Investigations of Modified Classical Dynamical Models: Melnikov’s Approach, Simulations and Applications, and Probabilistic Control of Perturbations

Abstract

We suggest a few kinds of extended classical oscillators in this study. We present a few specific modules for examining these oscillators’ behavior. This will be an essential component of a broader web-based scientific computing platform that is in the works. The modeling and synthesis of radiating antenna designs is also taken into consideration as a potential use case for Melnikov functions. Additionally, we discuss strategies for achieving probabilistic control over system perturbations.

1. Introduction

A number of authors have researched the classical differential model:

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx+ϵF(x,y)y,\hfill \end{array}\right.$$

(1)

where $0\le ϵ<1$. The literature on this subject is extensive and diverse. The outstanding studies of Tricomi [1], Stoker [2], Levi, Hoppensteadt and Miranker [3], Perko [4], and Guckenheimer and Holmes [5] provide the reader with comprehensive information. A universal escape nonlinear oscillator model describes a variety of oscillatory physical phenomena that occur in various fields, including mechanics, quantum optics, acoustics, hydrodynamics, electronics, and engineering. The equation of motion for the sinusoidally driven escape oscillator, which includes nonlinear damping terms as a power series on the velocity, has been examined by Sanjuan [6]. See [7,8,9,10,11,12,13,14] for additional results. We examine the following model in [15].

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx+ϵ\left(a_1+{\displaystyle \sum _{i=0}^{\frac{n}{2}-1}}{a}_{n-2i}{x}^{n-2i}+{\displaystyle \sum _{i=0}^{\frac{n}{2}-2}}{b}_{n-2-2i}{y}^{n-2-2i}\right)y,\hfill \end{array}\right.$$

(2)

where $0\le ϵ<1$ and n is even. The classical differential model—the damped mathematical pendulum excited by horizontal harmonics—is the focus of several authors’ studies [16,17,18,19,20,21]:

$$\ddot{x}+ϵ\delta \dot{x}+sinx+ϵ\gamma cos\left(\omega t\right)cosx=0$$

(3)

or

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-ϵ(\delta y+\gamma cos(\omega t)cosx),\hfill \end{array}\right.$$

(4)

where $0\le ϵ<1$. In this paper, we suggest some classes of extended classical oscillators. Studies are examined in the context of Melnikov’s methodology [22]. We offer several simulations. Additionally, we present a few specialized modules for examining the dynamics of these fictitious oscillators. The obtained results can be included into a much broader scientific computing application—see [23,24,25,26,27,28] for additional information. The following is the paper’s plan. In Section 2.1 and Section 2.2, we provide our models. Section 2.1.1 and Section 2.2.1 examine investigations in the context of Melnikov’s methodology. Section 2.1.2 and Section 2.2.2 contain a few simulations. Section 2.1.3 and Section 2.2.3 address one potential use of Melnikov functions, which is in the modeling of radiating antenna designs. Section 3 brings us to a close. Appendix A and Appendix B provide probabilistic control over the perturbations.

2. The Models

2.1. The Model A

The following novel class of extended oscillators is examined.

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

(5)

where A is the damping level, $p\ge 1$ is the damping exponent, N is an integer, and $0\le ϵ<1$. Specifically, we take into account the following model:

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

(6)

The equation $H(x,y)={\displaystyle \frac{1}{2}}{y}^2+1-cosx$ represents the system’s total energy ($ϵ=0$). Some details can be found in monograph [5]. The orbit is provided by (refer to Figure 1):

$$\left(x_0\left(t\right)=\pm 2\arctan \left(\sinh t\right);y_0\left(t\right)=\pm 2\mathrm{sech}t\right).$$

(7)

2.1.1. Observations in View of Melnikov’s Methodology

The Melnikov function can be used to determine the transverse intersection of the stable and unstable manifolds and provides a measure of the leading order distance between them when $ϵ\ne 0$. By definition, the Melnikov integral is provided by

$$M\left(t_0\right)={\displaystyle {\int }_{-\infty }^{\infty }}{y}_0\left(t\right)\left(A{y}_0\left(t\right){\left|y_0\left(t\right)\right|}^{p-1}-{\displaystyle \sum _{j=1}^N}{g}_{i}sin\left(j\omega (t+t_0)\right)\right)\mathrm{d}t,$$

(8)

where Equation (7) defines the functions $x_0\left(t\right)$ and $y_0\left(t\right)$. The task of determining the root of $M\left(t_0\right)$ is more intriguing from a numerical perspective as the parameters that appear in the suggested differential model are subject to certain practical and physical constraints.

