## 1. Introduction

Models of oscillatory systems (oscillators) are used in various fields of knowledge from mechanics to economics and biology [

1,

2,

3]. From the point of view of mathematics, these models are traditionally described using ordinary differential equations of the second order and the corresponding initial conditions, i.e., the Cauchy problem is posed [

1]. It should be noted that such mathematical models cannot take into account the properties of the environment, for example, heredity or memory effect. This effect is characterized by the fact that the oscillating medium can “remember” the impact on it for a long time.

For the first time, the model of the hereditary oscillator was presented in his work by the Italian mathematician V. Volterra [

4]. He proposed to take into account heredity in the linear oscillator model using an integro-differential equation with a difference kernel, which he later called the function of heredity or memory. It should be noted that V. Volterra derived the total energy conservation law for this generalized oscillator, in the formula of which an additional term appeared, which is responsible for the dissipation of its energy. This important result was confirmed in subsequent works on this topic.

If we choose a power-law memory function, then we can, using the mathematical apparatus of fractional calculus [

5,

6,

7], go to other model equations that contain derivatives of fractional orders. In this case, the orders of fractional derivatives, as shown by the results of [

8,

9], will be responsible for the intensity of energy dissipation and are related to the Q-factor of the oscillator. Oscillators with such a description are usually called fractional.

Research methods for mathematical models of fractional oscillators can be divided into exact and numerical. Exact methods, for example, include integral transformations [

10] or decomposition methods [

11], and numerical methods include the theory of finite-difference schemes [

12], variational-iterative methods [

11].

In this paper, we will carry out a numerical analysis of mathematical models of linear fractional oscillators (LFO) using elements of the theory of finite-difference schemes. As methods of numerical analysis, we will choose a method based on an explicit non-local finite-difference scheme (ENFDS) studied in the author’s work [

13] and the fractional ABM method, which was investigated in [

14,

15,

16]. Let us analyze the errors of the methods using Runge’s rule, and obtain the calculated curves of oscillograms and phase trajectories of the Mathieu, Airy LFOs and an analogue of a harmonic oscillator. Let us investigate the forced oscillations of the LFO and give an interpretation of the research results.

## 3. Generalization of Newton’s Second Law

Consider a mechanical system that takes into account dynamic memory in the presence of dissipation. Let us write the equation of motion for such a system in the form:

where

$F\left(x,\tau \right)$ is the resultant of all forces acting on a material body with mass

m,

$G\left(t-\tau \right)$ is a memory function that characterizes the change impulse of a mechanical system as a reaction to the initial action of a force.

In the case when the mechanical system does not have dissipation, the memory function is the Heaviside function, which corresponds to ideal memory. In this case, the momentum of the system does not change upon the initial action of the force. The presence of dissipation in the system leads to a gradual forgetting, the initial impact exerted on it, while the simplest form of the dynamic memory function can be described by the relation [

17]:

where

$\alpha $ is the intensity of dissipation, for

$\alpha =1$ there is no dissipation.

Substituting relation (

4) into (

3) and taking into account the definition of the fractional Riemann–Liouville integral (

1), we have

Next, we invert the fractional integral in (

5) using the composition property:

As a result, taking into account relation (

6), we arrive at the equation:

Equation (

7) is a generalization of Newton’s second law and coincides with it for

$\beta =2$.

Depending on the right side of this equation—the form of the function

$F\left(x,t\right)$, we will obtain model equations of oscillatory systems with memory. In this paper, we will consider a wide class of oscillatory systems with dynamic memory, when the right-hand side can be represented as:

here

$f\left(t\right)$ is the external action,

$\omega \left(t\right)$ is the natural oscillation frequency,

$\lambda $ is the viscous friction coefficient.

Let

$m=1$ for simplicity. Taking into account (

8), let us formulate the following problem statement.

