## 2 Main Results-Simulations
### The New Model in the Light of [PERSON]'s Considerations  
The case \(n=5\). Let us explore the model  
\[\left\{\begin{array}{rcl}\frac{dx}{dt}&=&y-\epsilon(32x^{5}-32x^{3}+\mu x)\\ \frac{dy}{dt}&=&-x\end{array}\right. \tag{6}\]  
where \(\mu>0\), \(\epsilon>0\). The following is correct:  
**Proposition 1**.: _The Lienard-type system (6) for \(n=5\), and for all small enough \(\epsilon\
eq 0\) for \(\mu=7.2\) has a limit cycle with multiplicity-2._  
Proof.: For the Melnikov polynomial in \(r^{2}\) (see Figure 6), we have:  
\[P(r^{2},2)=\frac{\mu}{2}-12r^{2}+10r^{4}. \tag{7}\]  
The roots are  
\[\frac{\sqrt{6-\sqrt{36-5\mu}}}{\sqrt{10}};\ \sqrt{\frac{3}{5}+\frac{1}{10} \sqrt{36-5\mu}}.\]  
The radii \(r_{i}\); \(i=1,2\) for some values of \(\mu\) are tabulated in Table 1.  
Evidently, for example \(\mu=7.2\) we obtain a limit cycle with multiplicity-2 (see Table 1). This ends the proof of the proposition. The solution of the system (6) for \(\epsilon=0.001\), \(n=5\), \(\mu=7.2\) with \(x_{0}=0.7\), \(y_{0}=0.3\) is visualized on Figure 7.  
The case \(n=7\). Let us explore the model  
\[\left\{\begin{array}{rcl}\frac{dx}{dt}&=&y-\epsilon(128x^{7}-192x^{5}+80x^{3} -\mu x)\\ \frac{dy}{dt}&=&-x\end{array}\right. \tag{8}\]  
where \(\mu>0\), \(\epsilon>0\). The following is correct:  
\begin{table}
\begin{tabular}{c c c} \hline \(\mu\) & \(\mathbf{r_{1}}\) & \(\mathbf{r_{2}}\) \\ \hline
5 & 0.518013 & 0.965227 \\
6 & 0.595862 & 0.919211 \\
7 & 0.707107 & 0.83666 \\
7.2 & 0.774597 & 0.774597 \\ \hline \end{tabular}
\end{table}
Table 1: The radii \(r_{i}\); \(i=1,2\) for some values of \(\mu\).  
Figure 6: \(P(r^{2},2)\) for \(n=5\) and \(\mu=7.2\).  
**Proposition 2**.: _The Lienard-type system (8) for \(n=7\), and for all small enough \(\epsilon\
eq 0\) for \(\mu=7.00805257\ldots\) has a simple limit cycle and limit cycle with multiplicity-2._  
Proof.: The Melnikov polynomial in \(r^{2}\) (see Figure 8) gives:  
\[P(r^{2},3)=-\frac{\mu}{2}+30r^{2}-60r^{4}+35r^{6}. \tag{9}\]  
The radii \(r_{i}\); \(i=1,2,3\) for some values of \(\mu\) are tabulated in Table 2.  
Figure 8: \(P(r^{2},3)\) for \(n=7\) and \(\mu=7.00805257\). The roots are: 0.409106—simple and 0.879462—with multiplicity 2.  
Figure 7: (**a**) The solutions of the system (6) (\(\epsilon=0.001\); \(n=5\); \(\mu=7.2\)); (**b**) the portrait \((x,y)\); (**c**) 3D plot.  
For an proper choice of the parameter \(\mu\) (for example \(\mu=7.00805257\)), we have a limit cycle and limit cycle with multiplicity 2 (see Table 2). The solution of the system (8) for \(\epsilon=0.001\), \(n=7\), \(\mu=7.00805257\) with \(x_{0}=0.6\), \(y_{0}=0.4\) is visualized in Figure 9.  
The case \(n=9\). Consider the model  
\[\left\{\begin{array}{rcl}\dfrac{dx}{dt}&=&y-\epsilon(512x^{9}-1024x^{7}+672x ^{5}-160x^{3}+\mu_{1}x)\\ \dfrac{dy}{dt}&=&-x\end{array}\right. \tag{10}\]  
where \(\mu_{1}>0\), \(\epsilon>0\). The following is correct:  
**Proposition 3**.: _The Lienard-type system (10) for \(n=9\), and for all small enough \(\epsilon\
eq 0\) for \(\mu_{1}=9.0482691\ldots\), has 2 simple limit cycles and limit cycle with multiplicity-2._  
The proof of existence of limit cycles is based on the ideas given in Propositions 1 and 2, and we will omit it here.  
\begin{table}
\begin{tabular}{c c c c} \hline \(\mu\) & \(\mathbf{r_{1}}\) & \(\mathbf{r_{2}}\) & \(\mathbf{r_{3}}\) \\ \hline
7.5 & 0.433391 & 0.808263 & 0.934435 \\
7.3 & 0.423255 & 0.826862 & 0.922735 \\
7.1 & 0.413469 & 0.851545 & 0.904544 \\
7.008805257 & 0.409106 & 0.878462 & 0.879469 \\ \hline \end{tabular}
\end{table}
Table 2: The values of \(\mu\) that proceed in 3 positive zeros of the Melnikov polynomial (Theorem 2); for \(\mu=7.00487909525\): a simple limit cycle and limit cycle with multiplicity–2.  
We note that for the Melnikov polynomial in \(r^{2}\) (see Figure 10) we obtain:  
\[P(r^{2},4)=\frac{\mu}{2}-60r^{2}+210r^{4}-280r^{6}+126r^{8}. \tag{11}\]  
The solution of the system (10) for \(\epsilon=0.001\), \(n=9\), \(\mu_{1}=9.0482691\) with \(x_{0}=0.5\), \(y_{0}=0.5\) is visualized in Figure 11.