A Note on the Dynamics of Modified rf-SQUIDs: Simulations and Possible Control over Oscillations

Abstract

The so-call SQUIDs (abbreviated from superconducting quantum interference device) are very sensitive apparatuses especially built for metering very low magnetic fields. These systems have applications in various practical fields—biology, geology, medicine, different engineering areas, etc. Their features are mainly based on superconductors and the Josephson effect. They can be differentiated into two main groups—direct current (DC) and radio frequency (RF) SQUIDs. Both of them were constructed in the 1960s at Ford Research Labs. The main difference between them is that the second ones use only one superconducting tunnel junction. This reduces their sensitivity, but makes them significantly cheaper. We investigate namely the rf-SQUIDs in the present work. A number of authors devote their research to the rf-SQUIDs driven by an oscillating external flux. We aim to enlarge the theoretical base of these systems by adding new factors in their dynamics. Several particular cases are explored and simulated. We demonstrate also some specialized modules for investigating the proposed model. One application for possible control over oscillations is also discussed. It is based on the Fourier transform and, as a consequence, on the characteristic function of some probability distributions.

1. Introduction

A number of authors devote their research to the rf superconducting quantum interference device (SQUID) driven by an oscillating external flux.

Direct current (DC) and radio frequency (RF) SQUIDs were constructed in the 1960s at Ford Research Labs—we refer to the original work [1] as well as to the seminal handbooks [2,3]. The reader can find detailed information about the odd and even subharmonics and chaos in rf SQUIDS and about homoclinic and heteroclinic bifurcations in rf SQUIDs in the leading studies [4,5,6,7,8,9,10,11,12]. We will explicitly note the important research related to the following: odd and even subharmonics and chaos in rf SQUIDS [4]; chaotic dynamics of periodically driven rf SQUIDs [5]; homoclinic and heteroclinic bifurcations in rf SQUIDs [6,7]; Melnikov’s method at a saddle-node [8,9]; and the dynamics of the forced Josephson junction [10]. Other interesting problems are discussed in [11,12]. The basic monographs [2,3] cited above provide a relatively complete bibliography on the subject under consideration. We will mention just some of the leading research on the following topics: theory of the signal transfer and noise properties of the rf SQUID [13,14,15,16,17,18,19,20]; SQUIDs and their applications [13,20,21,22,23,24,25]; performance factors in rf SQUIDs—high frequency limit [26,27,28,29,30]; signal characteristics for high Tc rf SQUID [31,32,33]. Some of the latest results concerning the control problems with real applications can be found in [34,35].

The dimensionless equations of motion for the rf SQUID can be written in the form (see, for example, [12]):

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-cx-ϵ\left(by-asin\left(\omega t\right)\right),\hfill \end{array}\right.$$

(1)

where $0\le ϵ<1$.

Evidently, this system is time-independent Hamiltonian with

$$H(x,y)={\displaystyle \frac{y^2}{2}}+{\displaystyle \frac{c{x}^2}{2}}-cosx$$

and the energy potential is

$$V={\displaystyle \frac{c{x}^2}{2}}-cosx.$$

Depending upon the potential parameter c, there may exist a great variety of separatrix solutions and associated hyperbolic fixed points in the phase space of (1) (for $ϵ=0$).

One obtains the hyperbolic fixed points $P=(x_0^{*},0)$ from the conditions

$$\left\{\begin{array}{c}sin{x}_0^{*}+c{x}_0^{*}=0\hfill \\ c+cos{x}_0^{*}<0.\hfill \end{array}\right.$$

(2)

Because the unperturbed system contains both heteroclinic and homoclinic orbits, the different cases are treated separately.

At first in [12], the author consider the case of an upper heteroclinic solution.

