Empirical Rules for Oscillation and Harmonic Approximation of Fractional Kelvin–Voigt Oscillators

Featured Application

The results provide simple approximate formulas to determine whether a fractional Kelvin–Voigt (FKV) oscillator exhibits oscillatory behavior and to identify the best-matching classical harmonic oscillator for a given set of FKV oscillator parameters. These formulas can assist in the analysis of vibrations in viscoelastic structures modeled by the fractional Kelvin–Voigt constitutive relation.

Abstract

Fractional Kelvin–Voigt (FKV) oscillators describe vibrations in viscoelastic structures with memory effects, leading to dynamics that are often more complex than those of classical harmonic oscillators. Since the harmonic oscillator is a simple, widely known, and broadly applied model, it is natural to ask under which conditions the dynamics of an FKV oscillator can be reliably approximated by a classical harmonic oscillator. In this work, we develop practical tools for such analysis by deriving approximate formulas that relate the parameters of an FKV oscillator to those of a best-fitting harmonic oscillator. The fitting is performed by minimizing a so-called divergence coefficient, a discrepancy measure that quantifies the difference between the responses of the FKV oscillator and its harmonic counterpart, using a genetic algorithm. The resulting data are then used to identify functional relationships between FKV parameters and the corresponding frequency and damping ratio of the approximating harmonic oscillator. The quality of these approximations is evaluated across a broad range of FKV parameters, leading to the identification of parameter regions where the approximation is reliable. In addition, we establish an empirical criterion that separates oscillatory from non-oscillatory FKV systems and employ statistical tools to validate both this classification and the accuracy of the proposed formulas over a wide parameter space. The methodology supports simplified modeling of viscoelastic dynamics and may contribute to applications in structural vibration analysis and material characterization.

1. Introduction

Fractional calculus has gained significant attention in recent decades due to its ability to accurately describe systems with memory and hereditary properties, which are commonly encountered in physics, engineering, biology, and materials science. In contrast to classical integer-order models, fractional derivatives provide a natural framework for modeling anomalous diffusion, viscoelasticity, heat conduction, etc. [1,2,3,4,5,6]. Their use has proven particularly valuable in viscoelastic modeling, where traditional models often fail to capture the complex time-dependent response of real materials [7]. Among the various fractional models, the fractional Kelvin–Voigt (FKV) model has emerged as a popular choice for describing damping in materials exhibiting both elastic and viscous behavior with memory effects [8,9,10].

The fractional Kelvin–Voigt (FKV) model has been extensively applied in the study of vibrations of various mechanical structures, including rods, beams, and plates [11,12,13]. To solve the governing dynamic equations arising from such models, methods like the Rayleigh–Ritz and Galerkin approaches are commonly employed [14,15]. These methods facilitate the decomposition of the governing equations into spatial and temporal components, allowing for analytical or numerical solutions of the resulting time-fractional differential equations. Usually, the spatial components obtained through these methods are orthogonal, reducing the problem to solving time-fractional differential equations.

A fundamental time-domain model resulting from such modal decompositions is commonly referred to in the literature as the fractional Kelvin–Voigt (FKV) oscillator. This terminology was introduced by Rossikhin and Shitikova in their works on the use of fractional derivatives for modeling damped vibrations in viscoelastic single-degree-of-freedom systems [16,17]. The FKV oscillator is governed by the following differential equation:

$$\ddot{x}\left(t\right)+\mu D^{\alpha }x\left(t\right)+{\omega }^{2}x\left(t\right)=f(t)$$

(1)

where $x(t)$ denotes the displacement, $f(t)$ is the external forcing function, $\mu >0$ is the fractional damping coefficient, $\omega$ is the undamped natural frequency, and $D^{\alpha }$ represents the fractional derivative of order $0<\alpha <1$.

The present work continues the research initiated in [18,19]. In [18], the FKV oscillator was obtained as a model describing the vibrations of a cantilever beam with a tip mass subjected to base excitation, corresponding to an assumed mode shape of the beam’s natural vibrations. That study investigated the approximation of the fractional model by a classical harmonic oscillator. Two methods were proposed for identifying the optimal parameters of the approximating harmonic oscillator: one based on a genetic algorithm minimizing the so-called divergence coefficient—a measure of discrepancy between the models introduced in [19]—and the other based on geometric similarity, using the shape of the response curve. The analysis was restricted to the Caputo derivative case, and it was demonstrated through several examples that the proposed approach yields satisfactory results.

In [19], the focus was placed on the Scott–Blair oscillator, which corresponds to the FKV oscillator with vanishing stiffness ($\omega =0$ in Equation (1)). Among the main contributions of that work was the derivation of approximate formulas that allow the determination of a harmonic oscillator corresponding to a given FKV oscillator. The results showed that, while the frequency approximation was highly accurate across a range of cases, the formula for the damping ratio tended to overestimate this quantity. Moreover, the study raised an important theoretical question: under what conditions does an FKV oscillator exhibit oscillatory behavior? This open question serves as one of the central motivations for the present investigation.

Several studies have addressed the problem of oscillation criteria in the context of fractional differential equations. In particular, the work [20] established sufficient conditions for oscillation based on generalized Riccati transformation techniques and inequality methods. These results apply primarily to scalar equations with fractional order derivatives and provide theoretical tools for distinguishing oscillatory from non-oscillatory behavior. More recently, in [21], the authors proposed novel oscillation criteria for kernel-function-dependent fractional dynamic equations using a general framework of nonlocal fractional derivatives. Their analysis, though general and mathematically rigorous, focuses primarily on abstract differential systems and does not directly address the specific structure of the FKV oscillator.

In contrast to these approaches, the methodology adopted in the present paper is empirical and application-oriented. It is based on fitting data obtained from numerical simulations of the FKV oscillator. A suitable classification criterion is proposed to distinguish oscillatory and non-oscillatory solutions, allowing for the construction of approximate boundaries in parameter space.

Another objective of this study is to derive approximate formulas that allow the estimation of the parameters of a classical harmonic oscillator corresponding to a given fractional Kelvin–Voigt (FKV) oscillator. This task was accomplished through empirical fitting to numerical data. To achieve this, for a selected range of FKV oscillator parameters, corresponding harmonic oscillators were identified by minimizing the divergence coefficient using a genetic algorithm, following the methodology introduced in [18]. The resulting harmonic oscillator parameters were then used to construct empirical relationships linking them to the parameters of the original FKV oscillator. Classical oscillator models themselves have numerous applications, for example, in modeling energy harvesting devices [22,23].

Throughout this analysis, the homogeneous form of the governing equation (i.e., Equation (1) with $f\left(t\right)=0$) is considered. Since the forcing term represents an external input rather than an inherent part of the FKV model, focusing on the unforced case provides a clearer basis for comparing the intrinsic dynamic properties of different systems.

Similar efforts to relate fractional and classical oscillator models have been reported in the literature. One study employed frequency-domain identification techniques to estimate fractional Kelvin–Voigt parameters based on harmonic response data [24]. Another work proposed energy-based equivalent differential equations for single-degree-of-freedom fractional systems, offering an interpretation of their dynamics in terms of classical analogs [25]. Fractional homogenization approaches were also applied to viscoelastic metamaterials, demonstrating how effective harmonic behavior can emerge from systems governed by fractional damping laws [26].

In the present study, the time-domain response of fractional Kelvin–Voigt (FKV) oscillators was computed over a systematically sampled range of model parameters. For each configuration, the solution was analyzed using a predefined classification criterion to determine whether the response is oscillatory or non-oscillatory. In the case of oscillatory behavior, a corresponding classical harmonic oscillator was identified by minimizing a mismatch measure between the two responses called divergence coefficient. The resulting data—including the FKV oscillator parameters, the classification outcome, the best-fitting harmonic oscillator parameters (when applicable), and the quality of fit—were recorded in a structured dataset using the HDF5 format. A reference to the dataset repository is provided later in the paper.

It is also demonstrated in this work that for small values of the order of the fractional derivative $\alpha$, the solution of the FKV oscillator with the Caputo derivative may fail to exhibit oscillations around zero during the initial phase of the response, depending on the values of the remaining parameters. This undesirable behavior is not observed in the case of the Riemann–Liouville derivative, which motivates its exclusive use in the present analysis. This contrasts with the approach adopted in [18], where the Caputo derivative was employed. For the examples considered in that study, the issue of initial non-oscillatory behavior did not occur.

