\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 68, 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 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/68\hfil Limit cycles of a Li\'enard differential equation] {Limit cycles of the generalized Li\'enard differential equation via averaging theory} \author[S. Badi, A. Makhlouf \hfil EJDE-2012/68\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 August 11, 2011. Published May 2, 2012.} \subjclass[2000]{37G15, 37C80, 37C30} \keywords{Limit cycle; averaging theory; Lienard differential equation} \begin{abstract} We apply the averaging theory of first and second order to a generalized Li\'enard differential equation. Our main result shows that for any $n,m \geq 1$ there are differential equations $\ddot{x}+f(x,\dot{x})\dot{x}+ g(x)=0$, with $f$ and $g$ polynomials of degree $n$ and $m$ respectively, having at most $[n/2]$ and $\max\{[(n-1)/2]+[m/2], [n+(-1)^{n+1}/2]\}$ limit cycles, where $[\cdot]$ denotes the integer part function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} One of the main topics in the theory of ordinary differential equations is the study of 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. The Second part of the 16th Hilbert's problem \cite{D} is related to the least upper bound on the number of limit cycles of polynomial vector fields having a fixed degree. Then there have been hundreds publications about the limit cycles of planar polynomial differential systems. The generalized polynomial Li\'enard differential equation was introduced in \cite{A}, and has the form \begin{equation}\label{L} \ddot{x}+f(x)\dot{x}+g(x)=0. \end{equation} where the dot denotes differentiation with respect to 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}). A lot of papers discusse the possible number of limit cycle of Li\'enard and 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 results on 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}} and {\cite{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,JL}. Now we shall describe briefly the main results about the limit cycles on Li\'enard differential systems. In 1928 Li\'enard {\cite{A}} proved that if $m=1$ and $F(x)=\int^x_0f(s)ds$ 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. In 1973 Rychkov {\cite{GSR}} proved that if $m=1$ and $F(x)=\int^x_0f(s)ds$ is an odd polynomial of degree five, then equation \eqref{L} has at most two limit cycles. In 1977 Lins, de Melo and Pugh {\cite{ADP}} proved that $H(1,1)=0$ and $H(1,2)=1$. In 1998 Coppel {\cite{WAC}} proved that $H(2,1)=1$. Dumortier, Li and Rousseau in {\cite{DR}} and {\cite{DL}} proved that $H(3,1)=1$. In 1997 Dumortier and Chengzhi {\cite{DChengzhi}} proved