Notes on Modified Planar Kelvin–Stuart Models: Simulations, Applications, Probabilistic Control on the Perturbations

Abstract

In this paper, we propose a new modified planar Kelvin–Stuart model. We demonstrate some 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. Investigations in light of Melnikov’s approach are considered. Some simulations and applications are also presented. The proposed new modifications of planar Kelvin–Stuart models contain many free parameters (the coefficients $g_i,i=1,2,\dots ,N$), which makes them attractive for use in engineering applications such as the antenna feeder technique (a possible generating and simulating of antenna factors) and the theory of approximations (a possible good approximation of a given electrical stage). The probabilistic control of the perturbations is discussed.

1. Introduction

A number of authors devote their research to the classical differential model of Kelvin–Stuart.

The publications on this topic are significant and varied. In [1], Bertozzi presented an extension of the planar Smale–Birkhoff homoclinic theorem to the case of a heteroclinic saddle connection containing a finite number of fixed points. An extension of Melnikov’s method is also given. More precisely, Bertozzi considers the following Kelvin–Stuart cat’s eye flow in the following plane:

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{a\sinh y}{a\cosh y+\sqrt{a^2-1}cosx}}-{\displaystyle \frac{ϵb\left(t\right)\sinh ycosx}{\sqrt{a^2-1}(a\cosh y+\sqrt{a^2-1}cosx)}}\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{\sqrt{a^2-1}sinx}{a\cosh y+\sqrt{a^2-1}cosx}}+{\displaystyle \frac{ϵb\left(t\right)sinx\cosh y}{\sqrt{a^2-1}(a\cosh y+\sqrt{a^2-1}cosx)}},\hfill \end{array}\right.$$

(1)

where $0\le ϵ<1$, the parameter $a>1$ controls the shape of the cat’s eye.

It is shown [1] that if $b\left(t\right)$ has the form $cos\left(kt\right)$, then the perturbation corresponds to the superposition of four waves.

H. Lamb [2] described the row-of-point-vortex system and proved that it is linearly unstable for double-periodic perturbations. R. E. Kelly [3] numerically observed that the Kelvin–Stuart vortex is unstable for double-periodic perturbations. For other results, see [4,5].

Kelvin–Stuart vortices are classical mixing layer flows with many applications in fluid mechanics and plasma physics.

Liao, Lin and Zhu [6] proved that the whole family of Kelvin–Stuart vortices is nonlinearly stable for co-periodic perturbations, and linearly unstable for multi-periodic or modulational perturbations.

Rodrigue [7] studied mixing and transport in the Kelvin–Stuart cat’s-eye-driven flow

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{\sinh y}{\cosh y+Acosx}}+ϵsin\left(\omega t\right)\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{Asinx}{\cosh y+Acosx}}+ϵsin\left(\omega t\right),\hfill \end{array}\right.$$

(2)

using the topological approximation method.

In the serious studies [6,7] cited above, the reader can find a considerable volume of literature devoted to this classic model.

The reader can find the module for calculating and plotting the internal maximum and minimum bifurcation curves for the model (2) and topological approximation method module implemented in CAS Mathematica in [7].

For mixing and the chaotic transport in the presence of diffusion see [8,9,10,11,12,13].

In this paper, we suggest a modified planar Kelvin–Stuart model.

Several simulations are composed. We also demonstrate 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 [14,15,16].

The plan of the article is as follows. We state our models in Section 2.1 and Section 2.4. Some simulations are presented in Section 2.2. Considerations in light of Melnikov’s approach are given in Section 2.5. Some applications are considered in Section 2.3 and Section 4. The probabilistic control of the perturbations is discussed in Section 3.