The structure of the paper is as follows. Section 1 introduces the problem, outlines the motivation, and provides a brief review of relevant literature. Section 2 describes the mathematical formulation of the problem and the numerical approach used to generate and analyze the data. It also clarifies the goals and scope of the study. Section 3 presents the main results, including the validation of the numerical method, the formulation and evaluation of an empirical criterion for oscillations, and the development of formulas for the parameters of an equivalent harmonic oscillator. Section 4 discusses the findings, and Section 5 summarizes the conclusions.

2. Problem Formulation and Numerical Methodology

2.1. Analytical Solutions

This section focuses on analytical solutions to the unforced fractional Kelvin–Voigt (FKV) oscillator equation. Specifically, we consider the homogeneous form of Equation (1), which reads:

$$\ddot{x}\left(t\right)+{\mu D}^{\alpha }x\left(t\right)+{\omega }^{2}x\left(t\right)=0$$

(2)

where $\mu >0$, $\omega >0$, and $\alpha \in (0,1)$. The initial conditions are assumed in the classical form: $x\left(0\right)=x_0$, $\dot{x}\left(0\right)=v_0$. The analytical results discussed here are derived using theorems presented in [27], which provide tools for solving linear fractional differential equations.

Based on Theorem 5.2 and Proposition 5.2 from [27] the solution of Equation (2) for the Riemann–Liouville fractional derivative takes the form:

$$x_{RL}\left(t\right)=x_0{x}_1^{\left(RL\right)}\left(t\right)+v_0{x}_2^{\left(RL\right)}\left(t\right)$$

(3)

where

$$x_1^{\left(RL\right)}\left(t\right)=\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-1\right)}^{n+k}\left(\begin{array}{c}n+k\\ n\end{array}\right){\omega }^{2n}{\mu }^k}{\Gamma \left(2\left(n+k\right)-\alpha k+1\right)}}{t}^{2\left(n+k\right)-\alpha k},$$

(4)

$$x_2^{\left(RL\right)}\left(t\right)=t\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-1\right)}^{n+k}\left(\begin{array}{c}n+k\\ n\end{array}\right){\omega }^{2n}{\mu }^k}{\Gamma \left(2\left(n+k\right)-\alpha k+2\right)}}{t}^{2\left(n+k\right)-\alpha k}$$

(5)

In the case of the Caputo fractional derivative, a similar representation can be obtained. Based on Theorem 5.13 and Proposition 5.8 from [27], the solution for Equation (2) in the case of the Caputo fractional derivative takes the form:

$$x_C\left(t\right)=x_0{x}_1^{\left(C\right)}\left(t\right)+v_0{x}_2^{\left(C\right)}\left(t\right)$$

(6)

where

$$x_1^{\left(C\right)}\left(t\right)=x_1^{\left(RL\right)}\left(t\right)+\mu t^{2-\alpha }\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-1\right)}^{n+k}\left(\begin{array}{c}n+k\\ n\end{array}\right){\omega }^{2n}{\mu }^k}{\Gamma (2\left(n+k\right)-\alpha (k+1)+3)}}{t}^{2\left(n+k\right)-\alpha k},$$

(7)

$$x_2^{\left(C\right)}\left(t\right)=x_2^{\left(RL\right)}\left(t\right).$$

(8)

Equation (2) is structurally similar to the equation governing the classical harmonic oscillator. In that case, the fractional derivative is replaced by a standard first-order time derivative, leading to the well-known equation:

$$\ddot{x}\left(t\right)+2\xi {\omega }_{0 }\dot{x} \left(t\right)+{\omega }_0^{2}x\left(t\right)=0,$$

(9)

where ${\omega }_0>0$ and $0<\xi <1$ are the natural frequency and damping ratio, respectively. For the reader’s convenience and to facilitate the interpretation of the results presented later in this paper, we briefly recall the form of the solutions to Equation (9) under the same initial conditions as those considered for the FKV oscillator:

$$x\left(t\right)=x_0{x}_1^{\left(HO\right)}\left(t\right)+v_0{x}_2^{\left(HO\right)}\left(t\right),$$

(10)

where

$$x_1^{\left(HO\right)}\left(t\right)=e^{-\xi {\omega }_{0}t}\left(cos\left({\omega }_0\sqrt{1-{\xi }^2} t\right)+{\displaystyle \frac{\xi \mathit{sin}\left({\omega }_0\sqrt{1-{\xi }^2} t\right)}{\sqrt{1-{\xi }^2}}}\right),$$

(11)

$$x_2^{\left(HO\right)}\left(t\right)={\displaystyle \frac{e^{-\xi {\omega }_{0}t}\mathit{sin}\left({\omega }_0\sqrt{1-{\xi }^2} t\right)}{{\omega }_0\sqrt{1-{\xi }^2}}}.$$

(12)

2.2. Asymptotic Behavior of the FKV Oscillator

In this section, we examine the structure of the solution and its behavior for asymptotic cases of the fractional order $\alpha$, namely $\alpha =0$ and $\alpha =1$.

By substituting $\alpha =0$ into Equation (2), we formally recover the equation of an undamped harmonic oscillator. It can be shown that, in this case, the solution for the Riemann–Liouville derivative (Equations (3)–(5)) reduces exactly to the solution of the classical harmonic oscillator (Equations (10)–(12)) with $\xi =0$ and ${\omega }_0=\sqrt{{\omega }^2+\mu }$. However, when examining the definition of $x_1^{\left(C\right)}\left(t\right)$ (see Equation (7)), we observe the presence of an additional factor compared to $x_1^{\left(RL\right)}$(t). Let us now analyze the structure of this factor for $\alpha =0$:

$$\begin{array}{c}{x}_1^{\left(C\right)}\left(t\right)-x_1^{\left(RL\right)}\left(t\right)=\mu t^2{\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \sum _{k=0}^{\infty }}\frac{{\left(-1\right)}^{n+k}\left(\begin{array}{c}n+k\\ n\end{array}\right){\omega }^{2n}{\mu }^k}{\left(2\left(n+k\right)+2\right)!}{t}^{2\left(n+k\right)}\\ =\mu t^2{\displaystyle \sum _{m=0}^{\infty }}{\displaystyle \sum _{k=0}^m}\frac{{\left(-1\right)}^m\left(\begin{array}{c}m\\ k\end{array}\right){\omega }^{2\left(m-k\right)}{\mu }^k}{(2(m+1))!}{t}^{2m}\\ =\mu t^2{\displaystyle \sum _{m=0}^{\infty }}\frac{{\left(-1\right)}^m{t}^{2m}}{\left(2(m+1)\right)!}{\left({\omega }^2+\mu \right)}^m=\mu {\displaystyle \sum _{l=1}^{\infty }}\frac{{\left(-1\right)}^{l-1}{t}^{2l}}{\left(2l\right)!}{\left({\omega }^2+\mu \right)}^{l-1}\\ =\frac{\mu }{{\omega }^2+\mu }\left(1-\mathit{cos}\left(\sqrt{{\omega }^2+\mu }\cdot t\right)\right).\end{array}$$

We first change the order of summation by introducing a new index $m=n+k$. The double sum is then written as a single sum over $m$ with an inner, finite sum over $k$, which is simplified using the binomial identity. Next, by shifting the summation index we arrive at a more compact single series. Finally, recognizing this series as the Taylor expansion of the cosine function allows us to obtain the closed-form expression. Taking into account the asymptotic form of $x_1^{\left(RL\right)}(t)$ for $\alpha =0$ (Equation (11) with $\xi =0$ and ${\omega }_0=\sqrt{{\omega }^2+\mu }$), we obtain the following final expression for $x_1^{\left(C\right)}(t)$ for $\alpha =0$:

$$x_{1,a}^{\left(C\right)}\left(t\right)={\displaystyle \frac{\mu }{{\omega }^2+\mu }}+{\displaystyle \frac{{\omega }^2}{{\omega }^2+\mu }}\mathit{cos}\left(\sqrt{{\omega }^2+\mu }·t\right).$$

(13)

To distinguish this limiting case from the general solution $x_1^{\left(C\right)}(t)$ we denote it as $x_{1,a}^{\left(C\right)}(t)$. Let us note that the expression contains a constant term. This suggests that for small values of $\alpha$, a similar constant component is likely to appear in the component $x_1^{\left(C\right)}(t)$. Consequently, one may infer that, in general, there exists no classical harmonic oscillator that can fully replicate the behavior of the FKV oscillator governed by the Caputo derivative. However, for small fractional orders $\alpha$ and in regimes where $\omega \gg \mu$, such as in the examples studied in [18], the discrepancy becomes negligible, and a meaningful approximation by a classical oscillator becomes possible.

