auxiliary quantity

$u(t) = |w(t)| + |w'(t)| + \cdots + |w^{(n-1)}(t)|,$

then, for some constant $\Upsilon$, the two-sided exponential estimate holds (Şen, 2013): $$e^{-2\Upsilon|t-t_0|} \leq \frac{u(t)}{u(t_0)} \leq e^{2\Upsilon|t-t_0|}.$$ This bound is obtained via the mean-value theorem and integrating factor techniques, encapsulating the maximum possible exponential growth or decay over finite intervals and serving as a key tool for existence and uniqueness proofs.

Setting $A(t) = 0$ recovers the analogous ODE estimate. These bounds are critical in establishing that sequences of approximate solutions (for example, via iteration or fixed-point techniques) remain uniformly bounded and converge—a cornerstone in well-posedness results for RDEs.

3. Spectral Theory and Multiplicity-Induced Dominancy

The spectral analysis of linear RDEs relies on the roots of the quasipolynomial characteristic function

$$\Delta(s) = s^n + \sum_{k=0}^{n-1} a_k s^k + e^{-s\tau} \sum_{k=0}^{n-1} \alpha_k s^k.$$

A distinctive phenomenon is multiplicity-induced dominancy (MID), wherein if the system is engineered such that a real root $s_0$ attains maximal algebraic multiplicity $2n$, then $s_0$ becomes the strictly dominant eigenvalue: all other roots $s$ satisfy $\mathrm{Re}(s) < s_0$ (Mazanti et al., 2020, Mazanti et al., 2020). The key steps are:

• Construction of coefficient sets yielding $s_0$ as a root of multiplicity $2n$.
• An explicit connection between this situation and Kummer's confluent hypergeometric function $M(n,2n+1,-z)$, based on an affine transformation of $\Delta(s)$ and integral identities.
• Proofs that, under this maximal multiplicity, the spectral abscissa (governing stability and decay rates) coincides with $s_0$, dramatically simplifying long-term system analysis and enabling pole-placement strategies in feedback control.

The paper (Mazanti et al., 2020) provides both necessary and sufficient algebraic conditions for coefficients guaranteeing this property, and demonstrates improved a priori bounds on the imaginary parts of any root with maximal real part, thus ruling out "quasi-dominance" by complex conjugate roots.

4. Oscillation Phenomena and Delay Structure

The dynamical behavior of RDEs—especially oscillatory solutions—is governed not just by coefficient sizes but critically by the structure of the delay functions. In first-order linear retarded equations with multiple delay terms,

$x'(t) + \sum_{i=1}^m p_i(t) x(\tau_i(t)) = 0,$

the analysis of oscillation can break down if the delays $\tau_i(t)$ are non-monotone. Novel criteria, based on limsup (rather than liminf) of weighted integrals involving the coefficients and the delay structure, have been developed to address this (Infante et al., 2013). The main result asserts that if

$$\limsup_{t\to\infty} \prod_{i=1}^m \left\{ \int_{T^*(t)}^t p_i(s) \exp\left\{\int_{\omega_i(t)}^{T_i(s)} p_i(\xi)d\xi\right\} ds \right\} > \frac{1}{m^m},$$

then all solutions oscillate, even in the presence of non-monotonic or "wiggling" delays. These results substantially extend the range of delay equations for which the presence of infinitely many solution zeros (oscillation) can be analytically guaranteed.

5. Numerical Methods and Computational Techniques

The infinite-dimensional character and discontinuity propagation in RDEs necessitate tailored numerical schemes.

Recursive ODE Embedding

The DelayDiffEq framework (Widmann et al., 2022) leverages recursive embedding of high-order ODE solvers—adapted from the OrdinaryDiffEq suite—using a method-of-steps approach. At each interval $[t_k, t_{k+1}]$, an ODE initial-value problem is solved with history input interpolated from previous intervals. To address error accumulation from history extrapolation and step sizes that may exceed $\tau$, a fixed-point iteration (possibly Anderson accelerated) is employed to achieve self-consistency in the delay arguments.

Discontinuity detection and management—critical for maintaining accuracy in the context of propagated derivative jumps—are handled by root-finding algorithms and dynamic step size control.

Collocation and Polynomial Expansion

For global approximations, collocation methods based on Laguerre polynomial expansions (Gürbüz, 2021) or general piecewise polynomials (andò et al., 2020) have been developed. These methods enforce the residual equation at selected nodes, leading to an algebraic system for the expansion coefficients. The explicit operational matrix representations of derivatives and delay-shifted terms in the Laguerre approach facilitate symbolic and efficient computation. Error estimates in various norms (e.g., $L_2$, max, RMS) confirm high accuracy and highlight convergence properties as the degree of expansion increases.

General Linear Methods and Uniform Stage Order

Two-Step General Linear Methods (TSRK) (Tuzov, 2017) address the order-reduction phenomenon in the presence of mild stiffness by balancing the uniform order and stage order through polynomial coefficient design. By maximizing the uniform stage order—i.e., ensuring high-order accuracy for the full function segment, not just at mesh points—TSRK methods maintain high convergence order even when the problem exhibits boundary or delay-induced stiffness.

Randomized Numerical Methods

For RDEs with time- or history-irregular right-hand sides ($f$ only Hölder continuous), randomized Runge–Kutta schemes (Difonzo et al., 22 Jan 2024) have been proposed. These schemes introduce randomness in time nodes at each integration step, which allows for averaging out irregularities and achieving higher convergence rates in the mean-square sense compared to deterministic Euler methods under weak smoothness assumptions of $f$.

6. Dynamical Systems, Attractors, and Dimension Reduction

A significant direction concerns the infinite-dimensional dynamics induced by RDEs. Exponential attractors—compact, invariant sets with explicit fractal dimensions—can be constructed in Banach spaces under squeezing property assumptions, with the fractal dimension estimate depending only on spectral gaps and Lipschitz constants rather than the entropy of embeddings (Hu et al., 2023). This advances the quantification of asymptotic complexity for retarded systems and retarded reaction-diffusion PDEs.

Furthermore, model reduction methodologies for systems driven by rough paths (rough differential equations) (Redmann et al., 2023) exploit covariance operators and Lyapunov analysis to identify invariant subspaces supporting all solution trajectories, enabling projection onto minimal-dimensional submanifolds. Though formulated for rough systems, this approach is adaptable to RDEs, particularly for compressing high-dimensional discretized models while preserving accuracy for quantities of interest.

7. Applications and Future Directions

RDEs underpin a broad spectrum of applied models:

• Biological and epidemiological systems: e.g., delayed feedback in disease modeling and physiological cycles.
• Control and engineering: design of feedback controllers with actuation/measurement delays, where spectral placement and decay rate optimization are critical (Mazanti et al., 2020).
• Quantum photonics: advanced-retarded equations engineered in photonic circuits for quantum feedback/quantum memory without measurement, via mapped waveguide arrays (Alvarez-Rodriguez et al., 2016).
• Lattice and continuum models: as found in retarded reaction-diffusion systems, economic models with gestational lags, and electromechanical systems with transmission delays.

Open challenges include rigorous extension of convergence proofs to spectral element methods for periodic RFDEs, further development of robust algorithms for strongly state-dependent delays, and the implementation of model reduction principles for semi-discretized or high-dimensional retarded and rough systems. Advances in these directions promise increased computational tractability and deeper theoretical insight into time-delayed dynamical systems.