Proposition 1.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions if $p=2$ and $N=1$

$$M\left(t_0\right)=2\pi \left(2A-g_1\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)sin\left(t_0\omega \right)\right)=0.$$

(9)

$M\left(t_0\right)=0$ (for $p=2;N=1;A=0.015;\omega =0.2;g_1=0.09$), for instance, is shown on Figure 2. In the interval $(0,16)$, the roots are: $1.78742;13.9205$. Melnikov’s condition for chaotic behavior of the dynamical model can be formulated by the reader from Proposition 1 (also see Figure 2).

Proposition 2.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions for $p=2$ and $N=2$

$$M\left(t_0\right)=2\pi \left(2A-g_1\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)sin\left(t_0\omega \right)-g_2\mathrm{sech}\left(\pi \omega \right)sin\left(2t_0\omega \right)\right)=0.$$

(10)

For instance, the formula $M\left(t_0\right)=0$ is shown in Figure 3 (for $p=2$; $N=2$; $A=0.015$; $\omega =0.6$; $g_1=0.07$; $g_2=0.3$). The following are the roots in the interval of $(0,10)$: $0.224036;2.7794;5.63804;$ $7.0665$. We can demonstrate the following claim when $p=4$ and $N=1$.

Proposition 3.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions for $p=4$ and $N=1$

$$M\left(t_0\right)=2\pi \left(6A-g_1\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)sin\left(t_0\omega \right)\right)=0.$$

(11)

For instance, Figure 4 shows the equation $M\left(t_0\right)=0$ (for $p=4;N=1;A=0.015;\omega =0.1;g_1=0.091$). The root in interval $(0,16)$ is: $15.708$ with a multiplicity of two.

Proposition 4.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions for $p=4$ and $N=2$

$$M\left(t_0\right)=2\pi \left(6A-g_1\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)sin\left(t_0\omega \right)-g_2\mathrm{sech}\left(\pi \omega \right)sin\left(2t_0\omega \right)\right)=0.$$

(12)

As an illustration, Figure 5 shows the equation $M\left(t_0\right)=0$ (for $p=4;N=2;A=0.011;\omega =0.92;g_1=0.091;g_2=0.32$). The roots (with a multiplicity of two) at an interval of $(0,10)$ are: $0.97718;7.92402$. $M\left(t_0\right)=0$ is the equation (for $p=4;N=2;A=0.02;\omega =0.7;g_1=0.005;g_2=0.2$) and has no roots (see Figure 6).

Proposition 5.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions if $p=4$ and $N=3$

$$\begin{array}{ccc}M\left(t_0\right)\hfill & =& {\displaystyle \frac{1}{-1+2\cosh \left(\pi \omega \right)}}{e}^{-2i{t}_0\omega }\pi \left(-i{e}^{2i{t}_0\omega }{g}_1+i{e}^{4i{t}_0\omega }{g}_1+6A{e}^{2i{t}_0\omega }\cosh \left({\displaystyle \frac{\pi \omega }{2}}\right)\right.\hfill \\ \hfill & & +i{e}^{2i{t}_0\omega }{g}_1\cosh \left(\pi \omega \right)-i{e}^{4i{t}_0\omega }{g}_1\cosh \left(\pi \omega \right)-i{g}_3\cosh \left(\pi \omega \right)\hfill \\ \hfill & & +i{e}^{6i{t}_0\omega }{g}_3\cosh \left(\pi \omega \right)-i{e}^{i{t}_0\omega }{g}_2\cosh \left({\displaystyle \frac{3\pi \omega }{2}}\right)+i{e}^{5i{t}_0\omega }{g}_2\cosh \left({\displaystyle \frac{3\pi \omega }{2}}\right)\hfill \\ \hfill & & -i{e}^{2i{t}_0\omega }{g}_1\cosh \left(2\pi \omega \right)+i{e}^{4i{t}_0\omega }{g}_1\cosh \left(2\pi \omega \right)\hfill \\ \hfill & & \left.+6A{e}^{3i{t}_0\omega }\cosh \left({\displaystyle \frac{5\pi \omega }{2}}\right)\right)\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)\mathrm{sech}\left(\pi \omega \right)=0.\hfill \end{array}$$