## 6. Forced Oscillations of Linear Fractional Oscillators

To determine the relationship of the orders of fractional derivatives in the model Equation (

9) with the characteristics of the LFO, we will investigate their forced oscillations. In Equation (

9), as the external force, we take a harmonic function with amplitude

$\delta $ and frequency

$\phi $:

$f\left(t\right)=\delta cos\left(\phi t\right)$. Based on this, the model equation will take the form:

**Theorem** **4.** Equation (32) is equivalent to the equation:where $m=-{\phi}^{\beta -2}cos\left(\beta \pi /\phantom{\beta \pi 2}\phantom{\rule{0.0pt}{0ex}}2\right),p={\phi}^{\beta -1}sin\left(\beta \pi /\phantom{\beta \pi 2}\phantom{\rule{0.0pt}{0ex}}2\right)+\lambda {\phi}^{\gamma -1}sin\left(\gamma \pi /\phantom{\gamma \pi 2}\phantom{\rule{0.0pt}{0ex}}2\right)$, ${s}^{2}=\omega \left(t\right)+\lambda {\phi}^{\gamma}cos\left(\gamma \pi /\phantom{\gamma \pi 2}\phantom{\rule{0.0pt}{0ex}}2\right)$, $x\left(t\right)=Acos\left(u\right),\dot{x}\left(t\right)=-A\phi sin\left(u\right),\ddot{x}\left(t\right)=-A{\phi}^{2}cos\left(u\right),u=\phi t+\mathsf{\Phi}$, $A,\mathsf{\Phi}$ is the amplitude and phase of steady-state oscillations. **Proof.** We will seek a solution to Equation (

32) in the form:

Taking into account (

2), Equation (

32) can be rewritten as:

Then the first term in (

35), taking into account representation (

34), can be represented as:

In the case of steady-state oscillations at

$t\to \infty $, the integrals in (

36) can be written as follows [

5]:

Taking into account relations (

37) and (

38), we get:

Similarly, the second term in (

35) will have the form:

Taking into account (

34), (

38) and (

40), Equation (

32) can be rewritten as:

Taking into account in Equation (

33) that

$\dot{x}\left(t\right)=-A\phi sin\left(u\right),\ddot{x}\left(t\right)=-A{\phi}^{2}cos\left(u\right)$, we arrive at Equation (

33). ☐

**Remark** **6.** Please note that Equation (33) is a classical linear oscillator with friction and external harmonic action, for which the relations for the amplitude–frequency (AFC), phase-frequency (PFC) characteristics and Q-factor are known:where $U={s}^{2}-{\phi}^{2}m,W=\phi p$. Formulas (

42) for the AFC and PFC can be rewritten as follows:

**Remark** **7.** Recall that the AFC determines the dependence of the amplitude of steady-state oscillations on the frequency of the harmonic input signal. The maximum value of the amplitude corresponds to the value of the resonant frequency. Therefore, the AFC curves are also called resonant. PFC is the dependence of the phase difference between the input and output signals on the frequency of the harmonic input signal. Q-factor is a parameter of an oscillatory system that determines the width of the resonance and characterizes how many times the energy reserves in the system are greater than the energy losses during the phase change by 1 radian.

Figure 4 shows the resonance curves AFC obtained by formula (

43), as well as using the ABM method, taking into account the parameter values [

18]:

$\lambda =0.5,\delta =50,\gamma =1,\omega \left(t\right)={\omega}_{0}=1$.

Figure 4 shows that in the classical case with the value of the parameter

$\beta =2$, the maximum amplitude is achieved at the value of the frequency

$\varphi =1$, i.e., resonance occurs (

$\varphi ={\omega}_{0}=1$). Furthermore, with decreasing parameter

$\beta $, the maximum amplitude decreases, and the resonant frequency shifts to the region of lower frequencies. Therefore, it can be concluded that the parameter

$\beta $ is responsible for the intensity of energy dissipation in an oscillatory system with memory.

From

Figure 4, we also see that the resonance curves obtained by the numerical ABM algorithm give an acceptable result. The numerical algorithm for calculating the resonance curves is based on the fact that over time the amplitude of the oscillations reaches a steady state. Therefore, we first calculate the initial LFO model using the ABM method for sufficiently large

t and different values of the frequency of external influence

$\phi $.

Figure 5 shows the curves of the phase-frequency characteristic

$\mathsf{\Phi}$ at different values of the parameters

$\beta $ and

$\gamma $.

From

Figure 5a, for the classical case at

$\beta =2$ and

$\gamma =1$, the limiting value of the phase shift at

$\phi \to \infty $ is

$\mathsf{\Phi}=-\pi $. With a decrease in the values of the parameter

$\beta $, the limiting phase shift will be

$\mathsf{\Phi}=-\beta \pi /2$, i.e., will decrease. Please note that the PFC curves are rearranged in reverse order, which is typical for memory effects.