From (1), (2) one obtains

$$t=t_0+{\int }_0^{x_0}{\displaystyle \frac{\mathrm{d}z}{\sqrt{2V-c{z}^2+2cosz}}}.$$

The Melnikov function [36] is given by (see, for example, [12]):

$$\begin{array}{ccc}M\left(t_0\right)\hfill & =& {\int }_{-\infty }^{\infty }(a{y}_{0}sin\left(\omega t\right)-b{y}_0^2)\mathrm{d}t\hfill \\ \hfill & =& -2bF\left(c\right)+2aG(c,\omega )sin\left(\omega t_0\right),\hfill \end{array}$$

(3)

where

$$F\left(c\right)={\int }_0^{x_0^{*}}\sqrt{2V-c{z}^2+2cosz}\mathrm{d}z,$$

$$G(c,\omega )={\int }_0^{x_0^{*}}cos\left(\omega {\int }_0^{x_0}{\displaystyle \frac{\mathrm{d}z}{\sqrt{2V-c{z}^2+2cosz}}}\right)\mathrm{d}{x}_0.$$

In this paper, we suggest a new modified model.

Several simulations are composed. We demonstrate also some specialized modules for investigating the dynamics of these hypothetical oscillators.

The derived results can be used as an integral part of a much more general application for scientific computing—for some details, see [37,38,39,40].

The plan of the paper is as follows. We state our model in Section 2. Some simulations are also presented. One application for possible control over oscillations is discussed in Section 3. We conclude with Section 4.

2. A New Model: Some Simulations

We consider the following new modified model of the form:

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-cx-ϵ\left(by-{\displaystyle \sum _{i=1}^N}{g}_{i}cos\left(i\omega t\right)\right),\hfill \end{array}\right.$$

(4)

where $0\le ϵ<1$, $g_i\ge 0,i=1,2,\dots N$, and N is integer.

We note that the Melnikov function [36] corresponding to Model (4) is of the form

$$M\left(t_0\right)={\int }_{-\infty }^{\infty }\left(y_0{\displaystyle \sum _{i=1}^N}{g}_{i}cos\left(i\omega t\right)-b{y}_0^2\right)\mathrm{d}t.$$

(5)

The study of the dynamics of such models and the explicit representation of the Melnikov function can be done in the way described, for example, in [12] and in the previous Section 1, and we will omit it here.

Bruhn and Koch discussion [12] shows that the Melnikov method can successfully be applied to systems with complicated potentials, for which an analytic calculation of the associated integrals is not possible.

The expense in numerical calculation is relatively low, so the authors [12] recommend the numerical implementation of the Melnikov method, to determine parameter regions of complicated behaviour.

Because of the complicated potential the resulting integrals have to be evaluated numerically (for example, applying a fourth order Runge–Kutta method). However, we note that Runge–Kutta methods are not symplectic, so they are not suited well for simulating Hamiltonian and chaotic systems. There are several modern ODE solvers, see [41,42,43] as alternatives, namely, semi-implicit CD methods, which are suitable for boundary value problems, as well as the semi-explicit midpoint method and composition semi-implicit schemes based on Suzuki/Yoshida/Suzuki-Trotter or Casas formulas. Otherwise, the numerical effects may affect the results of analysis, and some features can be suppressed (see [44]) or induced (see [45] on inducing multistability in initially monostable systems), which may confuse the final results of the study. The reader can find additional bibliography on numerical methods for chaotic ODE in [46].

We will look at some interesting simulations on Model (4):

Example 1. 

Given $N=4$, $c=0.07$, $\omega =0.04$, $b=-0.9$, $g_1=0.13$, $g_2=0.62$, $g_3=0.25$, $g_4=0.58$, and $ϵ=0.0085$, the simulations on the system (4) for $x_0=2.5$; $y_0=0$ are depicted on Figure 1.

Example 2. 

Given $N=6$, $c=0.1$, $\omega =0.2$, $b=-0.8$, $g_1=0.03$, $g_2=0.12$, $g_3=0.05$, $g_4=0.15$, $g_5=0.1$, $g_6=0.01$, and $ϵ=0.0085$, the simulations on the system (4) for $x_0=2.5$; $y_0=0$ are depicted on Figure 2.