that $H(2,2)=1$. Blows, Lloyd {\cite{BLL}} and Lynch ({\cite{LLL}} and {\cite{SLynch}}) used inductive arguments 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 \emph{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 \emph{averaging theory of second order}. For more information about the averaging theory see \cite{SV,V}}. \section{Proof of statement (a) of Theorem \ref{thm1}} For applying the first-order averaging method, we write system \eqref{B} with $k=1$, in polar coordinates $(r, \theta)$ where $x=r\cos(\theta)$, $y=r\sin(\theta)$, $r>0$. In this way system \eqref{B} is written in the standard form for applying the averaging theory. If we write $f_{n}^1(x,y)=f(x,y)=\sum^n_{i+j=0}a_{ij}x^iy^j$ and $g_{m}^1(x)=g(x)=\sum^m_{i=0}b_ix^i$, system \eqref{B} becomes \begin{equation}\label{1.3} \begin{gathered} \dot{r}=-\epsilon [\sum^n_{i+j=0}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)], \\ \dot{\theta}=-1- \frac{\epsilon}{r}[ \sum^n_{i+j=0}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)]. \end{gathered} \end{equation} Taking $\theta$ as the new independent variable, system \eqref{1.3} becomes \begin{equation*} \frac{dr}{d \theta}= \epsilon \Big( \sum^n_{i+j=0}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) + O(\epsilon^2) \end{equation*} and \begin{equation*} F_{10}(r)=\frac{1}{2\pi} \int^{2\pi}_0 \Big( \sum^n_{i+j=0}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) d\theta. \end{equation*} To calculate the exact expression of $F_{10}$ we use the following formulas: \begin{gather*} \int^{2\pi}_0 \cos^{i}(\theta) \sin^{j+2}(\theta) d\theta =\begin{cases} 0 & \text{if $i$ is odd, or $j$ is odd} \\ \alpha_{ij} &\text{if $i$ is even and $j$ is even}, \end{cases} \\ \int^{2\pi}_0 \cos^{i}(\theta) \sin(\theta) d\theta =0 ,\quad \text{for } i=0,1,\dots \end{gather*} Hence \begin{equation}\label{1.4} F_{10}(r)=\frac{1}{2\pi} \sum^n_{i+j=0} a_{ij} \alpha_{ij} r^{i+j+1} \quad \text{when $i$ is even and $j$ is even}. \end{equation} Then the polynomial $F_{10}(r)$ has at most $[\frac{n}{2}]$ positive roots, and we can choose the coefficients $a_{ij}$ with $i$ even and $j$ even in such a way that $F_{10}(r)$ has exactly $[\frac{n}{2}]$ simple positive roots. Hence statement (a) of Theorem \ref{thm1} is proved. \begin{example} \rm We consider the system \begin{equation}\label{a} \begin{gathered} \dot{x}=y, \\ \dot{y}=-x-\epsilon ( 3-x+2y+{x}^2-y^2-2xy^2)y+(x-y) . \end{gathered} \end{equation} The corresponding system \eqref{1.3} associated with \eqref{a} is \begin{align*} \dot{r}&=-\epsilon r \sin(\theta) (2\sin(\theta)+\cos(\theta) -r(\cos(\theta)\sin(\theta)+2\sin(\theta)^2)\\ &\quad +r^2(\sin(\theta) \cos(\theta)^2-\sin(\theta)^3)-2r^3 \cos(\theta) \sin(\theta)^3), \\ \dot{\theta}&=-1-\epsilon[2r \sin(\theta)\cos(\theta) -r^2\cos(\theta)^2\sin(\theta)\\ &\quad +2r^2\sin(\theta)^2\cos(\theta)+r^3\sin(\theta)\cos(\theta)^3\\ &\quad -r^3\sin(\theta)^3 \cos(\theta)-2r^4\cos(\theta)^2\sin(\theta)^3 +r\cos(\theta)^2]. \end{align*} To look for limit cycles, we must solve the equation \begin{equation}\label{1.8} F_{10}=\frac{1}{2\pi} \Big( 2r \pi -\frac{1}{4}r^3 \pi \Big) =0, \end{equation} This equation possesses the positive root $r=2$. According with statement (a) of Theorem \ref{thm1}, that system \eqref{a} has exactly one limit cycle bifurcating from the periodic orbits of the linear differential system \eqref{a} with $\epsilon=0$, using the averaging theory of first order. \end{example} \section{Proof of statement (b) of Theorem \ref{thm1}} For proving statement (b) of Theorem \ref{thm1} we shall use the second-order averaging theory. If we write $f_{n}^1(x,y)=\sum^n_{i+j=0}a_{ij}x^iy^j$, $f_{n}^2(x,y)=\sum^n_{i+j=0}c_{ij}x^iy^j$, $g_{m}^1(x)=\sum^m_{i=0}b_ix^i$ and $g_{m}^2(x)=\sum^m_{i=0}d_ix^i$ then system \eqref{B} with $k=2$ in polar coordinates $(r,\theta)$, $r>0$ becomes \begin{equation} \label{1.5} \begin{aligned} \dot{r}&=-\epsilon \Big( \sum^n_{i+j=0}a_{ij}r^{i+j+1}\cos^i(\theta) \sin^{j+2}(\theta)+ \sum^m_{i=0}b_ir^i \cos^i(\theta)\sin(\theta)\Big)\\ &\quad -\epsilon^2 \Big( \sum^n_{i+j=0}c_{ij}r^{i+j+1}\cos^i(\theta) \sin^{j+2}(\theta)+\sum^m_{i=0}d_ir^i \cos^i(\theta)\sin(\theta) \Big), \\ \dot{\theta}&=-1- \frac{\epsilon}{r}\Big( \sum^n_{i+j=0}a_{ij}r^{i+j+1}\cos^{i+1} (\theta) \sin^{j+1}(\theta)+ \sum^m_{i=0}b_ir^i \cos^{i+1}(\theta) \Big) \\ &\quad -\frac{\epsilon^2}{r} \Big( \sum^n_{i+j=0}c_{ij}r^{i+j+1}\cos^{i+1}(\theta) \sin^{j+1}(\theta)+ \sum^m_{i=0}d_ir^i \cos^{i+1}(\theta)\Big). \end{aligned} \end{equation} Taking $\theta$ as the new independent variable in the system \eqref{1.5}, it becomes \begin{equation*} \frac{dr}{d \theta}=\epsilon F_{1}(\theta,r)+\epsilon^2 F_{2}(\theta,r) + O(\epsilon^3), \end{equation*} where \begin{gather*} F_{1}(\theta,r) = \sum^n_{i+j=0}a_{ij}r^{i+j+1}\cos^i(\theta) \sin^{j+2}(\theta) + \sum^m_{i=0}b_ir^i \cos^i(\theta)\sin(\theta) , \\ \begin{aligned} &F_{2}(\theta,r)\\ &= \Big[\sum^n_{i+j=0}c_{ij}r^{i+j+1}\cos^i(\theta) \sin^{j+2}(\theta)+\sum^m_{i=0}d_ir^i \cos^i(\theta)\sin(\theta)\Big] \\ &\quad -r \cos(\theta)\sin(\theta) \Big[ \sum^n_{i+j=0}a_{ij}r^{i+j}\cos^i(\theta) \sin^{j+1}(\theta)+\sum^m_{i=0}b_ir^{i-1} \cos^i(\theta)\sin(\theta)\Big]^2. \end{aligned} \end{gather*} Now we determine the corresponding function $F_{20}$. For this we put $F_{10}\equiv 0$ which is equivalent to $a_{ij}=0$ for all $i$ and $j$ even, and we compute \begin{equation*} \frac{d}{dr}F_{1}(\theta,r)= \sum^n_{i+j=0}(i+j+1) a_{ij}r^{i+j} \cos^i(\theta) \sin^{j+2}(\theta)+\sum^m_{i=1} i b_ir^{i-1} \cos^i(\theta)\sin(\theta), \end{equation*} and $$ \int^{\theta}_0F_{1}(\phi,r) d \phi=y_{1}^1+ y_{1}^2 $$ where \begin{align*} y_{1}^1 &=\int^{\theta}_0\sum^n_{i+j=0}a_{ij}r^{i+j+1}\cos^i(\phi) \sin^{j+2}(\phi)\\ &= a_{10} r^2 \big( \alpha_{110} \sin(\theta)+\alpha_{210}\ \sin(3 \theta) \big) +\dots + a_{lb} r^{l+b+1} \Big( \alpha_{1lb} \sin(\theta)+\alpha_{2lb}\sin(3 \theta)\\ &\quad +\dots +\alpha_{\frac{(l+b+2)+1}{2}lb} \sin((l+b+2)\theta)\Big) + a_{01} r^2 \big( \alpha_{101} +\alpha_{201} \cos(\theta) \\ &\quad +\alpha_{301} \cos(3 \theta) \big) +\dots + a_{cd} r^{c+d+1} \Big( \alpha_{1cd} + \alpha_{2cd} \cos(\theta) +\alpha_{3cd} \cos(3 \theta)+\dots \\ &\quad +\alpha_{\frac{(c+d+2)+3}{2}cd} \cos((c+d+2)\theta) \Big) + a_{11} r^3 \big(\alpha_{111} +\alpha_{211} \cos(2 \theta) \\ &\quad +\alpha_{311} \cos(4 \theta) \big) +\dots + a_{ld}r^{l+d+1} \Big( \alpha_{1ld} +\alpha_{2ld} \cos(2 \theta) +\alpha_{3ld} \cos(4 \theta)+\dots \\ &\quad + \alpha_{\frac{(l+d+2)+2}{2} l d} \cos((l+d+2) \theta) \Big), \end{align*} such that $l$ is the greatest odd number and $b$ is the greatest even number so that $l+b$ is less than or equal to $n$. $c$ is the greatest even number and $d$ is the greatest odd number so that $c+d$ is less than or equal to $n$. $\alpha_{ijk}$ are real constants exhibited during the computation of $\int^{\theta}_0\cos^i(\phi) \sin^{j+2}(\phi) d\phi$ for all $i$ and $j$. and $$ y_{1}^2=\int^{\theta}_0\sum^m_{i=0}b_ir^i \cos^i(\phi)\sin(\phi) =b_0(1-\cos(\theta))+\dots +b_{m}r^m \frac{1}{m+1}(1-\cos^{m+1}(\theta). $$ We know from \eqref{1.4} that $F_{10}$ is identically zero if and only if $a_{ij}=0$ for all $i$ even and $j$ even. Moreover \begin{gather*} \int^{2\pi}_0\cos^i(\theta) \sin^{j+2}(\theta)\sin((2k+1)\theta)d\theta =\begin{cases} 0 &\text{if $i$ is odd and $j \in \mathbb{N}$}, \\ A^{2k+1}_{ij} &\text{if $i$ is even and $j$ is odd},\\ & $k=0,1,\dots$ \end{cases} \\ \int^{2\pi}_0\cos^i(\theta) \sin^{j+2}(\theta)d\theta \neq 0, \quad \text{if and only if $i$ is even and $j$ even}, \\ \int^{2\pi}_0\cos^i(\theta) \sin^{j+2}(\theta)\cos((2k+1)\theta)d\theta =\begin{cases} 0 &\text{if $j$ is odd and $i \in \mathbb{N}$}, \\ B^{2k+1}_{ij} &\text{if $i$ is odd and $j$ is even},\\ & k=0,1,\dots \end{cases} \\ \int^{2\pi}_0\cos^i(\theta) \sin^{j+2}(\theta)\cos((2k)\theta)d\theta =0, \quad\text{ for $i$ odd or $j$ odd, $k=0,1,\dots$} \\ \int^{2\pi}_0\cos^i(\theta) \sin^{j+2}(\theta)\cos^{m+1}(\theta)d\theta =\begin{cases} 0 &\text{if $j$ is odd and }i,m \in \mathbb{N}, \\ M_{ij}^m &\text{if $i$ is odd, $j$ is even and $m$ is even}, \end{cases} \\ \int^{2\pi}_0\cos^i(\theta) \sin(\theta)\sin((2k+1)\theta)d\theta =\begin{cases} 0 &\text{if $i$ is odd, } k=0,1,\dots \\ N_i^{2k+1} &\text{if $i$ is even, } k=0,1,\dots \end{cases} \\ \int^{2\pi}_0\cos^i(\theta) \sin(\theta)d\theta =0,\quad \forall i \in \mathbb{N}, \\ \int^{2\pi}_0\cos^i(\theta) \sin(\theta) \cos((2k+1)\theta)d\theta =0,\quad \forall i,k \in \mathbb{N}, \\ \int^{2\pi}_0\cos^i(\theta) \sin(\theta) \cos((2k)\theta)d\theta =0,\quad \forall i,k \in \mathbb{N}, \\ \int^{2\pi}_0\cos^i(\theta) \sin(\theta) \cos^{m+1}(\theta)d\theta =0,\quad \forall i,m \in \mathbb{N}, \end{gather*} So \begin{align*} &\int^{2\pi}_0 \frac{d}{dr}F_{1}(\theta,r)y_{1}(\theta,r)d\theta \\ &=\int^{2\pi}_0 [\sum^n_{i+j=0}(i+j+1) a_{ij}r^{i+j} \cos^i(\theta) \sin^{j+2}(\theta)\\ &\quad +\sum^m_{i=1} i b_ir^{i-1} \cos^i(\theta)\sin(\theta)] (y_{1}^1+y_{1}^2) d\theta\\ &=\sum^n_{i+j=0}(i+j+1) a_{ij}r^{i+j} \int^{2\pi}_0 \cos^i(\theta) \sin^{j+2}(\theta) (y_{1}^1+y_{1}^2) d\theta\\ &\quad + \sum^m_{i=1} i b_ir^{i-1} \int^{2\pi}_0 \cos^i(\theta) \sin(\theta) (y_{1}^1+y_{1}^2) d\theta \\ &=\sum^n_{i+j=1_{i \ even, \ j odd}}(i+j+1) a_{ij}r^{i+j} [ a_{10} r^2 ( \alpha_{110} A^1_{ij} + \alpha_{210} A^3_{ij} )+\dots \\ &\quad + a_{lb} r^{l+b+1}( \alpha_{1lb} A^1_{ij} + \alpha_{2lb} A^3_{ij}+\dots + \alpha_{\frac{(l+b+2)+1}{2}lb} A^{l+b+2}_{ij} ) ] \\ &\quad +\sum^n_{i+j=1,i \text{ odd}, j\text{ even}} (i+j+1) a_{ij}r^{i+j} [ a_{01} r^2 (\alpha_{201} B^1_{ij} +\alpha_{301} B^3_{ij} ) +\dots \\ &\quad +a_{cd} r^{c+d+1} (\alpha_{2cd} B^1_{ij} +\alpha_{3cd} B^3_{ij} +\dots +\alpha_{\frac{(c+d+2)+3}{2}cd} B^{c+d+2}_{ij} ) ] \\ &\quad +\sum^n_{i+j=1,i\text{ odd}, j\text{ even}, m\text{ even}} (i+j+1) a_{ij}r^{i+j} [-b_0 M_{ij}^0 -\dots -b_{m}r^m \frac{1}{m+1}M_{ij}^m] \\ &\quad +\sum^m_{i=2_{i even}} i b_ir^{i-1} [a_{10}r^2(\alpha_{110}N_i^1+\alpha_{210}N_i^3)+\dots \\ &\quad +a_{lb}r^{l+b+1}(\alpha_{1lb}N_i^1+\alpha_{2lb}N_i^3 +\dots +\alpha_{\frac{(l+b+2)+1}{2}lb}N_i^{l+b+2})]. \end{align*} Moreover, \begin{align*} &\int^{2\pi}_0F_{2}(\theta,r)d \theta\\ & = \int^{2\pi}_0 [\sum^n_{i+j=0} c_{ij} r^{i+j+1} \cos^i(\theta) \sin^{j+2}(\theta)+ \sum_{i=0}^m d_ir^i \cos^i(\theta)\sin(\theta)] d \theta \\ &\quad - \int^{2\pi}_0r \cos(\theta) \sin(\theta) \Big[ \sum^n_{i+j=0} a_{ij} r^{i+j}\cos^i(\theta) \sin^{j+1}(\theta) + \sum_ {i=0}^m b_ {i} r^{i-1} \cos^i(\theta) \Big]^2 d \theta, \end{align*} but $$ \int^{2\pi}_0 \cos^i(\theta) \sin^{j+2}(\theta) d \theta =\begin{cases} 0 &\text{if $i$ is odd or $j$ is odd}, \\ F_{ij}\neq 0 &\text{if $i$ is even and $j$ even.