(13)

For instance, the equation $M\left(t_0\right)=0$ (for $p=4;N=3;A=0.05;\omega =0.09;g_1=0.6;g_2=0.3;g_3=0.1$) is depicted on Figure 7.

Proposition 6.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions if $p=6$ and $N=2$

$$M\left(t_0\right)=2\pi \left(20A-g_1\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)sin\left(t_0\omega \right)-g_2\mathrm{sech}\left(\pi \omega \right)sin\left(2t_0\omega \right)\right)=0.$$

(14)

Melnikov’s criterion for the occurrence of the intersection between the perturbed and unperturbed separatrixes can be formulated by the reader using Propositions 1 through 6. The analogous approximation issue for randomly selected p and N can be examined by the reader.

2.1.2. Some Simulations

We will concentrate on a few interesting simulations here:

1. For the given $p=2;N=2;A=0.015;ϵ=0.1;\omega =0.6;g_1=0.07;g_2=0.3$, the simulations on the system (6) for $x_0=0.6$; $y_0=0.2$ are depicted in Figure 8.

2. For the given $p=4;N=2;A=0.011;ϵ=0.1;\omega =0.92;g_1=0.9;g_2=0.2$, the simulations on the system (6) for $x_0=0.7$; $y_0=0.3$ are depicted in Figure 9.

3. For the given $p=6;N=3;A=0.011;ϵ=0.1;\omega =0.98;g_1=0.96;g_2=0.1;g_3=0.6$, the simulations on the system (6) for $x_0=0.8$; $y_0=0.4$ are depicted in Figure 10.

2.1.3. The Modeling and Synthesis of Radiating Antenna Patterns Is One Potential Use for Melnikov Functions

This brings us to $M\left(t\right)$. The hypothetical normalized antenna factor is defined as follows: The formula is $M^{\ast }\left(\theta \right)={\displaystyle \frac{1}{D}}\left|M(Kcos\theta +k_1)\right|$, where $\theta$ is the azimuth angle, $K=kd;$ $k={\displaystyle \frac{2\pi }{\lambda }};\lambda$ is the wave length, d is the distance between emitters, and $k_1$ is the phase difference.

For fixed $A=0.0001;\omega =0.27;g_1=0.6;g_2=0.001;K=19.91;k_1=0.001$ (from Proposition 4), see Figure 11.

For fixed $A=0.002;\omega =0.27;g_1=0.07;g_2=0.001;K=20.91;k_1=0$ (from Proposition 6), see Figure 12.

Naturally, the hypothetical Melnikov array that we provided can be viewed as an addition to the Array Antenna Theory, following careful evaluation by experts in this field of study.

2.2. The Model B

The following novel class of extended oscillators is examined

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

(15)

where A is the damping level, $p\ge 1$ is the damping exponent, N is an integer, and $0\le ϵ<1$. Specifically, we take into account the following model

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

(16)

2.2.1. Taking into Account Melnikov’s Methodology

The integral of Melnikov is defined as follows

$$M\left(t_0\right)={\displaystyle {\int }_{-\infty }^{\infty }}{y}_0\left(t\right)\left(A{y}_0\left(t\right){\left|y_0\left(t\right)\right|}^{p-1}+cos\left(x_0\left(t\right)\right){\displaystyle \sum _{j=1}^N}{g}_{i}cos\left(j\omega (t+t_0)\right)\right)\mathrm{d}t,$$

(17)

where Equation (7) defines the functions $x_0\left(t\right)$ and $y_0\left(t\right)$.