Example 3. 

Given $N=6$, $c=0.1$, $\omega =0.2$, $b=0.6$, $g_1=0.6$, $g_2=0.5$, $g_3=0.4$, $g_4=0.3$, $g_5=0.2$, $g_6=0.1$, and $ϵ=0.0085$, the simulations on the system (4) for $x_0=3.5$; $y_0=0$ are depicted on Figure 3.

3. Possible Control over Oscillations

3.1. Approximation with Restrictions

The new Model (4) has many free parameters that make it attractive for engineering calculations, for example for possible oscillation control—a user-preset level (or fork) for the oscillations of the y-component of the differential system solution. Suppose that we need to constrain the oscillator in the interval $\left(F\right(t),-F(t\left)\right)$, where the function $F(·)$ is of the type (the most common constraint)

$$F\left(t\right)=a_1+{\displaystyle \frac{a_2}{1+a_3{e}^{a_{4}t}}}.$$

(6)

Therefore, once we define the oscillator by the parameters c, $\omega$, b, $ϵ$, N, $g_i,i=1,2,\dots ,N$, and the initial condition $\left\{x_0;y_0\right\}$, we need to find these values of $a_i,i=1,2,\dots ,N$ that minimize the distance between the y-component and the functions $F\left(t\right)$ and $-F\left(t\right)$.

Example 4. 

Given $N=6$, $c=0.1$, $\omega =0.25$, $b=0.6$, $g_i=0.6-0.1(i-1),i=1,2,3,4,5,6$, $ϵ=0.0085$, $x_0=2.5$, and $y_0=0$, it turns out that the optimal values for $a_i,i=1,2,3,4,5,6$ are $a_1=-1.6$, $a_2=-1.8$, $a_3=2$, and $a_4=0.016$. The simulations on the system (4) together with the constraints $\left(F\right(t),-F(t\left)\right)$ are depicted on Figure 4 and Figure 5.

Example 5. 

Given $N=8$, $c=0.01$, $\omega =0.28$, $b=0.5$, $g_1=0.6$, $g_2=0.5$, $g_3=0.4$, $g_4=0.3$, $g_5=0.2$, $g_6=0.1$, $g_7=0.02$, $g_8=0.01$, $ϵ=0.0085$, $x_0=3$, and $y_0=0$, the optimal values are $a_1=-1.5$, $a_2=-1.7$, $a_3=3.6$, $a_4=0.0014\times 16$. The simulations are depicted on Figure 6 and Figure 7.

Remarks

(i) We will explicitly note that classical optimization techniques were essentially used in solving the problem (Examples 4 and 5) and we will not dwell on them here.

(ii) In some cases (from purely physical and mechanical constraints), the choice of function $F\left(t\right)$ may be of the logistic, log–logistic, “cut”, “step”, “U–and S–shaped”, or “activation” function type [47].

In our previous articles, we considered some models under the assumption that the parameters are generated from some discrete probability distribution.

3.2. Fourier-Probabilistic Construction to Possible Control over the Oscillations

The task of generating a probabilistic construction with a view to possible control over the oscillations of the dynamic model proposed in this article is also interesting. It is based on the characteristic functions which in fact are the Fourier transforms of the probability density. Having in mind the well known exponential presentation of the cos-function

$$cos\left(x\right)={\displaystyle \frac{e^{ix}+e^{-ix}}{2}},$$

(7)

we rewrite y-dynamics of (4) as

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}& =-sinx-cx-ϵ\left(by-\sum _{n=1}^N{g}_{n}cos\left(n\omega t\right)\right)\hfill \\ \hfill & =-sinx-cx-ϵ\left(by-\sum _{n=1}^N{g}_n{\displaystyle \frac{e^{in\omega t}+e^{-in\omega t}}{2}}\right).\hfill \end{array}$$

