\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 168, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2013/168\hfil Maximum number of limit cycles] {Maximum number of limit cycles for generalized Li\'enard differential equations} \author[S. Badi, A. Makhlouf \hfil EJDE-2013/168\hfilneg] {Sabrina Badi, Amar Makhlouf} % in alphabetical order \address{Sabrina Badi \newline Department of Mathematics, University of Guelma \\ P.O. Box 401, Guelma 24000, Algeria} \email{badisabrina@yahoo.fr} \address{Amar Makhlouf \newline Department of Mathematics, University of Annaba \\ P.O. Box 12, Annaba 23000, Algeria} \email{makhloufamar@yahoo.fr} \thanks{Submitted February 17, 2013. Published July 22, 2013} \subjclass[2000]{34C25, 34C29, 54D10, 34G15} \keywords{Limlit cycle; averaging theory; Li\'enard equation} \begin{abstract} Applying the averaging theory of first and second order to a class of generalized polynomial Li\'enard differential equations, we improve the known lower bounds for the maximum number of limit cycles that this class can exhibit. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \allowdisplaybreaks \section{Introduction and statement of the main results} One of the main problems in the theory of ordinary differential equations is the study of their limit cycles, their existence, their number and their stability. A limit cycle of a differential equation is a periodic orbit in the set of all isolated periodic orbits of the differential equation. These last years hundreds of papers studied the limit cycles of planar polynomial differential systems. The Second part of the 16th Hilbert's problem \cite{D} is related with the least upper bound on the number of limit cycles of polynomial vector fields having a fixed degree. The generalized polynomial Li\'enard differential equation $$\label{L} \ddot{x}+f(x)\dot{x}+g(x)=0.$$ was introduced in \cite{A}. Here the dot denotes differentiation with respect to the time $t$, and $f(x)$ and $g(x)$ are polynomials in the variable $x$ of degrees $n$ and $m$ respectively. The Li\'enard equation, which is often taken as the typical example of nonlinear self-excited vibration problem, can be used to model resistor-inductor-capacitor circuits with nonlinear circuit elements. It can also be used to model certain mechanical systems which contain the nonlinear damping coefficients and the restoring force or stiffness. Limit cycles usually arise at a Hopf bifurcation in nonlinear systems with varying parameters. In mechanical systems, the varying parameter is frequently a damping coefficient (see \cite{AA,DingL}). Lots of papers discussed the possible number of limit cycle of Li\'enard or generalized mixed Rayleigh-Li\'enard oscillators. Ding and Leung \cite{DingL} investigated