For $\alpha =1$, analogous calculations to those performed for $\alpha =0$ become difficult to carry out analytically. However, the results of numerical experiments presented in [18] for $\alpha =1$ indicate a reversed situation: in this regime, the Caputo-based component $x_1^{\left(C\right)}(t)$ converges to the classical harmonic oscillator solution. The Riemann–Liouville-based component $x_1^{\left(RL\right)}(t)$ also exhibits harmonic behavior, but with different parameters (see Figures 2 and 3 in [18], and the corresponding discussion).

To illustrate the behavior described above for $\alpha =0$, we now present two representative examples of the functions $x_1^{\left(C\right)}(t)$ and $x_1^{\left(RL\right)}(t)$ for the Caputo and Riemann–Liouville derivatives, respectively. In particular, we highlight cases where the Caputo-based solution does not oscillate around zero, in contrast to its Riemann–Liouville counterpart. The solutions were computed using the Wynn–epsilon acceleration method, as employed in [18], which ensures reliable convergence of the series representations.

Figure 1 shows the functions $x_1^{\left(RL\right)}(t)$ and $x_1^{\left(C\right)}(t)$ for two different parameter sets. The blue solid line represents $x_1^{\left(RL\right)}(t)$, while the red dashed line corresponds to $x_1^{\left(C\right)}(t)$. In Figure 1a, for $\mu =10$ and $\omega =1$, the constant term in the asymptotic Caputo-based solution for $\alpha =0$, i.e., $x_{1,a}^{\left(C\right)}(t)$, equals $\mu /\left(\mu +{\omega }^2\right)=10/11$, which is significant. As expected, a similar effect is observed for small $\alpha$ (here $\alpha =0.05$): the solution $x_1^{\left(C\right)}(t)$ does not oscillate around zero. In contrast, Figure 1b corresponds to $\mu =1$ and $\omega =10$, where the constant term in the asymptotic solution is $1/101$, which is negligible. In this case, the two curves $x_1^{\left(RL\right)}(t)$ and $x_1^{\left(C\right)}(t)$ almost overlap throughout the entire time interval. These results confirm that for small $\alpha$, the behavior of the Caputo and Riemann–Liouville models becomes similar when $\omega \gg \mu$, while a clear difference persists when $\mu$ is large compared to $\omega$.

2.3. Computational Approach

In this section, we describe the computational methods used in our analysis of the FKV oscillator. The following aspects are addressed: (a) the procedure for determining a classical harmonic oscillator that best approximates a given FKV oscillator, (b) the criterion used to classify solutions as oscillatory or non-oscillatory, (c) the parameter range considered and the structure of the dataset created and used throughout the analysis, (d) the method for computing the solution components, and (e) the application of statistical tools to validate the proposed formulas and the classification criterion over a broader parameter range than the one directly covered by the created dataset.

To determine the classical harmonic oscillator that best approximates a given FKV oscillator, we follow the procedure introduced in [18]. The optimal harmonic oscillator is defined as the one that minimizes the so-called divergence coefficient $I({\omega }_0,\xi )$, which quantifies the difference between the two solutions. The minimum of this function is found using a genetic algorithm. The corresponding minimum value of the divergence coefficient is denoted by ${\epsilon }_m$, while the parameters of the harmonic oscillator that achieve this minimum are denoted by ${\omega }_0^{\left(m\right)}$ and ${\xi }^{\left(m\right)}$. The definition of the divergence coefficient $I({\omega }_0,\xi )$ can be found in [18,19]. The divergence coefficient measures the difference between two oscillators: $I(O_1$,$O_2)$ is smaller when the oscillators $O_1$ and $O_2$ are more similar. It is defined using the supremum norm, which allows a simple interpretation as the maximum relative difference between the solution components. Above we write $I({\omega }_0,\xi )$ because the FKV oscillator is fixed, while the parameters of the harmonic oscillator are to be determined. In general, we use the notation $I(O_1,O_2)$ to compare two oscillators.

To classify a given FKV oscillator as oscillatory or non-oscillatory, we apply a simple criterion based on the zero crossings of the solution components. Specifically, if either $x_1^{\left(RL\right)}(t)$ and $x_2^{\left(RL\right)}(t)$ does not attain a zero (with the exception of $x_2^{\left(RL\right)}\left(0\right)=0$), the solution $x(t)$ defined in Equation (3) is classified as non-oscillatory. Otherwise, the solution is considered to exhibit oscillatory behavior. The zeros of the components are computed numerically.

In our computations, we considered FKV oscillators with parameters $(\mu ,\alpha ,\omega )$ taken from the ranges $0<\mu \le 100$, $0<\alpha <1$, $0<\omega \le 100$. More precisely, we constructed a dataset containing values ${\mu }_i=1+\left(i-1\right)·1.5$, ${\alpha }_j=0.05·j$, ${\omega }_k=1+\left(k-1\right)·1.5$, where $i,k=\mathrm{1,2},\dots ,67, j=\mathrm{1,2},\dots ,19$. For each FKV oscillator with parameters $({\mu }_i,{\alpha }_j,{\omega }_k)$, the dataset stores a binary label isVibrating indicating whether the corresponding FKV oscillator exhibits oscillatory behavior (isVibrating = 1) or not (isVibrating = 0), according to the criterion described above. In addition, for all parameter combinations classified as oscillatory (isVibrating = 1), the dataset also includes the minimum of divergence coefficient ${\epsilon }_m$ and the corresponding optimal harmonic oscillator parameters ${\omega }_0^{\left(m\right)}$ and ${\xi }^{\left(m\right)}$. The full dataset is provided in HDF5 format and is available for download as part of the supplementary materials in [28]. It also includes an example MATLAB script illustrating how to access and use the data.

To compute the approximate solution of the FKV oscillator equation with the Riemann–Liouville derivative, previous work [18] employed a method based on the analytical form (3), combined with the Wynn–epsilon convergence acceleration technique. While this method is highly accurate, it is not suitable for our purposes due to two major limitations. First, the method requires careful tuning of parameters for each individual case (e.g., numerical precision, the number of terms in the truncated series), making it impractical for large-scale automated computations. Second, it is computationally expensive and time-consuming.

In this work, we adopt a fully automatic and significantly faster numerical method based on a finite-difference discretization of the Grünwald–Letnikov definition of the fractional derivative. For reference, the Grünwald–Letnikov derivative of order $0<\alpha <1$ is defined as:

$$D_{GL}^{\alpha }x\left(t\right)=\underset{n\to \infty }{\mathit{lim}}{\left({\displaystyle \frac{n}{t}}\right)}^{\alpha }\sum _{r=0}^n{\left(-1\right)}^r\left(\begin{array}{c}\alpha \\ r\end{array}\right)x\left(t-r{\displaystyle \frac{t}{n}}\right),$$

(14)

where the generalized binomial coefficient is given by

$$\left(\begin{array}{c}\alpha \\ r\end{array}\right)={\displaystyle \frac{\alpha \left(\alpha -1\right)\dots (\alpha -r+1)}{r!}}.$$

(15)

The subscript GL indicates the Grünwald–Letnikov fractional derivative. For functions $x(t)$ that are continuous and possess a derivative, the Grünwald–Letnikov derivative of order $0<\alpha <1$ is equivalent to the Riemann–Liouville fractional derivative [29].

To solve the FKV oscillator equation numerically using the Grünwald–Letnikov approximation, we first rewrite the original second-order equation as a system of two coupled first-order equations—a standard approach for such problems. We then apply the Grünwald–Letnikov approximation for the fractional derivative, omitting the limit in Equation (14) and using a fixed number of steps $n$. This leads to a discrete-time recurrence scheme in which the values of the solution (3) $x\left(t_i\right)$ and its derivative $\dot{x}\left(t_i\right)$ at time step $t_i=i·\Delta t$, with $\Delta t=T/n$, depend on their values at all previous steps $t_0=0, t_1,\dots ,t_{i-1}$. Here, $T$ denotes the total simulation time, and $i=0,1,\dots ,n$. To obtain $x_1^{\left(RL\right)}(t)$, we set the initial conditions $\left(x_0,v_0\right)=(1, 0)$; to compute $x_2^{\left(RL\right)}(t)$, we use $\left(x_0,v_0\right)=(0, 1)$.