Figure 5b shows PFC curves plotted at various values of the parameter

$\gamma $. It can be noted that in this case the limiting value of the phase shift remains practically unchanged, but the regrouping of the curves remains.

Figure 6a shows the Q-factor plotted for various values of beta and fixed frequency

$\phi =0.5,1,1.5$, as well as

$\gamma =1$. It is seen that with decreasing values of the parameter

$\beta $, the Q-factor decreases for each fixed value of the frequency

$\phi $. This fact confirms that the beta parameter is responsible for the intensity of the LFO energy dissipation.

Figure 6b shows the Q-factor obtained for different values of the parameter gamma and fixed values of the frequency

$\phi =0.5,1,1.5$, as well as

$\beta =1.8$. Here we can see that the Q-factor increases with decreasing gamma parameter values. Therefore, the

$\gamma $ parameter, as well as the beta parameter, is responsible for the intensity of energy dissipation. Please note that the

$\beta $ and

$\gamma $ parameters act in different directions, i.e., if the

$\beta $ parameter decreases the Q-factor, then the

$\gamma $ parameter, on the contrary, leads to its increase.

In the previous examples in Formula (

43), we chose the natural frequency

$\omega \left(t\right)$ as a constant. Consider more general cases when

$\omega \left(t\right)=t$ and

$\omega \left(t\right)=1+3cos\left(2t\right)$. The first case corresponds to the fractional Airy oscillator, and the second to the fractional Mathieu oscillator. We will explore them in more detail in the next section.

Figure 7 shows AFC surfaces and Q-factor surfaces plotted against

$\beta $ and

t for a fractional Airy oscillator. Here the parameter values were chosen: gray—

$\gamma =0.8$, blue—

$\gamma =0.6$, red—

$\gamma =0.4$,

$\phi =0.5$, the rest of the parameter values are taken from the previous example.

In

Figure 7a, we can see that as the values of the

$\beta $ parameter decrease over the entire time interval [0,2], AFC also decreases for each value of

$\gamma $.

Figure 7b confirms this pattern. We see that the Q-factor decreases with decreasing values of the

$\beta $ parameter for each value of

$\gamma $ over the entire time interval.

Figure 8 shows the surfaces of the frequency response and Q-factor of a fractional Airy oscillator depending on

$\gamma $ and

t. Here the values of the

$\beta $ parameter were chosen: 1.8—gray, 1.6—blue, 1.4—red. Leave the rest of the parameters unchanged.

In

Figure 8a,b, we see that the AFC and Q-factor decrease with decreasing

$\gamma $ values for various fixed

$\beta $ values over the entire time interval of

t.

Figure 9 and

Figure 10, by analogy with

Figure 7 and

Figure 8, show the AFC and Q-factor surfaces for the Mathieu fractional oscillator at different values of the parameters

$\beta $ and

$\gamma $.

Figure 9 and

Figure 10 confirm the conclusions drawn earlier: the parameters

$\beta $ and

$\gamma $ are responsible for the intensity of energy dissipation and act in different directions. With a decrease in the parameter

$\beta $, the intensity of energy dissipation increases, and with a decrease in the parameter

$\gamma $, the intensity of energy dissipation decreases.

## 8. Discussion

The article considered a wide class of linear fractional oscillators in the case when the memory function was chosen as a power-law one (

4). However, the memory function can be chosen differently, based on the conditions of the problem or experimental data. Therefore, the class of hereditary oscillators is much wider and richer than the class of fractional oscillators. On the other hand, the definition of the memory function can give a new definition of the derivative of fractional order, which will make it possible to develop fractional calculus in the future.

In this work, several numerical methods for solving the Cauchy problem were chosen: ABM and ENFD, which were implemented in the Maple computer environment. We did not set ourselves the task of improving the accuracy of these methods, therefore, it is possible to use other more accurate methods.

It is of interest to study the asymptotic stability of the equilibrium points of the fractional dynamical system (

9).

Also of interest is the study of LFO with variable memory. In this case, derivatives of fractional variable orders appear in the model equation of the Cauchy problem (

9). Some aspects of the numerical analysis of the generalized Cauchy problem (

9) were considered by the author in [

25].

In developing the research topic, it is of particular interest to consider the Cauchy problem (

9) in terms of the fractional Riemann–Liouville derivative with non-local initial conditions [

26,

27].