# On a Class of Orthogonal Polynomials as Corrections in
## 2 Main Results  
Let \(a=1\). Then  
\[\left\{ \begin{array}{rcl}P_{3}(x)&=&x^{3}-\frac{5}{6}x\\ P_{5}(x)&=&x^{5}-\frac{7}{60}x^{3}+\frac{7}{24}x\\ P_{7}(x)&=&x^{7}-\frac{223}{140}x^{5}+\frac{59}{90}x^{3}-\frac{1}{16}x\\ P_{9}(x)&=&x^{9}-\frac{4609}{2520}x^{7}+\frac{10219}{10,080}x^{5}-\frac{2519} {12,960}x^{3}+\frac{11}{1152}x\\ P_{11}(x)&=&x^{11}-\frac{559,991}{27,720}x^{9}+\frac{227,11}{16,800}x^{7}- \frac{67,171}{181,440}x^{5}+\frac{2041}{515,840}x^{3}-\frac{13}{11,520}x.\end{array} \right.\]  
Let \(Poly(x)\) (in (1)) coincides with polynomials \(P_{n}(x)\). Without going into details, we will note some more interesting level curves. _The level curves_  
_The case (1): \(n=9\)._  
The Hamiltonian of system (1) (\(\epsilon=0\)) is  
\[H(x,y)=\frac{y^{2}}{2}-\frac{1}{10}x^{10}+\frac{4609}{20,160}x^{8}-\frac{10,21 9}{60,480}x^{6}+\frac{2519}{51,840}x^{4}-\frac{11}{2304}x^{2}.\]  
The level curves \(L_{h_{i}}=\{H(x,y)=h_{i}\}\) are depicted at Figure 2.  
Figure 1: The polynomials \(P_{n}(x)\) for fixed \(a=1\); \(n=1,3,5,7,9\). _The case (2): \(n=7\)._  
The Hamiltonian of system (1) (\(\epsilon=0\)) is  
\[H(x,y)=\frac{y^{2}}{2}-\frac{1}{8}x^{8}+\frac{223}{840}x^{6}-\frac{59}{360}x^{4 }+\frac{1}{32}x^{2}.\]  
The level curves \(L_{h_{i}}=\{H(x,y)=h_{i}\}\) are depicted at Figure 3.  
_The model in the light of [PERSON]'s considerations_  
The Melnikov polynomial [5] for the system  
\[\left\{ \begin{array}{rcl}\frac{dx}{dt}&=&y-\epsilon\big{(}a_{1}x+a_{2}x^{2}+ \cdots+a_{2n+1}x^{2n+1}\big{)}\\ \frac{dy}{dt}&=&-x\end{array} \right. \tag{2}\]  
Figure 3: Level curves (the case 2). Figure 2: Level curves (the case 1). is defined as  
\[M(r^{2},n)=\frac{a_{1}}{2}+\frac{3}{8}a_{3}r^{2}+\cdots+\left( \begin{array}{c}2n +2\\ n+1\end{array} \right)\frac{a_{2n+1}}{2^{2n+2}}r^{2n}. \tag{3}\]  
It is known [7; 8] that the system for sufficiently small \(\epsilon\
eq 0\) has at most \(n\) limit cycles asymptotic to circles of radii \(r_{j}\), \(j=1,2,\ldots,n\) as \(\epsilon\to 0\) if and only if the \(n\)th degree polynomial \(M(r^{2},n)\) has \(n\) positive roots \(r^{2}=r_{j}^{2}\), \(j=1,2,\ldots,n\). _The case \(n=7\)._  
Consider the model  
\[\left\{ \begin{array}{rcl}\frac{dx}{dt}&=&y-\epsilon(x^{7}-\frac{223}{140}x^ {5}+\frac{59}{90}x^{3}-\mu x)\\ \frac{dy}{dt}&=&-x\end{array} \right. \tag{4}\]  
where \(\mu>0,\epsilon>0\). The following is valid. Proposition 1: The Lienard-type system for \(n=7\), and for all sufficiently small \(\epsilon\
eq 0\). (a) for \(\mu=\frac{1}{16}=0.0625\) has three hyperbolic limit cycles \(0.444513\), \(0.727036\) and \(1.04606\). (b) for \(\mu=0.0735880318901\) has a simple limit cycle \(1.06318\) and limit cycle \(0.587389\) with multiplicity two. Proof: For the Melnikov polynomial in \(r^{2}\) (see Figure 4) we have:  
\[M(r^{2},3)=-\frac{\mu}{2}+\frac{359}{720}r^{2}-\frac{223}{448}r^{4}+\frac{35}{ 128}r^{6}. \tag{5}\]  
Evidently, for example \(\mu=0.0735880318901\) we have a simple limit cycle and cycle with multiplicity two. Figure 4: **(a)** The Melnikov polynomial \(M(r^{2},3)\) for \(n=7\) and \(\mu=\frac{1}{16}=0.0625\) (three limit cycles); **(b)** The Melnikov polynomial \(M(r^{2},3)\) for \(n=7\) and \(\mu=0.0735880318901\) (simple limit cycle \(1.06318\) and limit cycle \(0.587389\) with multiplicity two). _The case \(n=11\)._  
Consider the model  
\[\left\{ \begin{array}{rcl}\frac{dx}{dt}&=&y-\epsilon(x^{11}-\frac{55.991}{27,720}x ^{9}+\frac{22.711}{16,800}x^{7}-\frac{67,171}{181,440}x^{5}+\frac{2041}{51,840}x ^{3}-\mu x)\\ \frac{dy}{dt}&=&-x\end{array} \right.\]  
where \(\mu>0\), \(\epsilon>0\). The following is valid. **Proposition 2**. : _The Lienard-type system for \(n=11\), and for all sufficiently small \(\epsilon\
eq 0\)._  
1. _for_ \(\mu=\frac{13}{11,520}=0.00112847222221\) _has five hyperbolic limit cycles_ \(0.25518\)_,_ \(0.442797\)_,_ \(588203\)_,_ \(0.729379\) _and_ \(1.03167\)_._
2. _for_ \(\mu=0.0012953041893\) _has simple limit cycles_ \(0.639205\)_,_ \(0.706901\)_,_ \(1.03221\) _and limit cycle_ \(0.338942\) _with multiplicity two._  
Proof.