Unfortunately, the discrete-time values of the components $x_1^{\left(RL\right)}\left(t_i\right)$ and $x_2^{\left(RL\right)}(t_i)$ obtained using the above method are highly sensitive to the choice of the time step $\Delta t$. In order to avoid selecting individually $\Delta t$ for each case, we implement an iterative refinement strategy described in the pseudocode below. This approach automatically adjusts the time step to ensure sufficient accuracy across a wide range of parameter values. Additionally, the algorithm returns a binary flag indicating whether the corresponding solution is classified as oscillatory or not, according to the criterion defined earlier. For simplicity, we define a helper function getComponents ($\mu ,\alpha , \omega , \Delta t, T$), which applies the method described above to an FKV oscillator with given parameters $\mu , \alpha , \omega$, time step $\Delta t$, and final simulation time $T$. It returns the components $x_1^{\left(RL\right)}\left(t_i\right)$ and $x_2^{\left(RL\right)}(t_i)$, and the classification flag isVibrating $\in \{0,1\}$.

The formula for estimating the initial simulation time $T$ in line 2 is based on the expression for the effective frequency introduced in [19]. In line 2, the tolerance parameter $\epsilon =0.05$ is set as a fixed value used to control the convergence of the iterative procedure. The quantity $r$ in line 6 is a divergence coefficient used in [18,19], but here it is applied to compare two successive approximations of the solution obtained using the function getComponents (…). The function getZeros (tab) is used to detect approximate zero crossings of the input array tab. It returns a list of time points $t_k$, such that the sign of two consecutive values changes between steps $i$ and $i+1$. The zero crossing is then approximated by the midpoint between the two successive time steps, i.e., $t_k=(t_i+t_{i+1})/2$.

The Wynn–epsilon-based method for computing the solution components was used in this work solely for validation purposes and for generating the reference data shown in Figure 1. All computations using this method were performed in Mathematica (version 12.1). In contrast, all numerical computations related to Algorithm 1, including iterative refinement and solution classification, were implemented in MATLAB (version R2024b).

Algorithm 1. Iterative computation of components $x_1^{\left(RL\right)}(t)$ and $x_2^{\left(RL\right)}(t)$ and classification flag.
line	pseudocode
1	Input: $\mu ,\alpha ,\omega$; Output: arrays x1RL, x2RL, flag isVibrating $\in \{0,1\}$
2	${\omega }_0\leftarrow \sqrt{{\omega }^2+{\mu }^{{\displaystyle \frac{2}{2-\alpha }}}}$, $T\leftarrow 20\pi /{\omega }_0$, $n\leftarrow {10}^2$, $\epsilon \leftarrow 0.05$
3	(x1prev,x2prev) $\leftarrow$ getComponents ($\mathrm{\mu },\mathrm{\alpha },\mathrm{\omega }$, T/n, T)
4	repeat
5	$\hspace{1em}\hspace{0.17em}n\leftarrow 10·n$, (x1curr,x2curr) $\leftarrow$ getComponents ($\mathrm{\mu },\mathrm{\alpha },\mathrm{\omega }$, T/n, T)
6	$\hspace{1em}\hspace{0.17em}r\leftarrow {\displaystyle \frac{max\{{‖x1curr-x1prev‖}_{\infty }, { ‖x2curr-x2prev‖}_{\infty }\} }{max\{{‖x1curr‖}_{\infty },{ ‖x2curr‖}_{\infty }\} }}$
7	x1prev $\leftarrow$ x1curr, x2prev $\leftarrow$ x2curr
8	until$r<\epsilon$
9	z1$\leftarrow$ getZeros(x1curr), z2 $\leftarrow$ getZeros(x2curr)
10	if length(z1) = 0 or length(z2) ≤ 1 then
11	isVibrating$\leftarrow$0, x1RL$\leftarrow$x1curr, x2RL $\leftarrow$ x2curr
12	else
13	isVibrating $\leftarrow$ 1, $T_0\leftarrow$2 times the average of the mean spacing between zero crossings in z1 and in z2, $T\leftarrow 10·T_0$
14	(x1RL,x2RL) $\leftarrow$ getComponents ($\mathrm{\mu },\mathrm{\alpha },\mathrm{\omega }$, T/n, T)
15	end if

Due to computational cost, the database [28] includes only cases with $\mu$ and $\omega$ limited to the range $[1, 100]$. For applications to the dynamics of mechanical structures such as rods, beams, and plates, the frequency range of the first three natural modes can be estimated as $\left[{0, 10}^4\right]$ rad/s [30]. The range of variation for the parameter $\mu$ can be taken as the same as for $\omega$, which follows from the asymptotic behavior of this coefficient for $\alpha =1$.

To verify the empirical formulas, two methods were used: the confusion matrix and the coefficient of determination $R^2$ [31]. In the confusion matrix, predictions are compared with exact results, giving counts of true positives (TP), false positives (FP), true negatives (TN), and false negatives (FN). For example, TP denotes the number of cases correctly identified by the model as belonging to the class that they actually belong to. The main measures are

$$\begin{array}{c}\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{y}={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}}, \mathrm{P}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}={\displaystyle \frac{\mathrm{T}\mathrm{P}}{TP+FP}},\hfill \\ \mathrm{R}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{l}={\displaystyle \frac{TP}{TP+FN}}, \mathrm{F}1\mathrm{-s}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}={\displaystyle \frac{2·Precision·Recall}{Precision+Recall}}.\hfill \end{array}$$

(16)

Accuracy gives the proportion of all correct predictions. Precision measures the proportion of correctly identified positive cases among all cases predicted as positive, while Recall indicates the proportion of correctly identified positives among all actual positives. The F1-score, defined as the harmonic mean of Precision and Recall, provides a balanced measure that accounts for both correctness and completeness of the positive predictions. To verify the empirical formulas, the agreement was evaluated using the coefficient of determination:

$${\mathrm{R}}^2=1-{\displaystyle \frac{\sum _{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}{\sum _{i=1}^N{\left(y_i-\overline{y}\right)}^2}},$$

(17)

where $y_i$ are the reference values, ${\widehat{y}}_i$ are the predicted values, and $\overline{y}$ is the mean of the reference values, $N$ is sample size. In addition, the relative difference was considered, both its maximum and mean value.

2.4. Aims and Scope of the Study

Based on the computational tools introduced in the previous section, we now formulate the main goals of this study. The specific aims are listed below.

• Formulate a criterion for classifying an FKV oscillator as oscillating or non-oscillating based on its parameters $(\mu ,\alpha ,\omega ).$ To this end, we seek a surface $F\left(\mu ,\alpha ,\omega \right)=0$ that separates the two types of behavior: oscillations occur when $F\left(\mu ,\alpha ,\omega \right)<0$, and no oscillations are present when $F\left(\mu ,\alpha ,\omega \right)>0.$ For comparison, in the classical harmonic oscillator, the analogous transition is defined by the condition ${\xi }^2-1=0$, where $\xi$ is the damping ratio.
• Derive simple formulas for estimating the parameters ${\omega }_0^{\left(m\right)}=f\left(\mu ,\alpha , \omega \right)$ and ${\xi }^{\left(m\right)}=g(\mu ,\alpha ,\omega )$ of the classical harmonic oscillator that best approximates a given FKV oscillator. These expressions should provide a direct way to match the behavior of the FKV model using standard oscillator parameters.
• Identify the region in the parameter space $(\mu ,\alpha ,\omega )$ where the FKV oscillator can be accurately approximated by a classical harmonic oscillator. This will help determine when the use of a simpler classical model is justified.

It should be noted that for the Caputo derivative, oscillations generally do not occur around zero but rather around a shifted equilibrium position (see the discussion at the end of Section 2.2), which limits its suitability for achieving Aim 2. This issue does not appear in the Riemann–Liouville case. On the other hand, both formulations share the same component related to the initial condition $\dot{x}\left(0\right)=v_0$ (see Equation (8)), so results obtained for the Riemann–Liouville case are expected to remain valid for the Caputo case as well. Therefore, in this study we focus exclusively on the Riemann–Liouville case.