2. The Models

2.1. The Model A

We consider the following modified planar Kelvin–Stuart model

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{a\sinh y}{a\cosh y+\sqrt{a^2-1}cosx}}\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{\sqrt{a^2-1}sinx}{a\cosh y+\sqrt{a^2-1}cosx}}-ϵ{\displaystyle \sum _{j=1}^N}{g}_{j}sin\left(j\omega t\right),\hfill \end{array}\right.$$

(3)

where $0\le ϵ\le 1$, $g_i\ge 0$, and N is the integer.

Without community restriction, we will fix the parameter $a=2$.

The Hamiltonian of this system ($ϵ=0$) is

$$H(x,y)=log(2\cosh y+\sqrt{3}cosx).$$

Following the ideas given in [1], the form of the heteroclinic orbit for the unperturbed dynamical system (3) is (see Figure 1):

$$\begin{array}{c}{x}_0\left(t\right)=\pm \mathrm{Arcos}\left(1-{\displaystyle \frac{16}{2+\sqrt{3}}}{\displaystyle \frac{1}{e^{\gamma t}+\beta +e^{-\gamma t}}}\right),\hfill \\ y_0\left(t\right)=\pm \mathrm{Arcosh}\left({\displaystyle \frac{2+\sqrt{3}-\sqrt{3}cos\left(x_0\left(t\right)\right)}{2}}\right),\hfill \end{array}$$

(4)

where

$$\gamma ={\displaystyle \frac{\sqrt{3}}{2+\sqrt{3}}}\sqrt{{\displaystyle \frac{8}{\sqrt{3}}}}$$

and

$$\beta =2{\displaystyle \frac{2-\sqrt{3}}{2+\sqrt{3}}}.$$

2.2. Some Simulations

We will look at some interesting simulations of model (3):

Example 1. 

For the given $N=1$, $\omega =0.092$, $g_1=0.9$, $ϵ=0.0095$ the simulations of the system (3) for $x_0=0.3$, $y_0=0.1$ are depicted in Figure 2.

Example 2. 

For the given $N=2$, $\omega =0.15$, $g_1=0.3$, $g_2=0.8$, $ϵ=0.0095$ the simulations of the system (3) for $x_0=0.3$, $y_0=0.1$ are depicted in Figure 3.

Example 3. 

For the given $N=3$, $\omega =0.35$, $g_1=0.3$, $g_2=0.8$, $g_3=0.2$, $ϵ=0.0095$ the simulations of the system (3) for $x_0=0.3$, $y_0=0.1$ are depicted in Figure 4.

Example 4. 

For the given $N=4$, $\omega =0.35$, $g_1=0.3$, $g_2=0.8$, $g_3=0.2$, $g_4=0.4$, $ϵ=0.0095$ the simulations of the system (3) for $x_0=0.3$, $y_0=0.1$ are depicted in Figure 5.

Example 5. 

For the given $N=5$, $\omega =0.35$, $g_1=0.3$, $g_2=0.85$, $g_3=0.2$, $g_4=0.4$, $g_5=0.6$, $ϵ=0.0095$ the simulations of the system (3) for $x_0=0.3$, $y_0=0.089$ are depicted in Figure 6.

2.3. Some Applications

• In some cases, after a suitable change of the variable $t=kcos\theta +k_1$ in the y component of the differential system (3), the expression $\left|y\right(\theta \left)\right|$ can be used to model a characteristic antenna factor in confidential intervals (see, for example, Figure 7).
• In a number of cases, the x component of the solution of the differential system (3) can be used to approximate electrical stages (see Figure 8).

Figure 8 actually depicts the approximation of the function:

$$s\left(t\right)=\left\{\begin{array}{c}-15.6,-200\le t\le -95,\hfill \\ 3,-95<t<140,\hfill \\ 22,140\le t\le 270\hfill \end{array}\right.$$

by using the $x\left(t\right)$ component of the system (3) for fixed $N=4,\omega =0.35,ϵ=0.0095$, $g_1=0.3,g_2=0.2,g_3=0.2,g_4=0.4$ with initial conditions $x_0=0.3,y_0=0.1$ in interval $(-200,270)$.