(8)

Note that we can assume without any restrictions ${\sum }_{n=1}^N{g}_n=1$. Therefore, we can view these coefficients as the probabilities of some distribution—note that $g_n\ge 0$. Hence, if we denote by $\xi$ a random variable distributed under this probability law and by $\mathrm{\Psi }\left(x\right)$ its characteristic function, then we can transform dynamics (8) into

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}& =-sinx-cx-ϵ\left(by-\sum _{n=1}^N{g}_n{\displaystyle \frac{e^{in\omega t}+e^{-in\omega t}}{2}}\right)\hfill \\ \hfill & =-sinx-cx-ϵ\left(by-{\displaystyle \frac{\mathbb{E}\left[e^{i\omega t\xi }\right]+\mathbb{E}\left[e^{-i\omega t\xi }\right]}{2}}\right)\hfill \\ \hfill & =-sinx-cx-ϵ\left(by-{\displaystyle \frac{\mathrm{\Psi }\left(\omega t\right)+\mathrm{\Psi }\left(-\omega t\right)}{2}}\right).\hfill \end{array}$$

(9)

Above we use the symbol $\mathbb{E}$ for the expectation. This presentation of the y-dynamics shows that we can use not only finite number of coefficients $g_n$, but we can work directly with the distribution assuming that it is supported on an arbitrary domain. If we denote this domain by A and by $g\left(\mathrm{d}x\right)$ the probability measure generated by $\xi$, then the original dynamics can be written as

$${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}=-sinx-cx-ϵ\left(by-\underset{A}{\int }cos\left(x\omega t\right)g\left(\mathrm{d}x\right)\right).$$

(10)

For example, let us consider a normal distributed random variable with parameters $\left(\mu ,\sigma \right)$. Its support is $A=\mathbb{R}$ and the characteristic function is

$$\mathrm{\Psi }\left(x\right)=e^{i\mu x-{\displaystyle \frac{{\sigma }^2{x}^2}{2}}}.$$

(11)

Some simulations based on these assumptions can be seen in Figure 8a,b. The parameters used are the same as in Example 1—$c=0.07$, $\omega =0.04$, $b=-0.9$, $ϵ=0.0085$, $x_0=2.5$; $y_0=0$. In addition, the expectation of the normal distribution is $\mu =1$, whereas its standard deviation is $\sigma =3$.

On the other hand, if we use a $\left(\theta ,\alpha \right)$-gamma distribution, then its support is $A={\mathbb{R}}^{+}$ and the characteristic function is

$$\mathrm{\Psi }\left(x\right)={\left({\displaystyle \frac{\theta }{\theta -ix}}\right)}^{\alpha }.$$

(12)

Some simulations based on the values $\alpha =1$ and $\theta =2$ are presented in Figure 8c,d. The other parameters are similar to those for Example 3—$c=0.1$, $\omega =0.2$, $b=0.6$, $ϵ=0.0085$. The initial condition is taken at $x_0=1$ and $y_0=5$.

Under this construction, the underlying distribution drives the perturbations of the system via its characteristic function. Having in mind that it is the Fourier transform of the distribution’s density, we may influence the perturbations by an arbitrary function through its Fourier transform. Thus, if $f\left(·\right)$ is the underlying function, then $\mathrm{\Psi }\left(x\right)=\left(\mathcal{F}f\right)\left(x\right)$. For example, if the underlying function is $f\left(x\right)={\left|x\right|}^p$ for some p, then

$$\mathrm{\Psi }\left(x\right)=-{\displaystyle \frac{2sin\left(\frac{\pi p}{2}\right)\mathrm{\Gamma }\left(p+1\right)}{{\left|2\pi x\right|}^{p+1}}}.$$

(13)

Figure 8e,f present some simulations based on the parameters similar to ones used in Example 5, namely $c=0.01$, $\omega =0.28$, $b=0.5$, and $ϵ=0.0085$. The power p is stated at $p=2$, whereas the initial condition is $x_0=1$ and $y_0=5$.