3. Results

3.1. Validation of the Numerical Algorithm

To assess the reliability of the proposed numerical approach, we validate the results obtained with Algorithm 1 against those produced by a highly accurate reference method. The latter, introduced in [18], combines the analytical series representation with the Wynn–epsilon convergence acceleration algorithm, which has been demonstrated to provide excellent accuracy. For further computational details of this reference method, we refer the reader to [18].

To validate the computational method introduced as Algorithm 1, we selected two FKV oscillators with parameters $\left(\mu ,\alpha ,\omega \right)=(10,0.1,1)$ and $\left(\mu ,\alpha ,\omega \right)=(10,0.5,1)$. The results obtained with Algorithm 1 are compared with those from the analytical series-based method.

Figure 1 and Figure 2 show very good agreement between the two methods, with the curves almost overlapping. A closer analysis based on the absolute difference between the corresponding components confirms this: the maximum absolute difference was on the order of ${10}^{-3}$ for $x_1^{\left(RL\right)}(t)$ and ${10}^{-4}$ for $x_2^{\left(RL\right)}(t)$, which is a satisfactory result. On average, Algorithm 1 was about 70 times faster than the method based on analytical formulas. Using a smaller value of tolerance parameter $\epsilon$ in Algorithm 1 reduces the absolute differences further but significantly increases the computation time.

To investigate the influence of the tolerance parameter $\epsilon$, additional computations were carried out for the case $\left(\mu ,\alpha ,\omega \right)=(10,0.1,1)$. For $\epsilon =0.1$, Algorithm 1 was about 230 times faster than the Wynn–$\epsilon$ method, but the errors reached ${10}^{-2}$ for $x_1^{\left(RL\right)}(t)$ and ${10}^{-3}$ for $x_2^{\left(RL\right)}(t)$. Reducing $\epsilon$ to 0.05 improved the accuracy by an order of magnitude (errors ${10}^{-3}$ and ${10}^{-4}$), while the algorithm was still about 70 times faster. For $\epsilon =0.025$, the accuracy improved only slightly, with errors remaining of the same order of magnitude as for $\epsilon =0.05$, whereas the speed-up decreased to about 17. Finally, for $\epsilon =0.01$ the errors were reduced to ${10}^{-4}$ and ${10}^{-5}$, but the computation became slower than the Wynn–$\epsilon$ method (speed-up < 1). These results indicate that $\epsilon =0.05$ provides the best compromise, ensuring accuracy while keeping the computation around two orders of magnitude faster than the Wynn–$\epsilon$ method.

Algorithm 1 classified the oscillator in Figure 2 as oscillating (isVibrating = 1) and the one in Figure 3 as non-oscillating (isVibrating = 0). It is worth noting that the behavior of the oscillator in Figure 3 differs from the classical case of overdamped harmonic motion. This highlights that the dynamics of the fractional model are not fully identical to those of the classical harmonic oscillator.

3.2. Oscillation Classification Criterion

To address Aim 1 from Section 2.4, we determined an approximate form of the surface $F\left(\mu ,\alpha ,\omega \right)=0$ that separates the oscillatory regime, defined by $F\left(\mu ,\alpha ,\omega \right)<0$, from the non-oscillatory regime, where $F\left(\mu ,\alpha ,\omega \right)>0$.

As a first step, we used the values of the array isVibrating(i,j,k) stored in the database [28], described in Section 2.3. In this database, each FKV oscillator with parameters $({\mu }_i,{\alpha }_j,{\mu }_k$), where $i,k=1,2,\dots ,67$ and $j=1,2,\dots 19$, is assigned a value isVibrating(i,j,k): 1 if the oscillator is classified as vibrating, and 0 otherwise. To approximate the surface, we considered each cuboid defined by the vertices $V_{000}=({\mu }_i,{\alpha }_j,{\mu }_k$), $V_{100}=({\mu }_{i+1},{\alpha }_j,{\mu }_k$), $V_{110}=({\mu }_{i+1},{\alpha }_{j+1},{\mu }_k$), $V_{010}=({\mu }_i,{\alpha }_{j+1},{\mu }_k$), $V_{001}=({\mu }_i,{\alpha }_j,{\mu }_{k+1}$), $V_{101}=({\mu }_{i+1},{\alpha }_j,{\mu }_{k+1}$), $V_{111}=({\mu }_{i+1},{\alpha }_{j+1},{\mu }_{k+1}$), $V_{011}=({\mu }_i,{\alpha }_{j+1},{\mu }_{k+1}$), where $i,k=1,2,\dots ,66$ and $j=1,2,\dots 18$. For each cuboid, we examined the diagonals (e.g., $V_{000}{V}_{111}$, $V_{100}{V}_{011}$, etc.). If a diagonal connected two vertices with different classifications (one with isVibrating = 1 and the other with isVibrating = 0), we added the midpoint of that diagonal to the set of points approximating the surface $F\left(\mu ,\alpha ,\omega \right)=0$.

Figure 4 illustrates the approximate shape of this surface obtained using the above procedure. Although the accuracy of the approximation is limited, several clear regularities can be observed:

• If $\alpha \le 0.4$, then the FKV oscillator with parameters $(\mu ,\alpha ,\omega )$ is vibrating.
• If an oscillator with $(\mu ,\alpha ,\omega )$ is vibrating, then so is the oscillator with $({\mu }_1,\alpha ,\omega )$ for ${\mu }_1<\mu$.
• If an oscillator with $(\mu ,\alpha ,\omega )$ is vibrating, then so is the oscillator with $(\mu ,{\alpha }_1,\omega )$ for ${\alpha }_1<\alpha$.
• If an oscillator with $(\mu ,\alpha ,\omega )$ is vibrating, then so is the oscillator with $\left(\mu ,\alpha ,{\omega }_1\right)$ for ${\omega }_1>\omega$.
• If $\omega >\mu$, then the oscillator with $(\mu ,\alpha ,\omega )$ is vibrating.

Figure 4. Aproximate shape of the surface $F\left(\mu ,\alpha ,\omega \right)=0$.

Figure 4. Aproximate shape of the surface $F\left(\mu ,\alpha ,\omega \right)=0$.

Observations 1 and 5 were tested in wider parameter ranges with higher accuracy.

For fixed values of $\mu$ and $\omega$, we define the critical value of the parameter $\alpha$, denoted ${\alpha }_{cr}$, such that oscillators with $\alpha <{\alpha }_{cr}$ are classified as oscillating, while those with $\alpha >{\alpha }_{cr}$ are classified as non-oscillating. We determined ${\alpha }_{cr}={\alpha }_{cr}\left(\mu \right)$ for $\omega =1$ and $\mu$ varying in the range $0\le \mu \le 1000$, with a step size of $\Delta \mu =1.5$. The following bisection-like procedure was applied:

(a)

identify values ${\alpha }_v$ and ${\alpha }_{nv}$ (${\alpha }_v<{\alpha }_{nv}$, ${\alpha }_{nv}-{\alpha }_v=0.1$) such that FKV oscillator with parameters $(\mu ,{\alpha }_v,\omega )$ vibrates, while FKV oscillator with $(\mu ,{\alpha }_{nv},\omega )$ does not.

(b)

while ${\alpha }_{nv}-{\alpha }_v>0.01$: compute ${\alpha }_0=0.5·({\alpha }_v+{\alpha }_{nv})$. If the oscillator with $(\mu ,{\alpha }_0,\omega )$ vibrates set ${\alpha }_v={\alpha }_0$; otherwise, set ${\alpha }_{nv}={\alpha }_0$.

After completing the calculations, we obtained the curve ${\alpha }_{cr}={\alpha }_{cr}\left(\mu \right)$, which was then approximated by a simple empirical formula:

$${\alpha }_{cr}\left(\mu \right)=0.4+{\displaystyle \frac{0.5}{\sqrt[4]{{\mu }^5}}}.$$

(18)

Formula (18) provides an excellent fit to the computed dependence, with the maximum relative difference between the numerical results and the formula being less than 0.5%. This result also confirms Observation 1.