We will explicitly note that the $x\left(t\right)$ component can be used in practice to approximate specific functions and point sets in the plane.

Figure 9 actually depicts the approximation of the cut function (red) by using $x\left(t\right)$ component of the system (3) for fixed $N=3,\omega =0.35,ϵ=0.0095,g_1=0.1,g_2=0.3,g_3=0.1$ with initial conditions $x_0=0.3,y_0=0.1$ in interval $(179,278)$.

2.4. Model B

Consider the generalized planar Kelvin–Stuart cat’s eye (perturbed system)

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{\sinh y}{\cosh y+Acosx}}+ϵ{\displaystyle \sum _{j=1}^N}{g}_{j}sin\left(j\omega t\right)\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{Asinx}{\cosh y+Acosx}}+ϵ{\displaystyle \sum _{j=1}^N}{g}_{j}sin\left(j\omega t\right),\hfill \end{array}\right.$$

(5)

where $0\le ϵ\le 1$, $g_i\ge 0$ and N is integer.

It is known that the unperturbed Hamiltonian is

$$H(x.y)=ln(\cosh y+Acosx)-ln(1+A)$$

and the heteroclinic orbit can be expressed as [7]

$$\begin{array}{c}{\Gamma }_0\left(t\right)=(x_0\left(t\right),y_0\left(t\right));\hfill \\ x_0\left(t\right)=\mathrm{Arcos}\left({\displaystyle \frac{{\sinh }^2\left(\lambda t\right)-\frac{\lambda }{\sqrt{A}}}{{\cosh }^2\left(\lambda t\right)-\sqrt{A}\lambda }}\right);\hfill \\ y_0\left(t\right)=\mathrm{Arcosh}\left({\displaystyle \frac{{\cosh }^2\left(\lambda t\right)+\sqrt{A}\lambda }{{\cosh }^2\left(\lambda t\right)-\sqrt{A}\lambda }}\right),\hfill \end{array}$$

where $\lambda ={\displaystyle \frac{\sqrt{A}}{1+A}}$.

Example 6. 

For the given $N=6$, $\omega =0.2$, $g_1=0.3$, $g_2=0.3$, $g_3=0.4$, $g_4=0.3$, $g_5=0.3$, $g_6=0.1$, $ϵ=0.1$, $A=0.3$ the simulations of the system (5) for $x_0=0.3$, $y_0=0.089$ are depicted on Figure 10.

2.5. Considerations in Light of Melnikov’s Approach

Denote f and g as

$$f=\left(\begin{array}{c}{\displaystyle \frac{\sinh y}{\cosh y+Acosx}}\hfill \\ {\displaystyle \frac{Asinx}{\cosh y+Acosx}}\hfill \end{array}\right);g=\left(\begin{array}{c}{\displaystyle \sum _{j=1}^N}{g}_{j}sin\left(j\omega t\right)\hfill \\ {\displaystyle \sum _{j=1}^N}{g}_{j}sin\left(j\omega t\right)\hfill \end{array}\right).$$

Then the Melnikov function [17] is of the form

$$M\left(t_0\right)={\displaystyle {\int }_{-\infty }^{\infty }}{e}^{-\int \nabla .f\left({\Gamma }_0\left(s\right)\right)ds}f\left({\Gamma }_0\left(t\right)\right)\wedge g({\Gamma }_0\left(t\right),t+t_0)dt$$

(we note that $\nabla .f\left({\Gamma }_0\left(s\right)\right)=0$ for the considered Kelvin–Stuart cat’s eye system).