the generalized mixed Rayleigh-Li\'enard oscillator with highly nonlinear terms. They consider mainly the number of limit cycle bifurcation diagrams of these systems. For the subclass of polynomial vector fields \eqref{L} we have a simplified version of Hilbert's problem, see \cite{ADP,SS}. Many of the results on the limit cycles of polynomial differential systems have been obtained by considering limit cycles which bifurcate from a single degenerate singular point, that are so called \emph{small amplitude limit cycles}, see \cite{NGL,SL}. We denote by $\hat{H}(m,n)$ the maximum number of small amplitude limit cycles for systems of the form \eqref{L}. The values of $\hat{H}(m,n)$ give a lower bound for the maximum number $H(m,n)$ (i.e. the Hilbert number) of limit cycles that the differential equation \eqref{L} with $m$ and $n$ fixed can have. For more information about the Hilbert's 16th problem and related topics see {\cite{YI}} and {\cite{JL}}. Now we shall describe briefly the main results about the limit cycles on Li\'enard differential systems as it is described in \cite{LMT}. \begin{itemize} \item In 1928 Li\'enard {\cite{A}} proved that if $m=1$ and $F(x)=\int^x_{0}f(s)ds$ is a continuous odd function , which has a unique root at $x=a$ and is monotone increasing for $x\geq a$, then equation \eqref{L} has a unique limit cycle. \item In 1973 Rychkov \cite{GSR} proved that if $m=1$ and $F(x)=\int^x_{0}f(s)ds$ is an odd polynomial of degree five, then equation \eqref{L} has at most two limit cycles. \item In 1977 Lins, de Melo and Pugh \cite{ADP} proved that $H(1,1)=0$ and $H(1,2)=1$. \item In 1990, 1996, Dumortier, Li and Rousseau in \cite{DR} and \cite{DL} proved that $H(3,1)=1$. \item In 1998 Coppel \cite{WAC} proved that $H(2,1)=1$. \item In 1997 Dumortier and Chengzhi \cite{DChengzhi} proved that $H(2,2)=1$. \item In 2010 Chengzhi Li and Llibre \cite{LiLlibrea} proved that $H(1,3)=1$. \end{itemize} Blows, Lloyd \cite{BLL} and Lynch \cite{LLL,SLynch} have used inductive arguments in order to prove the following results. \begin{itemize} \item If $g$ is odd then $\hat{H}(m,n)=[\frac{n}{2}]$. \item If $f$ is even then $\hat{H}(m,n)=n$, whatever $g$ is. \item If $f$ is odd then $\hat{H}(m,2n+1)=[\frac{(m-2)}{2}]+n$. \item If $g(x)=x+g_{e}(x)$, where $g_{e}$ is even then $\hat{H}(2m,2)=m$. \end{itemize} Christopher and Lynch \cite{CHLY} developed a new algebraic method for determining the Liapunov quantities of system \eqref{L} and proved the following: \begin{itemize} \item $\hat{H}(m,2)=[\frac{(2m+1)}{3}]$, \item $\hat{H}(2,n)=[\frac{(2n+1)}{3}]$, \item $\hat{H}(m,3)=2[\frac{(3m+2)}{8}]$ for all $10$ sufficiently small there exists a $T$-periodic solution $\varphi(., \epsilon)$ of the system \eqref{1.2} such