Analogously, following Observation 5, we define the critical value ${\mu }_{cr}$. Here, we study the dependence ${\mu }_{cr}={\mu }_{cr}\left(\omega \right)$ for $0\le \omega \le 100$ and fixed values of $\alpha \in \{0.45, 0.5, \dots , 1\}$. For each chosen pair $(\alpha ,\omega )$, the critical value ${\mu }_{cr}$ was determined using an analogous algorithm as the one applied to compute ${\alpha }_{cr}$. The results were particularly interesting: we observed that the dependence can be expressed as ${\mu }_{cr}\left(\omega \right)=f\left(\alpha \right)·{\omega }^{2-\alpha }$. To verify this and determine the form of $f\left(\alpha \right)$, we computed the ratios ${\mu }_{cr}\left(\omega \right)/{\omega }^{2-\alpha }$ for $0.4\le \alpha \le 1$ (step $\Delta \alpha =0.01$) and for $\omega \in \{1, 25, 50, 75, 100\}$. An empirical approximation of $f\left(\alpha \right)$ was then obtained by fitting these results:

$$f\left(\alpha \right)={\displaystyle \frac{{\alpha }^2-9.26723\alpha +8.36013}{24.8814{\alpha }^3-67.248{\alpha }^2+55.207\alpha -12.7939}}$$

(19)

Figure 5 presents the validation of the empirical formula for $f\left(\alpha \right)$. Figure 5a shows the computed values of $f\left(\alpha \right)$ over the range $0.4\le \alpha \le 1$ for different $\omega$. For small values of $\alpha$ ($\alpha <0.43)$, the computed values of $f\left(\alpha \right)$ are large and vary considerably with $\omega$ As $\alpha$ increases, the results stabilize, and in the range shown in Figure 5a the curves for different $\omega$ values almost coincide. This behavior is explained by the asymptotic trend of $f\left(\alpha \right)$ which, according to Observation 1, tends to infinity as $\alpha$ approaches 0.4 from the right. At the other endpoint $\alpha =1$, it is expected that $f\left(1\right)=2$. However, due to the asymptotic mismatch of the Riemann–Liouville derivative model for $\alpha =1$, a sharp discontinuity can be observed between $\alpha =0.95$ and $\alpha =1$. Figure 5b shows the distribution of relative differences between the values of $f\left(\alpha \right)$ computed for $1\le \omega \le 100$ and $0.5\le \alpha \le 1$, and those obtained from the empirical Formula (19). In this regime, the agreement is very good: the relative error does not exceed 1.5%.

Using the fitted function $f\left(\alpha \right)$ given in Equation (19), we introduce the formula for the classification function, announced at the beginning of this subsection:

$$F\left(\mu ,\alpha ,\omega \right)=\mu -f\left(\alpha \right)·{\omega }^{2-\alpha }$$

(20)

This expression is valid for $\alpha >0.4$. As an illustration, consider the FKV oscillators shown in Figure 3. For parameters $\left(\mu ,\alpha ,\omega \right)=(10,0.5,1)$ we obtain $F\left(10,0.5,1\right)\approx 6.41>0$and the oscillator is, therefore, classified as non-oscillating, which is consistent with the behavior shown in Figure 3.

In summary, the following criterion for oscillations can be formulated. If $\alpha \le 0.4$, the oscillator is classified as vibrating regardless of the values of $\mu$ and $\omega$. Otherwise, the classification is based on the function $F(\mu ,\alpha ,\omega )$ defined in Equation (20): if $F\left(\mu ,\alpha ,\omega \right)<0$, the oscillator is classified as vibrating, while if $F\left(\mu ,\alpha ,\omega \right)>0$, it is classified as non-oscillating.

Within the parameter range covered by the database, the proposed criterion gives correct results. To test its performance in a wider range ($\mu ,\omega \in [{0,10}^4],\alpha \in (0,1)$), $N=1000$ random parameter sets $(\mu ,\alpha ,\omega )$ were generated. In this case, the confusion matrix showed Accuracy = Precision = Recall = F1-score = 1, i.e., 100% effectiveness. To verify the criterion more precisely, another $N=1000$ sets were generated near the boundary surface $F\left(\mu ,\alpha ,\omega \right)=0$, with $\mu$ determined using Equation (20) with a random error of $\pm 10$. Here the results were Accuracy = 0.69, Precision = 0.68, Recall = 0.83, F1-score = 0.75. These values indicate that, close to the boundary, about 30% of cases are misclassified: the criterion detects oscillations relatively well (high Recall), but more often mislabels non-oscillatory cases as oscillatory (lower Precision). The F1-score of 75% confirms that the overall balance between correct and incorrect classifications is still acceptable. In summary, the criterion works perfectly far from the boundary, while near the boundary it remains fairly effective but reflects the increased complexity of FKV oscillator dynamics compared to classical harmonic oscillators (see Figure 3).

3.3. Empirical Formulas for Equivalent Harmonic Oscillator Parameters

This subsection discusses the formulas for the natural frequency ${\omega }_0^{\left(m\right)}$ and the damping ratio ${\xi }^{\left(m\right)}$ of the harmonic oscillator corresponding to a given FKV oscillator with parameters $(\mu ,\alpha ,\omega )$. The expressions proposed in [19] are as follows:

$${\omega }_0^{\left(m\right)}\left(\mathrm{\mu },\mathrm{\alpha },\mathrm{\omega }\right)=\sqrt{{\mathrm{\omega }}^2+{\mathrm{\mu }}^{{\displaystyle \frac{2}{2-\alpha }}}}$$

(21)

$${\xi }^{\left(m\right)}\left(\mathrm{\mu },\mathrm{\alpha },\mathrm{\omega }\right)={\displaystyle \frac{\alpha }{\sqrt{1+{\left(\frac{\omega }{{\mu }^{\frac{1}{2-\alpha }}}\right)}^2}}}.$$

(22)

These formulas were derived as an application of relations originally obtained for the Scott–Blair oscillator, under the assumption that $\alpha$ is close to zero. The examples presented in [19] showed that Equation (21) provides a good approximation of the actual frequency, while Equation (22) overestimates the true values.

By fitting to the data stored in the database [28], the following formulas were obtained:

$$\begin{gathered} {\omega }_0^{\left(m\right)}\left(\mathrm{\mu },\mathrm{\alpha },\mathrm{\omega }\right)={\displaystyle \frac{\alpha -0.924947}{0.708291\alpha -0.921832}}{\omega }_0\left(\mu ,\alpha ,\omega \right)\hspace{1em} \\ +{\displaystyle \frac{(1.840381\alpha -0.024497)(\omega +0.075836\mu )}{5.699401-4.338589\alpha }}, \end{gathered}$$

(23)

$$\begin{array}{cc}\hfill {\xi }^{\left(m\right)}\left(\mathrm{\mu },\mathrm{\alpha },\mathrm{\omega }\right)=& 0.00097\hfill \\ & \hspace{0.17em}+{\displaystyle \frac{2.7243 {\alpha }^2+0.3797\alpha +0.00352}{{\left(1+\frac{\left(0.22667+8.73586 \alpha \right){\omega }^{2.28811-0.85655 \alpha }}{{\mu }^{1.06979+0.348072\alpha }}\right)}^{0.72911}}},\end{array}$$

(24)

where ${\omega }_0\left(\mu ,\alpha ,\omega \right)$ is defined by Equation (21).

To evaluate the quality of Formulas (21) and (23) for the frequency, the database [28] was searched for FKV oscillators classified as vibrating (isVibrating = 1) and having a divergence coefficient not greater than 0.1 (${\epsilon }_m\le 0.1$). For these oscillators, the relative differences were calculated between the values of the ${\omega }_0^{\left(m\right)}$ stored in the database (obtained using the genetic algorithm) and those predicted by Formulas (21) and (23). In a similar way, the Formulas (22) and (24) for the damping ratio ${\xi }^{\left(m\right)}$ were verified, but in this case absolute differences were used due to the small magnitude of the damping ratios.

For the frequency approximation given by Formula (21), the maximum relative difference is 0.32 (32%), observed for the FKV oscillator with parameters $\left(\mu ,\alpha ,\omega \right)=(97,0.95,80.5)$, for which ${\epsilon }_m\approx 0.1$. Although this is a large deviation, the mean relative difference is only 0.012 (about 1%), and the coefficient of determination is $R^2=0.9926$, indicating very good agreement. The largest error occurs at $\alpha =0.95$ i.e., close to 1, where difficulties in asymptotic convergence of the FKV oscillator with the Riemann-Liouville derivative to a harmonic oscillator are known (see the discussion at the end of Section 2.2). When the additional condition $\alpha \le 0.6$ is imposed, the maximum relative difference drops to 0.017 (for $\left(\mu ,\alpha ,\omega \right)=(71.5,0.5,89.5)$), the mean relative difference decreases to 0.007, and the coefficient of determination improves to $R^2=0.9996$, which reflects excellent agreement.