For the wedge product, we have

$$\begin{array}{c}f\left({\Gamma }_0\left(t\right)\right)\wedge g({\Gamma }_0\left(t\right),t+t_0)=\hfill \\ {\displaystyle \frac{1}{1+A}}\left(\sqrt{{\left({\displaystyle \frac{{\cosh }^2\left(\lambda t\right)+\sqrt{A}\lambda }{{\cosh }^2\left(\lambda t\right)-\sqrt{A}\lambda }}\right)}^2-1}-A\sqrt{1-{\left({\displaystyle \frac{{\sinh }^2\left(\lambda t\right)-\frac{\lambda }{\sqrt{A}}}{{\cosh }^2\left(\lambda t\right)-\sqrt{A}\lambda }}\right)}^2}\right)\times \hfill \\ \times {\displaystyle \sum _{j=1}^N}{g}_{j}sin\left(j\omega (t+t_0)\right).\hfill \end{array}$$

Using the known equality ${\cosh }^2\left(\lambda t\right)-\sqrt{A}\lambda ={\sinh }^2\left(\lambda t\right)+{\displaystyle \frac{\lambda }{\sqrt{A}}}$, we obtain

$$\begin{array}{c}f\left({\Gamma }_0\left(t\right)\right)\wedge g({\Gamma }_0\left(t\right),t+t_0)=\hfill \\ {\displaystyle \frac{2\sqrt{\lambda \sqrt{A}}}{1+A}}\left({\displaystyle \frac{\cosh \left(\lambda t\right)-\sqrt{A}\sinh \left(\lambda t\right)}{{\cosh }^2\left(\lambda t\right)-\lambda \sqrt{A}}}\right){\displaystyle \sum _{j=1}^N}{g}_{j}sin\left(j\omega (t+t_0)\right).\hfill \end{array}$$

Using

$$sin\left(i\omega (t+t_0)\right)=sin\left(i\omega t\right)cos\left(i\omega t_0\right)+cos\left(i\omega t\right)sin\left(i\omega t_0\right)$$

for the Melnikov integral, we have

$$\begin{array}{c}M\left(t_0\right)={\displaystyle \frac{2\sqrt{\lambda \sqrt{A}}}{1+A}}{\displaystyle {\int }_{-\infty }^{\infty }}\left({\displaystyle \frac{\cosh \left(\lambda t\right)-\sqrt{A}\sinh \left(\lambda t\right)}{{\cosh }^2\left(\lambda t\right)-\lambda \sqrt{A}}}\right)\times \hfill \\ {\displaystyle \sum _{j=1}^N}{g}_j\left(sin\left(j\omega t\right)cos\left(j\omega t_0\right)+cos\left(j\omega t\right)sin\left(j\omega t_0\right)\right)dt.\hfill \end{array}$$

(6)

Using the even and odd symmetries of the trigonometric and hyperbolic functions, it is easy to see that if we define

$$\begin{array}{c}{I}_i={\displaystyle \frac{2\sqrt{\lambda \sqrt{A}}}{1+A}}{g}_{i}sin\left(i\omega t_0\right){\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle \frac{\cosh \left(\lambda t\right)cos\left(i\omega t\right)}{{\cosh }^2\left(\lambda t\right)-\lambda \sqrt{A}}}dt\hfill \\ i=1,2,\dots ,N\hfill \end{array}$$

and

$$\begin{array}{c}{I}_i^{*}={\displaystyle \frac{2\sqrt{\lambda \sqrt{A}}\sqrt{A}}{1+A}}{g}_{i}cos\left(i\omega t_0\right){\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle \frac{sinh\left(\lambda t\right)sin\left(i\omega t\right)}{{\cosh }^2\left(\lambda t\right)-\lambda \sqrt{A}}}dt\hfill \\ i=1,2,\dots ,N\hfill \end{array}$$

then the Melnikov function can be represented as follows

$$M\left(t_0\right)=\sum _{i=1}^N{I}_i-\sum _{i=1}^N{I}_i^{*}.$$

(7)

From a numerical point of view, the task of finding the roots of $M\left(t_0\right)$ is more interesting, given that the parameters appearing in the proposed differential model are subject to a number of restrictions of a physical and practical nature.

The explicit representation of the Melnikov integral [17] corresponding to the differential system (5) can be obtained using the known technique described in [1,7,18].