that $\varphi(0, \epsilon)=a_{\epsilon}$. The expression $d_{B}(F_{10}+ \epsilon F_{20},V,a_{\epsilon})\neq 0$ means that the Brouwer degree of the function $F_{10}+ \epsilon F_{20}: V \to \mathbb{R}^n$ at the fixed point $a_{\epsilon}$ is not zero. A sufficient condition for the inequality to be true is that the Jacobian of the function $F_{10}+ \epsilon F_{20}$ at $a_{\epsilon}$ is not zero. If $F_{10}$ is not identically zero, then the zeros of $F_{10}+ \epsilon F_{20}$ are mainly the zeros of $F_{10}$ for $\epsilon$ sufficiently small. In this case the previous result provides the \textit{averaging theory of first order}. If $F_{10}$ is identically zero and $F_{20}$ is not identically zero, then the zeros of $F_{10}+ \epsilon F_{20}$ are mainly the zeros of $F_{20}$ for $\epsilon$ sufficiently small. In this case the previous result provides the \textit{averaging theory of second order}. For more information about the averaging theory see \cite{SV,V}. \section{Proof of statement (a) of Theorem 1} We shall need the first order averaging theory to prove statement (a) of Theorem 1. In order to apply the first order averaging method we write system \eqref{C} with $k=1$, in polar coordinates $(r, \theta)$ where $x=rcos(\theta), y=rsin(\theta), r>0$. In this way system \eqref{C} is written in the standard form for applying the averaging theory. If we write $f_{n}^1(x,y)=\sum^n_{i+j=0}a_{ij}x^iy^j$, $g_{m}^1(x)=\sum^m_{i=0}b_{i}x^i$ and $h_{l}^1(x)=\sum^l_{i=0}c_{i}x^i$ then system \eqref{C} becomes \label{1.3} \begin{gathered} \begin{aligned} \dot{r}&=\epsilon \Big[\sum^l_{i=0}c_{i}r^i \cos^{i+1}(\theta) -r \sin^2(\theta)\sum^n_{i+j=0}a_{ij} r^{i+j} \cos^i(\theta) \sin^{j}(\theta)\\ &\quad - \sin(\theta)\sum^m_{i=0}b_{i} r^i cos^i(\theta)\Big]+O(\epsilon^2), \end{aligned} \\ \begin{aligned} \dot{\theta}&=-1- \frac{\epsilon}{r}\Big[r \cos(\theta)\sin(\theta) \sum^n_{i+j=0}a_{ij} r^{i+j} \cos^{i}(\theta) sin^{j}(\theta)\\ &\quad + \cos(\theta)\sum^m_{i=0}b_{i} r^i \cos^{i}(\theta) + \sin(\theta) \sum^l_{i=0}c_{i} r^i \cos^{i}(\theta)\Big] +O(\epsilon^2). \end{aligned} \end{gathered} Now taking $\theta$ as the new independent variable, this system becomes \begin{align*} \frac{dr}{d \theta} &= -\epsilon \Big(\sum^l_{i=0}c_{i} r^i \cos^{i+1}(\theta) -r \sin^2(\theta)\sum^n_{i+j=0}a_{ij}r^{i+j}\cos^{i}(\theta) \sin^{j}(\theta)\\ &\quad -\sin(\theta) \sum^m_{i=0}b_{i} r^i \cos^i(\theta)\Big) + O(\epsilon^2) \\ &=\epsilon F_{1}(\theta,r)+O(\epsilon^2). \end{align*} Using the notation introduced in section 2 we have \begin{align*} F_{10}(r)&=\frac{-1}{2\pi} \int^{2\pi}_{0} \Big(\sum^l_{i=0}c_{i} r^i \cos^{i+1}(\theta) -r \sin^2(\theta)\sum^n_{i+j=0}a_{ij}r^{i+j}\cos^{i}(\theta) \sin^{j}(\theta)\\ &\quad -\sin(\theta) \sum^m_{i=0}b_{i} r^i \cos^i(\theta) \Big) d\theta. \end{align*} Since $$\int^{2\pi}_{0}\cos^{i+1}(\theta)d\theta=\begin{cases} 0 &\text{if i is even} \\ \alpha_{i}\neq 0 &\text{if i is odd}, \end{cases}$$ it follows that \begin{gather*} \int^{2\pi}_{0}\cos^{i}(\theta)\sin^{j+2}(\theta)d\theta =\begin{cases} 0 &\text{if $i$ odd and $j$ is odd} \\ \beta_{ij}\neq 0 &\text{if $i$ is even and $j$ even}, \end{cases} \\ \int^{2\pi}_{0}\sin(\theta)\cos^{i}(\theta)d\theta=0 \end{gather*} for $i=0,1,\dots,$ we have \begin{align*} F_{10}(r) &=\frac{-1}{2\pi} \int^{2\pi}_{0} \Big(\sum^l_{i=1,\,i \text{ odd}} c_{i} r^i \cos^{i+1}(\theta)\\ &-\sum^n_{i+j=0,\text{ $i$ even $j$ even}} a_{ij}r^{i+j+1}\cos^{i}(\theta) \sin^{j+2}(\theta)\Big) d\theta. \end{align*} We define $$M(l,n)=\begin{cases} \max\{l,n+1\} &\text{if l is odd, n is even} \\ \max\{l-1,n+1\} &\text{if l is even, n is even} \\ \max\{l,n\} &\text{if l is odd, n is odd} \\ \max\{l-1,n\} &\text{if l is even, n is odd}. \end{cases}$$ Therefore, $$M(l,n)=\max\{ O(l), O(n+1) \}$$ and $$\Big[ \frac{M(l,n)-1}{2} \Big]=\Big[\frac{\max\{O(l), O(n+1)\}-1}{2}\Big] = \max \Big\{\Big[\frac{l-1}{2}\Big], \Big[\frac{n}{2}\Big]\Big\}$$ finally, we have $$F_{10}(r)=\sum^{M(l,n)}_{k=1,\, k \text{ odd}} \sigma_{k}r^k,$$ with $$\sigma_{k}=\frac{-1}{2\pi} \int^{2\pi}_{0} \left(c_{k} \cos^{k+1}(\theta)-a_{(k-1-j)j}\cos^{k-1-j}(\theta) \sin^{j+2}(\theta)\right) d\theta,$$ where $k\geq 1$ is an odd integer number and $j\geq 0$ is an even one. Since $F_{10}(r)$ is an odd function, it has at most $[(M(l,n)-1)/2]$ simple positive real roots. From section 2 we obtain that for $|\epsilon|$ sufficiently small, the maximum number of limit cycles of system \eqref{C} which can bifurcate from the periodic orbits of the linear center $\dot{x}=y$, $\dot{y}=-x$ using the averaging theory of first order is $[(M(l,n)-1)/2 ]$. Hence statement (a) of Theorem 1 is proved. \section{Proof of statement (b) of Theorem 1} For proving statement (b) of Theorem 1 we shall use the second order averaging theory. In this section we consider the differential systems $$\label{D} \begin{gathered} \dot{x}=y+\epsilon h_{l}^1(x)+\epsilon^2 h_{l}^2(x)+O(\epsilon^3), \\ \dot{y}=-x-\epsilon(f_{n}^1(x,y)y+g_{m}^1(x))-\epsilon^2(f_{n}^2(x,y)y +g_{m}^2(x))+O(\epsilon^3). \end{gathered}$$ where $$h_{l}^2(x)=\sum^l_{i=0}\hat{c_{i}}x^i, \quad f_{n}^2(x,y)=\sum^n_{i+j=0}\hat{a_{ij}}x^iy^j, \quad g_{m}^2(x)=\sum^m_{i=0}\hat{b_{i}}x^i$$ Then system \eqref{D} in polar coordinates $(r,\theta), r>0$ becomes \begin{gather*} \dot{r} =\epsilon \frac{xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x)}{r} +\epsilon^2\frac{xh_{l}^2(x)-y^2f_{n}^2(x,y)-yg_{m}^2(x)}{r}+O(\epsilon^3), \\ \begin{aligned} \dot{\theta} &=-1- \epsilon \frac{xyf_{n}^1(x,y)+xg_{m}^1(x)+yh_{l}^1(x)}{r^2} -\epsilon^2 \frac{xyf_{n}^2(x,y)+xg_{m}^2(x)+yh_{l}^2(x)}{r^2}\\ &\quad +O(\epsilon^3). \end{aligned} \end{gather*} Taking $\theta$ as the new independent variable, this system becomes \begin{align*} \frac{dr}{d \theta} &=\epsilon \frac{xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x)}{r}-\epsilon^2 \Big[\frac{xh_{l}^2(x)-y^2f_{n}^2(x,y)-yg_{m}^2(x)}{r}\\ &\quad -\frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x))(xyf_{n}^1(x,y) +xg_{m}^1(x)+yh_{l}^1(x))}{r^3}\Big] \\ &\quad -\epsilon^3\Big[ \frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y) -yg_{m}^1(x))(xyf_{n}^2(x,y)+xg_{m}^2(x)+yh_{l}^2(x))}{r^3}\\ &\quad + \frac{(xh_{l}^2(x)-y^2f_{n}^2(x,y)-yg_{m}^2(x))(xyf_{n}^1(x,y) +xg_{m}^1(x)+yh_{l}^1(x))}{r^3}\\ &\quad - \frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x))(xyf_{n}^1(x,y) +xg_{m}^1(x)+yh_{l}^1(x))^2}{r^5} \Big]+O(\epsilon^4) \\ &=\epsilon F_{1}(\theta,r)+\epsilon^2 F_{2}(\theta,r) +\epsilon^3 F_{3}(\theta,r)+ O(\epsilon^4), \end{align*} Now we determine the corresponding function $$F_{20}= \frac{1}{2\pi}\int_{0}^{2\pi} \Big[\frac{d}{dr}F_{1}(\theta,r). \int^{\theta}_{0}F_{1}(\phi,r) d \phi + F_{2}(\theta,r) \Big]d\theta.$$ For this we put $F_{10}\equiv 0$ which is equivalent to \begin{gather*} c_{i}=0 \quad \text{for $i$ odd, and} \\ a_{ij}=0 \quad \text{for $i$ even and $j$ even} \end{gather*} First, we have \begin{align*} \frac{d}{dr}F_{1}(\theta,r) &=-\sum^l_{i=2, \text{ even}} i c_{i}r^{i-1} \cos^{i+1}(\theta)\\ &\quad + \sum^n_{i+j=2, \text{ $i$ odd or $j$ odd}} (i+j+1) a_{ij}r^{i+j} \cos^i(\theta) \sin^{j+2}(\theta)\\ &\quad +\sum^m_{i=1} i b_{i}r^{i-1} \cos^i(\theta)\sin(\theta), \end{align*} and \begin{align*} \int^{\theta}_{0}F_{1}(\phi,r) d \phi &=-\sum^l_{i=0,\, i \text{ even}} c_{i}r^i \int^{\theta}_{0}\cos^{i+1}(\phi)d\phi\\ &\quad + \sum^n_{i+j=1,\text{ $i$ odd or $j$ odd}} a_{ij}r^{i+j+1}\int^{\theta}_{0}\cos^i(\phi)\sin^{j+2}(\phi)d\phi \\ &\quad + \sum^m_{i=0}b_{i}r^i \int^{\theta}_{0}cos^i(\phi)sin(\phi)d\phi \\ &=-\sum^l_{i=0,\, i \text{ even}}c_{i}r^i A_{i+1}(\theta) +\sum^n_{i+j=1,\text{ $i$ odd or $j$ odd}} a_{ij}r^{i+j+1}A_{i,(j+2)}(\theta)\\ &\quad + \sum^m_{i=0}b_{i}r^i \Big( \frac{1-cos^{i+1}(\theta)}{i+1} \Big). \end{align*} where \begin{align*} A_{i}(\theta) &=\int^{\theta}_{0} \cos^{i}(\phi)d\phi\\ &=\sum^{i-2}_{k=1,\,k \text{ odd}}\frac{(i-k)!}{i!} \frac{(i-k)^2.(i-(k-2)))^2\dots(i-1)^2}{(i-k)^2}\sin(\theta)\cos^{i-k}(\theta)\\ &\quad + \frac{(i-1)^2(i-3)^2\dots(2)^2}{i!