Proposition 7.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions if $p=2$ and $N=1$

$$M\left(t_0\right)=2\pi \left(2A+g_1{\omega }^{2}cos\left(t_0\omega \right)\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)\right)=0.$$

(18)

As an illustration, the equation $M\left(t_0\right)=0$ (for $p=2;N=1;A=0.0025;\omega =0.3;$ $g_1=0.17$) is depicted in Figure 13. The roots in at an interval of $(0,20)$ are: $6.47699;14.467$.

The reader can derive Melnikov’s condition for the dynamical model’s chaotic behavior from Proposition 7 (also see Figure 13).

Proposition 8.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions for $p=4$ and $N=2$

$$M\left(t_0\right)=2\pi \left(6A+g_1{\omega }^{2}cos\left(t_0\omega \right)\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)+4g_2{\omega }^{2}cos\left(2t_0\omega \right)\mathrm{sech}\left(\pi \omega \right)\right)=0.$$

(19)

As an illustration, the equation $M\left(t_0\right)=0$ (for $p=4;N=2;A=0.0025;\omega =0.4;$$g_1=0.17;g_2=0.05$) is depicted in Figure 14. The root in interval $(0,5)$ is: $3.74239$.

Proposition 9.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions if $p=6$ and $N=3$

$$\begin{array}{ccc}M\left(t_0\right)\hfill & =& 2\pi \left(20A+g_1{\omega }^{2}cos\left(t_0\omega \right)\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)\right.\hfill \\ \hfill & & +4g_2{\omega }^{2}cos\left(2t_0\omega \right)\mathrm{sech}\left(\pi \omega \right)\hfill \\ \hfill & & \left.+9g_3{\omega }^{2}cos\left(3t_0\omega \right)\mathrm{sech}\left({\displaystyle \frac{3\pi \omega }{2}}\right)\right)=0.\hfill \end{array}$$

(20)

As an illustration, the equation $M\left(t_0\right)=0$ (for $p=6;N=3;A=0.001;\omega =0.7;$ $g_1=0.1;g_2=0.05;g_3=0.01$) has a root of $2.15038$ (in interval $(0,3)$ (see Figure 15).

For $p=6;N=3;A=0.001;\omega =0.35;g_1=0.08;g_2=0.06;g_3=0.01$, the equation $M\left(t_0\right)=0$ has no roots (see Figure 16).

2.2.2. Some Simulations

We will concentrate on a few interesting simulations here:

1. For the given $p=4;N=2;A=0.015;ϵ=0.01;\omega =0.6;g_1=0.07;g_2=0.3$, the simulations on the system (5.16) for $x_0=0.6$; $y_0=0.3$ are depicted in Figure 17.

2. For the given $p=6;N=3;A=0.005;ϵ=0.01;\omega =0.3;g_1=0.07;g_2=0.3;g_3=0.5$, the simulations on the system (5.16) for $x_0=0.7$; $y_0=0.4$ are depicted in Figure 18.

2.2.3. The Modeling and Synthesis of Radiating Antenna Patterns Is One Potential Use for Melnikov Functions (See Section 2.1.3 for Additional Information)

For fixed $A=0.0025;\omega =0.4;g_1=0.17;g_2=0.05;K=6;k_1=0.1$ (from Proposition 8), see Figure 19.

For fixed $A=0.001;\omega =0.6;g_1=0.16;g_2=0.1;g_3=0.01;K=3.44;k_1=0.01$ (from Proposition 9), see Figure 20.

For arbitrarily selected p and N, the reader can examine the corresponding approximation problem.

Proposition 10.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as solutions to the following equation if $p=8$ and $N=4$

$$\begin{array}{ccc}M\left(t_0\right)\hfill & =& 2\pi \left(70A+g_1{\omega }^{2}cos\left(t_0\omega \right)\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)\right.\hfill \\ \hfill & & +4g_2{\omega }^{2}cos\left(2t_0\omega \right)\mathrm{sech}\left(\pi \omega \right)\hfill \\ \hfill & & +9g_3{\omega }^{2}cos\left(3t_0\omega \right)\mathrm{sech}\left({\displaystyle \frac{3\pi \omega }{2}}\right)\hfill \\ \hfill & & \left.+16g_4{\omega }^{2}cos\left(4t_0\omega \right)\mathrm{sech}\left(2\pi \omega \right)\right)=0.\hfill \end{array}$$

(21)

As an illustration, equation $M\left(t_0\right)=0$ (for $p=8;N=4;A=0.0001;\omega =0.95;g_1=0.1;g_2=0.03;g_3=0.2;g_4=0.05$) is depicted in Figure 21).