The calculation of $M\left(t_0\right)$ is related to the application of the residue theorem.

This is the reason why we offer representation (7).

We note that with large values of the parameter N, the work of the dedicated modules implemented in the existing computer algebraic systems for scientific research for solving the Melnikov’s integrals is very difficult, and the user must also perform serious preliminary preparation and choose appropriate restrictions regarding the parameter $\omega$.

Using the identity ${\cosh }^2\left(\lambda t\right)={\displaystyle \frac{1}{2}}\left(\cosh \left(2\lambda t\right)+1\right)$, (following the ideas given in [7]) and with the substitution $t^{\prime }=2\lambda t$, the Equation (7) can be rewritten as

$$\begin{array}{c}M\left(t_0\right)={\displaystyle \sum _{i=1}^N}{I}_i-{\displaystyle \sum _{i=1}^N}{I}_i^{*},\hfill \\ I_i=2\sqrt{{\displaystyle \frac{\lambda }{\sqrt{A}}}}{g}_{i}sin\left(i\omega t_0\right){\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle \frac{\cosh \left(\frac{t}{2}\right)cos\left(\frac{i\omega t}{2\lambda }\right)}{\cosh t+\frac{1-A}{1+A}}}dt,\hfill \\ I_i^{*}=2\sqrt{\lambda \sqrt{A}}{g}_{i}cos\left(i\omega t_0\right){\displaystyle {\int }_{-\infty }^{\infty }}{\displaystyle \frac{\sinh \left(\frac{t}{2}\right)sin\left(\frac{i\omega t}{2\lambda }\right)}{\cosh t+\frac{1-A}{1+A}}}dt,\hfill \\ i=1,2,\dots ,N.\hfill \end{array}$$

(8)

Representation (8) is more appropriate compared to (7) and can be used directly by the users of the relevant specialized module, implemented for example in CAS Mathematica.

For example, the Melnikov function (7) for fixed $N=1$ and $g_1=1$, in case (a) $A=0.9$, $\omega =0.8$ and in case (b) $A=0.7$, $\omega =1.8$, is depicted in Figure 11.

Figure 12 illustrates the calculation of $I_n$ using CAS Mathematica 8.

$I_n^{*}$ is calculated in a similar way (see Figure 13).

Of course, the user must specify reasonable restrictions, for example $Im\left[{\displaystyle \frac{\omega n}{\sqrt{A}}}\right]+Im\left[\sqrt{A}\omega n\right]<1$.

Furthermore, for the explicit representation of $M\left(t_0\right)=0$, they must perform a number of additional calculations, including the $Simplify[\%]$ operator, since the calculated integrals (see Figure 11 and Figure 12) are expressed by hypergeometric function ${}_{2}F_1[.,.,.,.]$.

Remark. If $M\left(t_0\right)=0$ and ${\displaystyle \frac{M\left(t_0\right)}{\mathrm{d}{t}_0}}\ne 0$ for some $t_0$ and some sets of parameters, then chaos occurs. Nonstandard numerical methods connected to the investigation of the roots of equation $M\left(t_0\right)=0$ can be found in [19,20].

For example, the Melnikov functions (8) for fixed $N=2$ and (a) $A=0.2$, $\omega =0.95$, $g_1=0.8$, $g_2=0.9$; (b) $A=0.3$, $\omega =0.8$, $g_1=0.7$, $g_2=0.6$; (c) $A=0.35$, $\omega =0.7$, $g_1=0.6$, $g_2=0.5$ are depicted in Figure 14.