}\sin(\theta), \end{align*} \begin{align*} &A_{p,(2n+1)}\\ &=\int^{\theta}_{0} \cos^{p}(\phi) \sin^{2n+1}(\phi)d\phi \\ &=\frac{cos^{p+1}(\theta)}{2n+p+1}\Big\{\sin^{2n} + \sum^n_{k=1} \frac{2^k n(n-1)\dots(n-k+1) \sin^{2n-2k}(\theta)}{(2n+p-1) (2n+p-3)\dots(2n+p-2k+1)} \Big\}, \end{align*} \begin{align*} &A_{p,(2n)}\\ &=\int^{\theta}_{0} cos^{p}(\phi) sin^{2n}(\phi)d\phi\\ &=\frac{-cos^{p+1}(\theta)}{2n+p}\Big\{\sin^{2n-1} + \sum^{n-1}_{k=1} \frac{(2n-1)(2n-3)\dots(2n-2k+1) \sin^{2n-2k-1}(\theta)}{(2n+p-2)(2n+p-4)\dots(2n+p-2k)} \Big\}\\ &\quad + \frac{(2n-1)!!}{(2n+p).(2n+p-2)\dots(p+2)} \int^{\theta}_{0} \cos^{p}(\theta)d\theta; \end{align*} for more details see \cite{GraRys}. From the nine main products of $\frac{d}{dr}F_{1}(\theta,r)\int^{\theta}_{0}F_{1}(\phi,r) d \phi$, only the following five are not zero when we integrate them between $0$ and $2\pi$: \begin{gather*} \sum^l_{i=2,\,i \text{ even}} \sum^m_{k=0,\,k \text{ even}} \frac{i}{k+1} c_{i} b_{k} r^{i+k-1} \cos^{i+k+2}(\theta), \\ -\sum^n_{i+j=2,\text{$i$ even and $j$ odd}} \sum^l_{k=0,\,k \text{ even}} (i+j+1) a_{ij}c_{k} r^{i+j+k} \cos^i(\theta) \sin^{j+2}(\theta)A_{k+1}(\theta), \\ + \sum^n_{i+j=2} \sum^n_{k+h=1} (i+j+1) a_{ij} a_{kh} r^{i+j+k+h}\cos^i(\theta) \sin^{j+2}(\theta)A_{i,(j+2)}, \end{gather*} where if $i$ even $j$ is odd, and if $i$ odd $j$ even, and the same for $k$ and $h$, with $i+k$ odd and $j+h$ is odd too. \begin{gather*} \begin{aligned} &+\sum^n_{i+j=2,\text{ $i$ odd and $j$ even}} \sum^m_{k=0,\,k \text{ even}} (i+j+1) a_{ij} b_{k} r^{i+j+k} \cos^i(\theta)\\ &\times \sin^{j+2}(\theta)(\frac{1-cos^{k+1}(\theta)}{k+1}), \end{aligned} \\ -\sum^m_{i=2,\,i \text{ even}} \sum^l_{k=0,\,k \text{ even}} i b_{i}c_{k} r^{i+k-1} \cos^i(\theta) \sin(\theta) A_{k+1}(\theta). \end{gather*} Then the last five sums are odd polynomial in the variable $r$ of degree $O(n)+E(l)$, $2O(n)+1$, $O(n)+E(m)$,$E(l)+E(m)-1$, respectively. Therefore, $$\frac{1}{2\pi}\int_{0}^{2\pi}\big[\frac{d}{dr}F_{1}(\theta,r) \int^{\theta}_{0}F_{1}(\phi,r) d \phi \big]d\theta$$ is an odd polynomial in the variable $r$ and can contribute at most with $$\Big[ \frac{\max\{O(n)+E(l), 2O(n)+1, O(n)+E(m),E(l)+E(m)-1 \}-1}{2} \Big]$$ simple positive real roots to the roots of $F_{20}(r)$. Now we shall study the contribution of $\frac{1}{2\pi} \int^{2\pi}_{0}F_{2}(\theta,r)d\theta$ to $F_{20}(r)$. The first part, $$\frac{xh_{l}^2(x)-y^2f_{n}^2(x,y)-yg_{m}^2(x)}{r},$$ of $F_{2}(\theta,r)$, contributes at the roots of $F_{20}(r)$ exactly as the function $F_{1}(\theta,r)$ contributes to $F_{10}(r)$; i.e. it contributes at most with $$\Big[\frac{\max\{O(l), O(n+1) \} -1}{2} \Big]$$ simple positive roots to the roots of $F_{20}(r)$. Finally we shall study the contribution of the second part $$\frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x))(xyf_{n}^1(x,y) +xg_{m}^1(x)+yh_{l}^1(x))}{r^3}$$ of $F_{2}(\theta,r)$ to $F_{20}(r)$, which can be written as \begin{align*} &\frac{1}{r^2} \Big[\sum^l_{i=0,\,i \text{ even}} c_{i} r^i \cos^{i+1}(\theta) -\sum^n_{i+j=1,\text{ $i$ odd or $j$ odd}}a_{ij} r^{i+j+1} \cos^i(\theta)\sin^{j+2}(\theta)\\ &-\sum^m_{i=0}b_{i}r^i \cos^i(\theta)\sin(\theta) \Big], \end{align*} \begin{align*} &\Big[ \sum^n_{i+j=1,\text{ $i$ odd or $j$ odd}}a_{ij} r^{i+j+1} \cos^{i+1}(\theta)\sin^{j+1}(\theta) + \sum^m_{i=0}b_{i}r^i\cos^{i+1}(\theta)\\ &+\sum^l_{i=0,\, i \text{ even}} c_{i} r^i \cos^{i}(\theta)\sin(\theta) \Big]. \end{align*} From the nine products between the different sums, seven ones will not be zero after the integration with respect to $\theta$ between $0$ and $2\pi$, and two of these seven are equal. So the terms which will contribute to $F_{20}(r)$ are \begin{align*} &\frac{1}{r^2}\Big[ \sum^l_{k=0,\,k \text{ even}} \sum^n_{i+j=1,\text{ $i$ even and $j$ odd}}c_{k} a_{ij} r^{k+i+j+1} \cos^{k+i+2}(\theta) \sin^{j+1}(\theta)\\ &+ \sum^l_{k=0,\,k \text{ even}} \sum^m_{i=0,\, i \text{even}} c_{k} b_{i}r^{k+i} \cos^{k+i+2}(\theta) \\ &+ \sum^{2n}_{i+j=1, k+h=1,\text{ $i+k$ odd and $j+h$ odd}} a_{ij}a_{kh}r^{i+j+k+h+2}\cos^{i+k+1}(\theta)\sin^{j+h+3}(\theta) \\ &+2 \sum^n_{i+j=1,\text{ $i$ odd and $j$ even }} \sum^m_{k=0,\,k \text{ even}}a_{ij}b_{k}r^{i+j+k+1} \cos^{i+k+1}(\theta)\sin^{j+2}(\theta) \\ &+\sum^n_{i+j=1,\text{$i$ even and $j$ odd}} \sum^l_{k=0,\,k \text{ even}} a_{ij}c_{k} r^{i+j+k+1} \cos^{i+k}(\theta)\sin^{j+3}(\theta) \\ &+\sum^m_{i=0,\,i \text{ even}} \sum^l_{k=0,\,k \text{ even}} b_{i} c_{k} r^{i+k} \cos^{i+k}(\theta)\sin^{2}(\theta)\Big] \end{align*} So the integral between $0$ and $2\pi$ with respect to $\theta$ of this last expression is an odd polynomial in the variable $r$ of degree $\max \{ O(n)+O(m)+1, O(n)+E(l)+1, E(m)+E(l), 2 O(n)+2 \}$. Consequently the contribution of the second part, $$\frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x)) (xyf_{n}^1(x,y)+xg_{m}^1(x)+yh_{l}^1(x))}{r^3},$$ of $F_{2}(\theta,r)$ to the zeros of $F_{20}(r)$ is at most with $$\Big[ \frac{\{O(n)+O(m)+1, O(n)+E(l)+1, E(m)+E(l), 2 O(n)+2 \}-1}{2} \Big]$$ simple positive real roots. From the above results, we have that $F_{20}(r)$ has at most $$\Big[ \frac{\{O(n)+O(m)+1, O(n)+E(l)+1, E(m)+E(l), 2 O(n)+2, O(l), O(n+1) \}-1}{2} \Big]$$ simple positive real roots. So, from the results of section 2 statement (b) of Theorem 1 is proved. \begin{thebibliography}{10} \bibitem{AA}{N. Ananthkrishnan, K. Sudhakar, S. Sudershan, A. Agarwal}; \emph{Application of secondary bifurcations to large amplitude limit cycles in mechanical systems}, Journal of Sound and Vibration. (1998), \textbf{215} (1), 183-188. \bibitem{BM}{S. Badi, A. Makhlouf}; \emph{Limit cycles of the generalized Li\'enard differential equation via averaging theory}, Ann. of Diff. 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