For the frequency approximation given by Formula (23), the maximum relative difference is 0.024 (2.4%) observed for the FKV oscillator with parameters $\left(\mu ,\alpha ,\omega \right)=(1,0.5,2.5)$. The mean relative difference is 0.0016, and the coefficient of determination reaches $R^2=0.999987$, which indicates excellent agreement. Compared to Formula (21), this expression provides significantly higher accuracy, with both the maximum and mean relative differences being much smaller. It should be noted, however, that these results apply only within the parameter ranges covered by the database [28].

To verify the quality of these formulas over a wider parameter range, $N=1000$ sets of FKV oscillator parameters $(\mu ,\alpha ,\omega )$ were randomly generated with $\mu ,\omega \in [{1, 10}^4]$, $\alpha \in (0,1)$, such that the flag isVibrating returned by Algorithm 1 was equal to 1 and the parameters of the corresponding harmonic oscillator ${\omega }_0^{\left(m\right)}$ and ${\xi }^{\left(m\right)}$ satisfied ${\epsilon }_m\le 0.1$. This sample is also used later in this subsection to evaluate the accuracy of the formulas for ${\xi }^{\left(m\right)}$. For the frequency, the results were as follows:

• The maximum relative difference for relation (21) is 0.11, achieved for the FKV oscillator with parameters $\left(\mu ,\alpha ,\omega \right)=(5837.72,0.978,9170.07)$. The average relative difference is 0.006, with the coefficient of determination $R^2=0.998845.$
• The maximum relative difference for relation (23) is 0.16, achieved for $\left(\mu ,\alpha ,\omega \right)=(7434.585,0.117,12.627)$. The average relative difference is 0.018, with $R^2=0.998644$.

Figure 6 compares the frequency–$\alpha$ dependence obtained from the database with the approximations given by Equations (21) and (23) for different $\left(\mu ,\omega \right)$. In Figure 6a, Equation (21) shows the largest deviations, with a maximum relative error of around 32% but an average error reduced to about 3%; in contrast, Equation (23) is much more accurate, with both the maximum and mean errors below 0.5%. In Figure 6b, both formulas agree well with the database, yet Equation (23) again performs better (Equation (21): maximum around 0.8%, mean around 0.4%; Equation (23): maximum around 0.1%, mean around 0.06%). In Figure 6c, the accuracy of the two formulas is comparable (Equation (21): maximum around 0.8%, mean around 0.6%; Equation (23): maximum around 0.5%, mean around 0.4%), although the range of $\alpha$ is limited because for larger values ${\epsilon }_m>0.1$. Finally, Figure 6d, with $\left(\mu ,\omega \right)=\left(\mathrm{1,2.5}\right)$, illustrates that Equation (21) becomes independent of $\alpha$; here the errors are higher (maximum around 5.4%, mean around 1.7%), whereas Equation (23) still yields smaller deviations (maximum around 2.4%, mean around 1.5%).

Formula (22) for ${\xi }^{\left(m\right)}$ proposed in [19] considerably overestimates the damping ratio and does not provide satisfactory results. The maximum absolute difference with respect to the database values [28] is 0.22, the mean absolute difference is 0.013, and the coefficient of determination is $R^2=-0.399426$. In contrast, the new Formula (24) performs much better. It yields a maximum absolute difference of 0.012 (for $\left(\mu ,\alpha ,\mu \right)=(2,0.15,1))$, a mean absolute difference of 0.0009, and an $R^2$ of 0.999411.

The same sample of $N=1000$ parameter sets, previously generated to verify the formulas for ${\omega }_0^{\left(m\right)}$ outside the database range, was also used to evaluate the accuracy of the formulas for the damping ratio ${\xi }^{\left(m\right)}$. For Formula (22), the maximum absolute difference is 0.225 with an average absolute difference of 0.043 and a coefficient of determination $R^2=-1.02$. In contrast, Formula (24) provides much better accuracy, with a maximum absolute difference of 0.016 (for $\left(\mu ,\alpha ,\mu \right)=(8723.22,0.895,8898.4))$, a mean absolute difference of 0.0013, and an $R^2=0.996$. The results show that Formula (24) provides reliable estimates of the damping ratio also outside the database range, while Formula (22) fails to deliver satisfactory accuracy.

Figure 7 shows the dependence of ${\xi }^{\left(m\right)}$ on $\omega$, $\mu$ and $\alpha$. Formula (22) considerably overestimates the values of ${\xi }^{\left(m\right)}$, but it reproduces the correct qualitative trends: ${\xi }^{\left(m\right)}$ decreases with $\omega$, and increases with $\mu$ and $\alpha$. In case (a), where the maximum database value of ${\xi }^{\left(m\right)}$ is about 0.075, Equation (22) shows a maximum absolute difference of 0.053 (mean 0.008), while Equation (24) is much more accurate (maximum 0.011, mean 0.0006). In case (b), with database values up to 0.041, the deviations of Equation (22) reach 0.033 (mean 0.03), whereas Equation (24) remains very close to the database (maximum 0.001, mean 0.0004). In case (c), where the database values rise to about 0.1, Equation (22) shows the largest discrepancies (maximum 0.1, mean 0.04), while Equation (24) still provides an accurate approximation (maximum 0.0029, mean 0.0009). These results confirm that Equation (24) not only captures the correct qualitative trends but also provides a quantitatively reliable representation of the damping ratio across the considered cases.

We now examine how well the harmonic oscillators with parameters obtained from the proposed formulas fit the FKV oscillator. Two scenarios are considered: (I) ${\omega }_0^{\left(m\right)}$ given by (21) together with ${\xi }^{\left(m\right)}$ from (24); (II) ${\omega }_0^{\left(m\right)}$ given by (23) together with ${\xi }^{\left(m\right)}$ from (24). The fit between the FKV oscillator and the corresponding harmonic oscillator is measured using the divergence coefficient (see Section 2.2 and [18,19]).

Table 1. Comparison between FKV and harmonic oscillators using fitted frequency and damping.

$\mathbf{FKV} (\mathit{\mu },\mathit{\alpha },\mathit{\omega })$	$\mathit{H}{\mathit{O}}_{\mathit{D}\mathit{B}}({\mathit{\omega }}_0^{\left(\mathit{m}\right)},{\mathit{\xi }}^{\left(\mathit{m}\right)})$	${\mathit{\epsilon }}_{\mathit{m}}$	Scenario	$\mathit{H}{\mathit{O}}_{\mathit{A}}({\mathit{\omega }}_0^{\left(\mathit{m}\right)},{\mathit{\xi }}^{\left(\mathit{m}\right)})$	$\mathit{I}(\mathit{F}\mathit{K}\mathit{V},\mathit{H}{\mathit{O}}_{\mathit{A}})$	$\mathit{I}(\mathit{H}{\mathit{O}}_{\mathit{D}\mathit{B}},\mathit{H}{\mathit{O}}_{\mathit{A}})$
$(97, 0.95, 80.5)$	$(84.98, 0.45)$	0.1	(I)	$(112.1, 0.44)$	0.66	0.6
			(II)	$(84.71, 0.44)$	0.1	0.01
$(1, 0.5, 2.5)$	$(2.73, 0.07)$	0.09	(I)	$(2.69, 0.07)$	0.12	0.09
			(II)	$(2.67, 0.07)$	0.14	0.16
$(2.5, 0.15, 1)$	$(1.91, 0.074)$	0.095	(I)	$(1.92, 0.087)$	0.11	0.07
			(II)	$(1.88, 0.087)$	0.14	0.09
$(1, 0.8, 10)$	$(10.1, 0.0307)$	0.03	(I)	$(10.05, 0.029)$	0.077	0.073
			(II)	$(10.08, 0.029)$	0.04	0.04
$\left(10, 0.1, 1\right)$	$(3.48, 0.0644)$	0.08	(I)	$(3.5, 0.065)$	0.099	0.038
			(II)	$(3.45, 0.065)$	0.12	0.05
$(25, 0.5, 25)$	$(26.84, 0.0589)$	0.077	(I)	$(26.4, 0.0589)$	0.13	0.11
			(II)	$(26.6, 0.0589)$	0.09	0.06
$(5837.72, 0.978, 9170.07)$	$(8960.9, 0.257)$	0.07	(I)	$(10370.7, 0.243)$	0.25	0.31
			(II)	$(9318.08, 0.243)$	0.038	0.09
$(7434.585,0.117,12.627)$	$(113.42, 0.077)$	0.097	(I)	$(114.43, 0.085)$	0.12	0.06
			(II)	$(131.39, 0.085)$	0.57	0.64
$(7423.6,0.032,4823.2)$	$(4824.24, 0.0008)$	0.0008	(I)	$(4824.1, 0.0009)$	0.006	0.006
			(II)	$(4824.1, 0.0009)$	0.006	0.006
$(2977.9,0.659,1177.97)$	$(1254.7, 0.083)$	0.098	(I)	$(1240.8, 0.0897)$	0.12	0.06
			(II)	$(1312.5, 0.0897)$	0.23	0.21