Finally, let us discuss some limitations of the suggested approach for controlling the oscillations. First, Formula (9) is based on the characteristic functions of the probability distributions. The main criterion for $\mathrm{\Psi }\left(·\right)$ to be a characteristic function is the Bochner’s theorem. Unfortunately, the condition for positive definiteness is often hard to verify. Some alternatives give the Mathias and Khinchine criteria. Additionally, Pólya’s theorem uses very simple conditions, but they are only sufficient.

On the other hand, nevertheless, we use Formula (9) based on the characteristic functions or the more general approach of the Fourier transform, the function $\mathrm{\Psi }\left(·\right)$ may not exist in a closed form. Thus, some numerical alternatives have to be applied for approximation of $\mathrm{\Psi }\left(·\right)$—for example, Discrete Fourier Transform (DFT) or the Fast Fourier Transform (FFT).

4. Conclusions

In this article, we propose a new modified rf SQUID’s model. Several simulations are composed. We demonstrate also some specialized modules for investigating the dynamics of the proposed model. This will be included as an integral part of a planned much more general Web-based application for scientific computing. A number of authors devote their research on this topic to the case of ”strong damping” (see, for example, [9,12])

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-cx-by-ϵsin\left(\omega t\right).\hfill \end{array}\right.$$

1. The reader can formulate and explore the dynamics of modified models of the type

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-cx-by-ϵ{\displaystyle \sum _{i=1}^N}{g}_{i}sin\left(i\omega t\right)\hfill \end{array}\right.$$

(14)

using results obtained in the present paper. The case of highly damped systems is also of some interest and will be considered in future work.

The new model has many free parameters, which makes it attractive for engineering calculations, for example, for possible oscillation control—a user-preset level (or fork) for the oscillations of the y -component of the differential system solution.

The construction presented in Section 3.2 can be applied to Model (14) by using the exponential presentation of the sin-function

$$sin\left(x\right)={\displaystyle \frac{e^{ix}-e^{-ix}}{2i}}.$$

(15)

Thus dynamics (14) can be written as

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}& =y\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}& =-sinx-cx-by-ϵ{\displaystyle \frac{\mathrm{\Psi }\left(\omega t\right)-\mathrm{\Psi }\left(-\omega t\right)}{2i}}.\hfill \end{array}$$

(16)

We envisage future research on the current topic—modeling of chaotic systems (of the type (1) and (4)) with a desired set of properties—in the light of the interesting considerations in the articles [48,49].

Interesting results on the topic control of homoclinic chaos by weak periodic perturbations can be found in the monograph [50].

Other dynamic models from practice and their treatment in the light of the considerations in this paper.

2. The analytical study of controlling chaotic dynamics in spur gear systems is the subject of reflections by many authors working in the field of mechanisms and machine theory.

Gear systems have been widely used in many industrial applications due to their advantages of having accurate transmission ratios, compact dimensions, and high efficiency. With increasing requirements for gear performance in cutting-edge fields, many nonlinear dynamic characteristics that affect transmission performance must be considered in gear design.

Therefore, it is important to establish a relatively accurate dynamic model and investigate the complicated nonlinear behavior of the gear system subjected to various excitations. With the development of nonlinear dynamics theories, the nonlinear characteristics of these systems, such as stability, periodic responses, bifurcations, and chaos behaviors, have become the most interesting research areas.

A nonlinear dynamic model of a spur gear pair with backlash and static transmission error is formulated in [51].

More precisely, the conditions for existence of chaotic behavior in terms of homoclinic bifurcation by using Melnikov analysis are performed.