For $p=8;N=4;A=0.001;\omega =0.4;g_1=0.1;g_2=0.03;g_3=0.1;g_4=0.05$), equation $M\left(t_0\right)=0$ has no roots (see Figure 22).

The potential normalized antenna factor $M^{\ast }\left(\theta \right)$ in the case $p=8;N=4$ and fixed $A=0.0001;\omega =0.95;g_1=0.1;g_2=0.03;g_3=0.2;g_4=0.05;K=3.44;k_1=0.1$ (from Proposition 10), see Figure 23.

Proposition 11.

The roots of the Melnikov function $M\left(t_0\right)$ are provided as equation solutions if $p=10$ and $N=6$

$$\begin{array}{ccc}M\left(t_0\right)\hfill & =& 2\pi \left(252A+g_1{\omega }^{2}cos\left(t_0\omega \right)\mathrm{sech}\left({\displaystyle \frac{\pi \omega }{2}}\right)\right.\hfill \\ \hfill & & +4g_2{\omega }^{2}cos\left(2t_0\omega \right)\mathrm{sech}\left(\pi \omega \right)\hfill \\ \hfill & & +9g_3{\omega }^{2}cos\left(3t_0\omega \right)\mathrm{sech}\left({\displaystyle \frac{3\pi \omega }{2}}\right)\hfill \\ \hfill & & +16g_4{\omega }^{2}cos\left(4t_0\omega \right)\mathrm{sech}\left(2\pi \omega \right)\hfill \\ \hfill & & +25g_5{\omega }^{2}cos\left(5t_0\omega \right)\mathrm{sech}\left({\displaystyle \frac{5\pi \omega }{2}}\right)\hfill \\ \hfill & & \left.+36g_6{\omega }^{2}cos\left(6t_0\omega \right)\mathrm{sech}\left(3\pi \omega \right)\right)=0.\hfill \end{array}$$

(22)

The equation $M\left(t_0\right)=0$ (for $p=10;N=6;A=0.00001;\omega =0.8;g_1=0.07;g_2=0.03;g_3=0.17;g_4=0.05;g_5=0.77;g_6=0.02$) and normalized antenna factor $M^{\ast }\left(\theta \right)$ for $K=4.44;k_1=0$ are depicted in Figure 24.

3. Concluding Remarks

In this study, we have examined a few kinds of extended classical oscillators. We present a few specific modules for examining these oscillators’ behavior. For some of them, a cloud version is available that just needs a browser and an internet connection. This will be an essential component of a much broader web-based program for scientific computing that is in the works. In [29,30], numerical techniques for solving nonlinear algebraic equations are contained. The reader can formulate Melnikov’s criterion for the appearance of the intersection between the disturbed and unperturbed separatrixes using Propositions 1 through 6. For arbitrarily selected p and N, the reader can examine the corresponding approximation problem. We observe that arbitrary point sets in the plane about uniform metric can be approximated using Melnikov polynomials. These kinds of activities are pertinent to the general theory of radiation diagrams and digital filter fabrication and analysis. To determine $A_k$ coefficients, the standard N-element linear phased array factor is

$$AF\left(\theta \right)={\displaystyle \sum _{k=1}^{\frac{N}{2}}}{A}_{k}cos\left((2k-1)u\right)=M\left(x\right),$$

where $x=x_{0}cosu$ and $x_0$ is a design parameter, d is element separation, $\theta$ is the polar angle, and $u={\displaystyle \frac{\pi d}{\lambda }}cos\theta$. Soltis [31] used this concept in his creation of novel Gegenbauer-like and Jacobi-like antenna arrays. Naturally, this relatively new concept is a topic for defense, and experts in this field of science will conduct substantial research to support its legitimacy. The issue related to noise minimization (in decibels) also remains open.