3. Probabilistic Control of the Perturbations

Let us first discuss Model A, defined as (3). In [16], the coefficients $g_j$ are assumed to be the probabilities of some discrete distribution after scaling with ${\displaystyle \sum _{j=1}^N{g}_j}$. Using the complex presentation of the sin-function

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

(9)

the authors present the weighted sum of sins as

$$\begin{array}{cc}\hfill \sum _{j=1}^N{g}_{j}sin\left(j\omega t\right)& ={\displaystyle \frac{1}{2i}}\left(\mathbb{E}\left[e^{i\xi \omega t}\right]-\mathbb{E}\left[e^{-i\xi \omega t}\right]\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2i}}\left(\psi \left(\omega t\right)-\psi \left(-\omega t\right)\right),\hfill \end{array}$$

(10)

where $\psi \left(·\right)$ is the characteristic function of the related distribution. Using the same approach, we rewrite now the y-dynamics of (3) as

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}& ={\displaystyle \frac{\sqrt{a^2-1}sinx}{a\cosh y+\sqrt{a^2-1}cosx}}-ϵ{\displaystyle \sum _{j=1}^N}{g}_{j}sin\left(j\omega t\right)\hfill \\ \hfill & ={\displaystyle \frac{\sqrt{a^2-1}sinx}{a\cosh y+\sqrt{a^2-1}cosx}}-{\displaystyle \frac{ϵ}{2i}}\left(\psi \left(\omega t\right)-\psi \left(-\omega t\right)\right).\hfill \end{array}$$

(11)

In the current paper, we generalize the choice of possible distributions including the continuous models. To do this, we change the y-dynamics of (3) into

$${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}={\displaystyle \frac{\sqrt{a^2-1}sinx}{a\cosh y+\sqrt{a^2-1}cosx}}-ϵ\underset{D}{\int }g\left(u\right)sin\left(u\omega t\right)du,$$

(12)

where $g\left(·\right)$ is the probability density function (PDF) of the distribution and D is its domain. Note that presentation (12) with respect to the characteristic function still holds. We shall present several simulations based on two important distributions—gamma and beta. Their PDFs are

$$\begin{array}{cc}\hfill g_{\gamma }\left(u\right)& ={\displaystyle \frac{{\theta }^{\alpha }}{\Gamma \left(\alpha \right)}}{u}^{\alpha -1}{e}^{-\theta u},\hfill \\ \hfill g_{\beta }\left(u\right)& ={\displaystyle \frac{u^{\alpha -1}{\left(1-u\right)}^{\theta -1}}{B\left(\alpha ,\theta \right)}}.\hfill \end{array}$$

(13)

All driving parameters are positive reals. The domains are $D_{\gamma }=\left(0,\infty \right)$ and $D_{\beta }=\left(0,1\right)$. The characteristic functions are

$$\begin{array}{cc}\hfill {\psi }_{\gamma }\left(x\right)& ={\left({\displaystyle \frac{\theta }{\theta -ix}}\right)}^{\alpha },\hfill \\ \hfill {\psi }_{\beta }\left(x\right)& ={}_{1}F_1\left(\alpha ,\alpha +\theta ,ix\right).\hfill \end{array}$$

(14)

Note that these functions are always well defined. Some simulations are presented in Figure 15. The used parameters for dynamics (12) are $a=7$, $\omega =2$, $ϵ=1$, $x\left(0\right)=0.3$, and $y\left(0\right)=0.1$. The parameters for the gamma distribution are $\theta =10$ and $\alpha =5$, whereas the beta ones are $\alpha =1$ and $\beta =2$.

Next, we discuss Model B, defined by (5). Using the same approach, we modify it to

$$\left\{\begin{array}{ccc}{\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{\sinh y}{\cosh y+Acosx}}+{\displaystyle \frac{ϵ}{2i}}\left(\psi \left(\omega t\right)-\psi \left(-\omega t\right)\right)\hfill \\ {\displaystyle \frac{\mathrm{d}y}{\mathrm{d}t}}\hfill & =& {\displaystyle \frac{Asinx}{\cosh y+Acosx}}+{\displaystyle \frac{ϵ}{2i}}\left(\psi \left(\omega t\right)-\psi \left(-\omega t\right)\right).\hfill \end{array}\right.$$

(15)

We provide the dynamics in Figure 16, using the same value of the parameters together with $A=0.9$.