The planar model is of the form

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -f_h\left(x\right)-ϵ\left(Ay+g_{1}cos\left(\omega t\right)\right),\hfill \end{array}\right.$$

(17)

where backlash function $f_h\left(x\right)$ is a nonlinear displacement function and can be expressed as

$$f_h\left(x\right)=\left\{\begin{array}{c}x-(1-\alpha );1<x\hfill \\ \alpha x;-1\le x\le 1\hfill \\ x+(1-\alpha );1<-x,\hfill \end{array}\right.$$

where $\alpha$ is a mechanical parameter. The authors in [51] consider the following approximation (for $\alpha =0$): $f_h\left(x\right)\approx -0.1667x+0.1667x^3$ (see Figure 9) and study the dynamics of Model (17) using Melnikov analysis. Some investigations can be found in [52,53]. Dynamic response of a spur gear system with uncertain parameters is considered in [52]. Dynamic modeling and nonlinear analysis of a spur gear system considering a nonuniformly distributed meshing force is considered in [54].

The reader can formulate and explore the dynamics of modified model of the type

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -f_h\left(x\right)-ϵ\left(by-{\displaystyle \sum _{i=1}^N}{g}_{i}cos\left(i\omega t\right)\right).\hfill \end{array}\right.$$

(18)

We will explicitly note that specialists working in the subject under consideration may also use other approximations of the backlash function $f_h\left(x\right)$.

For example, the following approximation can also be considered for $\alpha =0.9$ (see Figure 10) $f_h\left(x\right)\approx sinx+0.28x$ and study the dynamics of Model (18) using considerations in this article.

3. Following the considerations in this article, the reader can successfully formulate and investigate the dynamics of the following extended hypothetical oscillator

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-cx-ϵ\left({by\left|y\right|}^{p-1}-{\displaystyle \sum _{i=1}^N}{g}_{i}sin\left(i\omega t\right)\right),\hfill \end{array}\right.$$

(19)

where $0\le ϵ<1$, b is the damping level, $p\ge 1$ is the damping exponent, and N is the integer.

At this stage, we will call Model (19) “hypothetical” until specialists and engineers working in this scientific field make a statement regarding its applicability.

Example 6. 

Let us assume that the parameters $p=4$ and $N=6$ in (19) are fixed.

Let us further assume that the user has fixed the desired function

$$F\left(t\right)=-1.5-{\displaystyle \frac{2.4}{1+4.6e^{0.0490909t}}}.$$

After applying an optimization technique, the desired control over the oscillations is achieved for the following values of the model parameters: $c=0.1$, $\omega =0.25$, $b=0.6$, $g_1=0.8$, $g_2=0.6$, $g_3=0.7$, $g_4=0.5$, $g_5=0.4$, $g_6=0.2$, and $ϵ=0.0085$ (see Figure 11 and Figure 12).

Remark 1. 

When more nonlinear damping terms are introduced, the following model is of interest to researchers:

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-cx-ϵ\left({\displaystyle \sum _{i=1}^M}{b}_{p_i}y{\left|y\right|}^{p_i-1}-{\displaystyle \sum _{i=1}^N}{g}_{i}sin\left(i\omega t\right)\right).\hfill \end{array}\right.$$

Following the ideas given in [55], the reader can investigate the dynamics of the following extended hypothetical oscillator:

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-cx-ϵ\left({by\left|y\right|}^{p-1}-cosx{\displaystyle \sum _{i=1}^N}{g}_{i}cos\left(i\omega t\right)\right).\hfill \end{array}\right.$$

(20)

The task of generating a probabilistic construction with a view to possible control over the oscillations of the dynamic models (19) and (20) is also interesting. We envisage future research on this topic.

4. Regarding the practical implementation and control of oscillations, specialists and engineers working in this scientific field have the say.

5. We envisage future research on Melnikov analysis of chaotic dynamics in generalized oscillatory systems (based on (4), (14), (18), (19), and (20)) in light of the considerations in [55,56,57,58,59,60].