Remark 1.

The monograph [32] is concerned with the control of chaos in dissipative non-autonomous system described by the differential equation

$$\ddot{x}+{\displaystyle \frac{\mathrm{d}U\left(x\right)}{\mathrm{d}x}}=-\mathrm{d}\left(x\dot{x}\right)+F\left(t\right),$$

where $U\left(x\right)$ is a nonlinear potential, $-\mathrm{d}\left(x\dot{x}\right)$ is a general dissipative force, and $F\left(t\right)$ is a general periodic function of period T. The planar system, corresponding to the damped pendulum, is

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-ϵ\left(Ay-g\mathit{cn}(\omega t;0)\right),\hfill \end{array}\right.$$

where $A,g>0$, and $cn(\omega t;m)$ is the Jacoby elliptic function of parameter m. In the book cited above, one can find a relatively precise bibliographic reference on the mentioned topic, as well as some modified but not sufficiently studied differential models. Following the considerations in this article, the reader can successfully formulate and investigate the dynamics of the following extended oscillators:

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-ϵ\left({Ay\left|y\right|}^{p-1}-{\displaystyle \sum _{j=1}^N}{g}_{j}cn(j\omega t;0)\right),\hfill \end{array}\right.$$

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& y\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& -sinx-ϵ\left({Ay\left|y\right|}^{p-1}-cosx{\displaystyle \sum _{j=1}^N}{g}_{j}cn(j\omega t;0)\right),\hfill \end{array}\right.$$

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

Proposition 12.

If $p=8$, $N=4$, and $\left|\mathrm{Im}\left(\omega \right)\right|<{\displaystyle \frac{1}{4}}$, then the roots of the corresponding Melnikov function $M\left(t_0\right)$ are given as solutions of the equation:

$$\begin{array}{ccc}M\left(t_0\right)\hfill & =& 70A-g_{1}cos\left(t_0\omega \right)\mathrm{sech}{\displaystyle \frac{\pi \omega }{2}}-g_{2}cos\left(2t_0\omega \right)\mathrm{sech}\left(\pi \omega \right)-\hfill \\ & & g_{3}cos\left(3t_0\omega \right)\mathrm{sech}{\displaystyle \frac{3\pi \omega }{2}}-g_{4}cos\left(4t_0\omega \right)\mathrm{sech}\left(2\pi \omega \right)=0.\hfill \end{array}$$

For example, the equation $M\left(t_0\right)=0$ (for $p=8$; $N=4$; $A=0.00$; $\omega =0.2$; $g_1=0.2$; $g_2=0.62$; $g_3=0.1$; $g_4=0.425$) is depicted in Figure 25.

Remark 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 (see [33,34,35,36,37,38]). Gear systems have been widely used in many industrial applications due to their advantages of having accurate transmission ratios, compact dimensions, and high efficiency. A nonlinear dynamic model of a spur gear pair with backlash and static transmission error is formulated in [33] as

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

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

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

where α is a mechanical parameter. The authors in [33] consider the following polynomial approximation (for $\alpha =0$): $f_h\left(x\right)\approx -0.1667x+0.1667x^3$ and study the dynamics of model using Melnikov analysis. The reader can formulate and explore the dynamics of the modified model of the type

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

in which a trigonometric approximation for backlash function $f_h\left(x\right)$ of type $sinx$ or $sinx+cx$ is used, and then use the apparatus proposed in this article.

Remark 3.

A number of authors have devoted their research to the RF superconducting quantum interference device (SQUID) driven by an oscillating external flux (see, for example, [39,40,41,42,43,44,45,46,47,48]). In this regard, we would like to note that the detailed study of modified dynamic models of type

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

will be the subject of our future research.

Remark 4.

For the purposes of our research (thankfully), the user does not need to have an explicit representation of the Melnikov polynomials corresponding to our differential model (for large values of the parameters), because the output of the first module used becomes the input for the second and most important module (implemented in the CAS Mathematica) used, designed to generate and visualize the corresponding diagrams of the radiating antenna array.