4. Concluding Remarks

1. In some cases, after a suitable change of the variable $t=kcos\theta +k_1$ in the y component of the differential system (5)—Model B, the expression $\left|y\right(\theta \left)\right|$ can be used to model a characteristic antenna factor or normalized factor ${\displaystyle \frac{\left|y\right(\theta \left)\right|}{N}}$ in confidential intervals.

Example 7. 

For $N=4,A=0.1,k=15.5,k_1=-0.25,ϵ=0.095,\omega =0.5,g_1=0.02,$$g_2=0.03,g_3=0.01,g_4=0.02$, the antenna factor is depicted in Figure 17.

Example 8. 

For $N=6,A=0.2,k=14.5,k_1=-0.15,ϵ=0.095,\omega =0.5,g_1=0.02$, $g_2=0.03,g_3=0.04,g_4=0.03,g_5=0.02,g_6=0.01$, the antenna factor is depicted in Figure 18.

Example 9. 

For $N=8,A=0.2,k=16,k_1=-0.85,ϵ=1,\omega =0.5,g_1=0.02$, $g_2=0.03,g_3=0.04,g_4=0.03,g_5=0.02,g_6=0.01,g_7=0.006,g_8=0.002$, the antenna factor is depicted in Figure 19.

Example 10. 

For $N=10,A=0.35,k=15.5,k_1=-0.4,ϵ=1,\omega =0.535,g_1=0.02$, $g_2=0.03,g_3=0.04,g_4=0.03,g_5=0.02,g_6=0.01,g_7=0.006,g_8=0.002,g_9=0.003$, $g_{10}=0.001$, the antenna factor is depicted in Figure 20.

2. The proposed new modifications of planar Kelvin–Stuart models—Models A and B—contain many free parameters (the coefficients $g_i,i=1,2,\dots ,N$), which makes them attractive for use in the engineering applications mentioned above (the antenna feeder technique and the theory of approximations).

Of course, the specialists working in these scientific fields have the last word.

As we have already noted in our previous publications (for more details, see [21]), with a suitable change of variable $t=kcos\theta +k_1$, the normalized expression is ${\displaystyle \frac{1}{D}}\left|M^{*}\left(\theta \right)\right|$, where $D=max{M}^{*}(.)$ can be used to model a characteristic antenna factor in confidential intervals. We will point out some examples.

Example 11. 

For $N=2,A=0.2,k=6,k_1=0.06,\omega =0.95,g_1=0.8,g_2=0.9$, the Melnikov antenna factor is depicted in Figure 21.

Example 12. 

For $N=4,A=0.5,k=9.6,k_1=0,\omega =0.94,g_1=g_2=g_3=g_4=0.1$, the Melnikov antenna factor is depicted in Figure 22.

The motivation in our paper is clear—proposing an extended model that allows us to explore similar classical models at a ”higher energy level” and generate Melnikov high-order polynomials (corresponding to the proposed extended model). We note that the general N-element linear phased array factor used to find $A_k$ coefficients 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 $u={\displaystyle \frac{\pi d}{\lambda }}cos\theta$, d is the element separation, $\theta$ is the polar angle, and $x=x_{0}cosu$, where $x_0$ is a design parameter. This idea was borrowed from Soltis [22], in his generation of new Gegenbauer-like and Jacobi-like antenna arrays.

Of course, practical antenna implementation, especially in the case of a large number of emitters, is a difficult task. In this regard, detailed studies by specialists working in this scientific direction are necessary.

Important issues related to the study of mixing and transport in Kelvin–Stuart cat’s-eye-driven flow (5) using the topological approximation method will be investigated in our future work.

We also envisage future research on the current topic—the modeling of chaotic systems (of the type (3) and (5)) with a desired set of properties [23,24], the study of the quasi-periodic structure of the disturbed system, and the existence of periodic orbits and associated invariant tori [25].