# On a Class of Orthogonal Polynomials as Corrections in
## 2 Main Results
: For the Melnikov polynomial in \(r^{2}\) (see Figure 5) we have:  
\[M(r^{2},5)=-\frac{\mu}{2}+\frac{6123}{414,720}r^{2}-\frac{335,855}{2,903,040}r ^{4}+\frac{794,885}{2,150,400}r^{6}-\frac{3,527,433}{7,096,320}r^{8}+\frac{231 }{1024}r^{10}.\]  
Evidently, for example \(\mu=0.0012953041893\) we have three simple limit cycles and limit cycle with multiplicity two. Numerical methods for finding zeros of polynomials can be found in [38; 39; 40]. Figure 5: **(a)** The Melnikov polynomial \(M(r^{2},5)\) for \(n=11\) and \(\mu=\frac{13}{11,520}\) (five limit cycles); **(b)** The Melnikov polynomial \(M(r^{2},5)\) for \(n=11\) and \(\mu=0.0012953041893\) (simple limit cycles \(0.639205\), \(0.706901\), \(1.03221\) and limit cycle \(0.338942\) with multiplicity two). We will note that some specifics of the amplitudes of these polynomials open up the possibility of modeling signals from the field of antenna-feeder technology. It is easy to take into account that the change of the variable \(t\) with \(t=b\cos\theta+c\) (\(\theta\) is the azimuthal angle and \(c\) is the phase difference) in the \(y(t)\)-component of the solution of the system (6) leads to radiation diagrams [41; 42]. Consider the following model in the light of [PERSON]'s approach:  
\[\left\{ \begin{array}{rcl}\frac{dx}{dt}&=&c(G(x)-y)\\ \frac{dy}{dt}&=&\frac{1}{c}x\end{array} \right. \tag{7}\]  
Figure 7: The simulations (system (6)) for \(x_{0}=1.1\), \(y_{0}=0.08\), \(b=1\), \(c=0.001\); \(\varepsilon=0.0001\): (**a**) the solution of the system; (**b**) \(y\)-component of the solution; (**c**) the portrait; (**d**) emitting chart. with \(c>0\) and  
\[G(x)=x^{9}-\frac{4609}{2520}x^{7}+\frac{10,219}{10,080}x^{5}-\frac{2519}{12,960}3 x^{3}+qx.\]  
The catastrophe surfaces \((x,y,p)=G(x)-y\) (\(q=5,10,15\)) for the model is depicted on Figure 8.  
Consider the model (7) with \(c>0\) and  
\[G(x)=x^{11}-\frac{55,991}{27,720}x^{9}+\frac{22,711}{16,800}x^{7}-\frac{67,17 1}{181,440}x^{5}+\frac{2041}{51,840}x^{3}-qx.\]  
The catastrophe surfaces \((x,y,p)=G(x)-y\) (\(q=1,2,3,4,5\)) for the model is depicted on Figure 9.  
Associated polynomials--\(P_{n}^{*}(x)\). We will explicitly note that for modeling specific radiation diagrams, the orthogonal polynomials associated with \(P_{n}(x)-P_{n}^{*}(x)\) can also be successfully used, generating from the following recursion [35]:  
\[(n+c+1)P_{n+1}^{*}(x)=(n+a+c)xP_{n}^{*}(x)-P_{n-1}^{*}(x)\] \[P_{0}^{*}(x)=1;\ \ P_{1}^{*}(x)=\frac{a+c}{c+1}x.\]  
Figure 8: The catastrophe surfaces in the light of Zeeman considerations. Figure 9: The catastrophe surfaces in the light of Zeeman considerations. The polynomials \(P_{n}^{*}(x)\) for \(n=1\),3,5,7,9 and fixed \(a=1\), \(c=3\)  
\[\left\{ \begin{array}{rcl}P_{3}^{*}(x)&=&x^{3}-\frac{11}{30}x\\ P_{5}^{*}(x)&=&x^{5}-\frac{533}{840}x^{3}+\frac{25}{336}x\\ P_{7}^{*}(x)&=&x^{7}-\frac{2131}{2520}x^{5}+\frac{7351}{3/800}x^{3}-\frac{107} {10/080}x\\ P_{9}^{*}(x)&=&x^{9}-\frac{28271}{27/720}x^{7}+\frac{1844,463}{554,400}x^{5}- \frac{3451}{89,100}3x^{3}+\frac{223}{190,080}x\end{array} \right.\]  
are shown in Figure 10.  
_Some simulations_  
The simulations on the Lienard-type system:  
\[\left\{ \begin{array}{rcl}\frac{dx}{dt}&=&y\\ \frac{dy}{dt}&=&-P_{5}^{*}(x)+eF(x)y\end{array} \right. \tag{8}\]  
where  
\[F(x)=x-x^{3}+x^{5}-\frac{1}{7}x^{7}\]  
with \(x_{0}=0.8\), \(y_{0}=0.6\), \(b=0.9\), \(c=0.7809\) are depicted on Figure 11.  
All experiments and algorithms were carried out using our own module in CAS Mathematica. We define the normalized factor as follows  
\[X(\theta)=\frac{|y(b\cos\theta+c)|}{N}.\]  
Figure 10: The polynomials \(P_{n}^{*}(x)\) for fixed \(a=1\); \(c=3\); \(n=1,3,5,7,9\).