} \end{cases} $$ Hence \begin{align*} &\int^{2\pi}_0 F_{2}(\theta,r)d \theta \\ &= \sum_{i+j=0, i \text{ even}, j \text{even}}^n C_{ij}F_{ij} r^{i+j+1} \\ &\quad -2 \sum_{i+j=1,i\text{ even},j\text{ odd}}^n \sum_{l+k=1,l\text{ odd}, k\text{ even}}^n a_{ij} a_{lk} r^{i+j+l+k+1} \int^{2\pi}_0 \cos^{i+l+1}(\theta)\sin^{j+k+3}(\theta) \\ &\quad -2 \sum_{i+j=0,i\text{ even}, j\text{ even}}^n \sum_{l+k=2,l\text{ odd}, k \text{ odd}}^n a_{ij} a_{lk} r^{i+j+l+k+1} \int^{2\pi}_0 \cos^{i+k+1}(\theta)\sin^{j+l+3}(\theta) \\ &\quad -2 \sum_{k=1, k\text{ odd}}^m \sum_{i+j=0,i \text{ even}, j\text{ even}}^n b_{k} a_{ij} r^{k+i+j} \int^{2\pi}_0 \cos^{k+i+1}(\theta)\sin^{j+2}(\theta) \\ &\quad -2 \sum_{k=0,k\text{ even}}^m \sum_{i+j=0,i\text{ even}, j\text{ odd}}^n b_{k} a_{ij} r^{k+i+j} \int^{2\pi}_0 \cos^{k+i+1}(\theta)\sin^{j+2}(\theta). \end{align*} Then $F_{20}(r)$ is the polynomial \begin{align*} &\sum^n_{i+j=1, i \text{ even}, j\text{ odd}}(i+j+1) a_{ij}r^{i+j} [ a_{10} r^2 ( \alpha_{110} A^1_{ij} + \alpha_{210} A^3_{ij} )+\dots \\ &\quad + a_{lb} r^{l+b+1}( \alpha_{1lb} A^1_{ij} + \alpha_{2lb} A^3_{ij}+\dots + \alpha_{\frac{(l+b+2)+1}{2}lb} A^{l+b+2}_{ij} ) ]\\ &\quad +\sum^n_{i+j=1, i \text{ odd}, j\text{ even}}(i+j+1) a_{ij}r^{i+j} \big[ a_{01} r^2 (\alpha_{201} B^1_{ij} +\alpha_{301} B^3_{ij} ) +\dots \\ &\quad +a_{cd} r^{c+d+1} (\alpha_{2cd} B^1_{ij} +\alpha_{3cd} B^3_{ij} +\dots +\alpha_{\frac{(c+d+2)+3}{2}cd} B^{c+d+2}_{ij} ) \big] \\ &\quad +\sum^n_{i+j=1,i \text{ odd}, j\text{ even}, m \text{ even}}(i+j+1) a_{ij}r^{i+j} \big[-b_0 M_{ij}^0 -\dots -b_{m}r^m \frac{1}{m+1}M_{ij}^m\big] \\ &\quad +\sum^m_{i=2,i\text{ even}} i b_ir^{i-1} \big[a_{10}r^2(\alpha_{110}N_i^1+\alpha_{210}N_i^3)+\dots \\ &\quad +a_{lb}r^{l+b+1}(\alpha_{1lb}N_i^1+\alpha_{2lb}N_i^3 +\dots +\alpha_{\frac{(l+b+2)+1}{2}lb}N_i^{l+b+2})\big] \\ &\quad + \sum_{i+j=0,i\text{ even}, j \text{ even}}^n C_{ij}F_{ij} r^{i+j+1} \\ &\quad -2 \sum_{i+j=1,i\text{ even}, j\text{ odd}}^n \sum_{l+k=1,l \text{odd}, k \text{ even}}^n a_{ij} a_{lk} r^{i+j+l+k+1} \int^{2\pi}_0 \cos^{i+l+1}(\theta)\sin^{j+k+3}(\theta) \\ &\quad -2 \sum_{i+j=0, i\text{ even}, j\text{ even}}^n \sum_{l+k=2,l \text{odd}, k \text{odd}}^n a_{ij} a_{lk} r^{i+j+l+k+1} \int^{2\pi}_0 \cos^{i+k+1}(\theta)\sin^{j+l+3}(\theta) \\ &\quad -2 \sum_{k=1,k\text{ odd}}^m \sum_{i+j=0,i\text{ even}, j\text{ even}}^n b_{k} a_{ij} r^{k+i+j} \int^{2\pi}_0 \cos^{k+i+1}(\theta)\sin^{j+2}(\theta) \\ &\quad -2 \sum_{k=0, k\text{ even}}^m \sum_{i+j=0,i \text{ even}, j \text{ odd}}^n b_{k} a_{ij} r^{k+i+j} \int^{2\pi}_0 \cos^{k+i+1}(\theta)\sin^{j+2}(\theta). \end{align*} We conclude that \begin{align*} F_{20}(r) &=\sum^n_{i+j=1, i \text{ even}, j\text{ odd}}(i+j+1) a_{ij}r^{i+j} \big[ a_{10} r^2 ( \alpha_{110} A^1_{ij} + \alpha_{210} A^3_{ij} )+\dots \\ &\quad + a_{lb} r^{l+b+1}( \alpha_{1lb} A^1_{ij} + \alpha_{2lb} A^3_{ij}+\dots + \alpha_{\frac{(l+b+2)+1}{2}lb} A^{l+b+2}_{ij} ) \big] \\ &\quad +\sum^n_{i+j=1, i\text{ odd}, j\text{ even}}(i+j+1) a_{ij}r^{i+j} \big[ a_{01} r^2 (\alpha_{201} B^1_{ij} +\alpha_{301} B^3_{ij} ) +\dots \\ &\quad +a_{cd} r^{c+d+1} (\alpha_{2cd} B^1_{ij} +\alpha_{3cd} B^3_{ij} +\dots +\alpha_{\frac{(c+d+2)+3}{2}cd} B^{c+d+2}_{ij} ) \big] \\ &\quad +\sum^n_{i+j=1, i \text{ odd}, j\text{ even}, m\text{ even}} (i+j+1) a_{ij}r^{i+j} [-b_0 M_{ij}^0 -\dots -b_{m}r^m \frac{1}{m+1}M_{ij}^m] \\ &\quad +\sum^m_{i=2,i\text{ even}} i b_ir^{i-1} \big[a_{10}r^2(\alpha_{110}N_i^1+\alpha_{210}N_i^3)+\dots \\ &\quad +a_{lb}r^{l+b+1}(\alpha_{1lb}N_i^1+\alpha_{2lb}N_i^3 +\dots +\alpha_{\frac{(l+b+2)+1}{2}lb}N_i^{l+b+2})\big] \\ &\quad + \sum_{i+j=0,i\text{ even}, j \text{ even}}^n C_{ij}F_{ij} r^{i+j+1}\\ &\quad -2 \sum_{i+j=1, i \text{ even}, j \text{ odd}}^n \sum_{l+k=1,l\text{ odd}, k \text{even}}^n a_{ij} a_{lk} r^{i+j+l+k+1} F_{(i+l+1)(j+k+1)} \\ &\quad -2 \sum_{i+j=0,i\text{ even}, j\text{ even}}^n \sum_{l+k=2,l \text{ odd}, k \text{ odd}}^n a_{ij} a_{lk} r^{i+j+l+k+1} F_{(i+k+1)(j+l+1)} \\ &\quad -2 \sum_{k=1,k\text{ odd}}^m \sum_{i+j=0,i\text{ even}, j\text{ even}}^n b_{k} a_{ij} r^{k+i+j} F_{(k+i+1)j} \\ &\quad -2 \sum_{k=0,k\text{ even}}^m \sum_{i+j=0, i\text{ even}, j\text{ odd}}^n b_{k} a_{ij} r^{k+i+j} F_{(k+i+1)j}. \end{align*} Note that to find the positive roots of $F_{20}$ we must find the zeros of a polynomial in $r^2$ of degree equal to the \begin{align*} \max \Big\{&\frac{i+j+l+b}{2},\frac{i+j+c+d}{2},\frac{i+j+m-1}{2}, \frac{l+b+m-1}{2},\\ &\frac{i+j}{2},\frac{i+j+l+k}{2},\frac{i+j+m-1}{2} \Big\} \end{align*} we conclude that $F_{20}$ has at most $\max\{[\frac{n-1}{2}]+[\frac{m}{2}] ,\, [n+\frac{(-1)^{n+1}}{2}]\}$ positive roots. Hence the statement (b) of Theorem \ref{thm1} follows. \begin{example} \rm We consider the system \begin{equation}\label{c} \begin{gathered} \dot{x}=y, \\ \dot{y}=-x-\epsilon[(x+2y+xy-y^3-2xy^2)y+x]-\epsilon^2[(3y+xy-x^2+x^3)y+x]. \end{gathered} \end{equation} To look for the limit cycles, we must solve the equation \begin{equation}\label{d} F_{20}=\frac{1}{2} \Big((\frac{-17}{96})r^7+(\frac{133}{96})r^5 -(\frac{1}{3})r^3 \Big) =0, \end{equation} This equation has two positive roots $r_{1}=2.752278171$ and $r_{2}=0.4984920115$. According with statement (b) of Theorem \ref{thm1}, that system \eqref{c} has exactly two limit cycles bifurcating from the periodic orbits of the linear differential system \eqref{c} with $\epsilon=0$, using the averaging theory of second order. \end{example} \begin{thebibliography}{00} \bibitem{AA}{N. Ananthkrishnan, K. Sudhakar, S. Sudershan, A. 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