contains 10 representative cases used to evaluate how well harmonic oscillators with fitted parameters reproduce the behavior of FKV oscillators. Each row corresponds to a selected set of parameters $(\mu ,\alpha ,\omega )$ for the FKV oscillator. The column $H{O}_{DB}$ lists the parameters of the best-fit harmonic oscillator obtained using a genetic algorithm, along with the corresponding divergence coefficient ${\epsilon }_m$ between the FKV oscillator and this harmonic oscillator. The column scenario indicates whether the approximate parameters ${\omega }_0^{\left(m\right)}$ and ${\xi }^{\left(m\right)}$ were computed using Formulas (21), (24)—scenario (I), or (23), (24)—scenario (II). The column $H{O}_A$ lists the parameters of the harmonic oscillator computed using the respective formulas. The next two columns show the divergence coefficient between the FKV and the approximated harmonic oscillator, and the divergence between the fitted (using genetic algorithm) and approximated harmonic oscillators, respectively. The first three rows correspond to the worst cases with respect to the frequency or damping ratio errors found in the database. The next three rows present randomly selected database examples. The last four rows correspond to cases outside the database range: the two worst cases for ${\omega }_0^{\left(m\right)}$ evaluated using Equations (21) and (23), and two randomly chosen examples.

Overall, a good agreement between the FKV oscillator and the corresponding harmonic oscillator with parameters obtained using scenario (I) or (II) is observed. Formula (21) for the natural frequency performs well over a broader range of α than originally suggested in [19], where it was proposed only for small α. Within the range of the database, Formula (23) gives more accurate frequency values, but outside this range, Formula (21) usually performs better. It is also important to note that a small relative difference between fitted and approximated parameters does not always imply a small divergence coefficient.

Figure 8 illustrates the distribution of the divergence coefficient ${\epsilon }_m$ across the $(\mu ,\omega )$ plane for selected values of $\alpha \in \{0.1,0.2,\dots ,0.9\}$. The values of ${\epsilon }_m$, which measure how well a harmonic oscillator approximates a given FKV oscillator, were taken from the dataset [28]. The contour line corresponding to ${\epsilon }_m=0.1$ is also plotted. This contour may be interpreted as a boundary separating regions where a good harmonic approximation exists (${\epsilon }_m\le 0.1$) from those where it does not (${\epsilon }_m>0.1$). It is evident that the area where no good approximation exists increases with $\alpha$, reflected by the upward shift in the ${\epsilon }_m=0.1$ contour. For fixed $\mu$ the minimal $\omega$ required to achieve a valid approximation increases with $\mu$. Yellow regions indicate oscillators classified as non-oscillating.

4. Discussion

The findings of this study give rise to several remarks and directions for further investigation. First, the asymptotic case $\alpha =1$ was not analytically investigated here for either the Riemann–Liouville or Caputo formulations; instead, reference was made to numerical examples from [18]. A detailed analysis should be carried out in future work.

Second, we introduced a geometric criterion to distinguish oscillatory and non-oscillatory regimes for FKV oscillators, inspired by the classical condition based on the sign changes in the solution components. As shown in Figure 3, the FKV oscillators exhibit a more complex structure—e.g., Figure 3b, the component $x_2^{\left(RL\right)}\left(t\right)$ does not cross zero, despite the oscillatory nature of $x_1^{\left(RL\right)}\left(t\right)$ (Figure 3a). This suggests that $x_2^{\left(RL\right)}\left(t\right)$ is more sensitive to the underlying oscillatory behavior.

All oscillators shown in Figure 9 are classified—according to the adopted criterion—as oscillatory. However, the behavior illustrated in Figure 9d closely resembles the non-oscillatory case shown in Figure 3b, indicating that the current classification is not fully reliable. These results should be considered as approximate, since, as discussed, it is not yet possible to formulate a strict criterion for the presence of oscillations.

Nevertheless, rather than relying on general trends such as the position of the $\epsilon ₘ = 0.1$ contour in Figure 8, one may formulate a more precise criterion for identifying whether a harmonic oscillator can effectively approximate a given FKV oscillator. Since Algorithm 1 computes the response over a time interval $[0, 10 T_0]$, where $T_0$ is the estimated fundamental period, it is expected that $x_2^{\left(RL\right)}\left(t\right)$ should have 20 zero crossings in this interval. If this condition is met, a harmonic approximation is likely possible; otherwise, it may fail. Developing and formalizing this criterion will be the subject of future research.

Third, an intriguing pattern emerged from the fitted parameters of equivalent harmonic oscillators. In all analyzed cases with good agreement (i.e., divergence coefficient ${\epsilon }_m\le 0.1$), the damping ratios were less than 0.5. For instance, in the database, the maximum damping ratio among these cases was 0.4926, while in the extended test set of 1000 randomly generated FKV oscillators, it was 0.2587. Whether this is a fundamental property of the FKV system or a coincidence remains an open question.

Finally, although the proposed empirical formulas for ${\omega }_0^{\left(m\right)}$ and ${\xi }^{\left(m\right)}$ show excellent agreement with database values in most cases, they are not derived from first principles. Further studies should aim to develop physically motivated expressions that better capture the dynamics of the FKV oscillator.

5. Conclusions

This study addressed the fractional homogeneous equation governing the FKV oscillator. The main objectives were twofold: (a) to introduce a criterion for determining whether an FKV oscillator exhibits oscillatory behavior and to derive an empirical formula for the boundary surface separating oscillatory and non-oscillatory regimes in the parameter space; (b) to develop formulas enabling the identification of an equivalent harmonic oscillator corresponding to a given FKV oscillator and to determine the region in parameter space where such an approximation is valid.

Asymptotic analysis for the case $\alpha =0$ revealed that goal (b) cannot always be achieved using the Caputo derivative formulation. Consequently, the study focused on the Riemann–Liouville case. For this formulation, a numerical algorithm was developed that automatically computes the solution components of the FKV oscillator. This approach enabled the systematic generation of a dataset used throughout the analysis.

To classify oscillatory behavior, a geometric criterion was introduced to distinguish between oscillatory and non-oscillatory FKV oscillators. Empirically derived conditions were formulated to implement this criterion, and numerical evidence confirmed their consistency with the geometric interpretation. The numerical algorithm incorporates this classification procedure by returning, in addition to the solution components, a flag that assigns each simulated oscillator to the oscillatory or non-oscillatory class. The criterion is not exact, since the dynamics of FKV systems do not allow for as sharp a distinction between oscillatory and non-oscillatory regimes as in the case of classical harmonic oscillators. Nonetheless, the validation demonstrated that the proposed framework provides sufficient accuracy for the intended applications.

To identify harmonic oscillators approximating FKV oscillators, a genetic algorithm was employed to minimize a divergence coefficient. This enabled the construction of a comprehensive database (referenced in the bibliography), containing the results of these optimizations and corresponding oscillation flags for FKV oscillators over a defined parameter range. The database facilitated the achievement of both main goals.

To assess generalization beyond the database, additional validation was conducted using randomly generated samples of 1000 FKV oscillators with parameters outside the original range. These tests yielded satisfactory results. Finally, the discussion section outlined possible directions for future research and presented one open question arising from this work.