\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 40, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/40\hfil Sharp bounds of the number of zeros] {Sharp bounds of the number of zeros of Abelian integrals with parameters} \author[X. Sun, J. Yang \hfil EJDE-2014/40\hfilneg] {Xianbo Sun, Junmin Yang} % in alphabetical order \address{Xianbo Sun \newline Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning, 530003 Guangxi, China} \email{xianbo01@126.com Tel +86 15977781786} \address{Junmin Yang \newline College of Mathematics and Information Science, Hebei Normal University \newline Shijiazhuang, 050024 Hebei, China} \email{jmyzhw@sina.com} \thanks{Submitted November 11, 2013. Published February 5, 2014.} \subjclass[2000]{34C05, 34C07, 34C08} \keywords{Limit cycle; Li\'enard system; Chebyshev system; bifurcation; \hfill\break\indent heteroclinc loop} \begin{abstract} In this article, we study four Abelian integrals over compact level curves of four sixth-degree hyper-elliptic Hamiltonians with parameters. We prove that the sharp bound of the number of zeros for each Abelian integral is 2. The proofs rely mainly on the Chebyshev criterion for Abelian integrals and asymptotic expansions of Abelian integrals. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and main result} The second part of Hilbert's 16th problem and its weak version are two open problems in the qualitative theory of planar differential equations. The first one asks for the maximal number of limit cycles and their distribution for the following planar polynomial differential equation of degree $n$, \begin{equation}\label{h1} \dot{x}=P_n(x,y), \ \ \dot{y}=Q_n(x,y). \end{equation} A special form of \eqref{h1} is \begin{equation}\label{eq11} \begin{array}{l} \dot{x}={H}_y + \varepsilon p(x,y,\delta),\quad \dot{y}=-{H}_x + \varepsilon q(x,y,\delta), \end{array} \end{equation} where $H(x,y)$, $p(x,y)$, $q(x,y)$ are polynomials of $x$ and $y$, and their degrees satisfy $\max\{\deg p, \deg q\}=n$ and $\deg(H)=n+1$, and $\varepsilon$ is a positive and sufficiently small parameter. The unperturbed form of \eqref{eq11} is \begin{equation}\label{eq11a} \dot{x}={H}_y,\quad \dot{y}=-{H}_x. \end{equation} The Hamiltonian function $H(x,y)$ defines at least one family of closed curves $L_h$ which form a period annulus of \eqref{eq11a} denoted by $\{L_h\}$, where $h$ is energy parameter on an open interval $J$. Corresponding to system \eqref{eq11}, the following integral is called Abelian integral or first order Melnikov function, \begin{equation}\label{AA} A_n(h)= \oint_{L_h} q(x,y)dx-p(x,y)dy, \quad h \in J, \end{equation} which plays an important role in studying the limit cycles of \eqref{eq11} (see the Poincar\'e-Pontryagin Theorem \cite{cl}), and finding the upper bound of the maximal number of zeros of $A_n(h)$ is the weak version of the second part of Hilbert's 16th problem (usually called weak Hilbert's 16th problem). Its research advances and the recent popular and efficient methods for special forms of \eqref{eq11} can be found in the survey works \cite{li2012,li2003}. Since both problems are difficult, mathematicians try to study special and simpler forms of \eqref{h1} and \eqref{eq11}. Smale 13th problem restricts Hilbert's 16th problem to the Li\'enard system $$ \dot{x}=y-f(x), \quad \dot{y}=-x. $$ To study the number of zeros of $A_n(h)$, many mathematicians concentrate on a simpler form of \eqref{eq11} as follows \begin{equation}\label{l0} \dot{x}=y, \quad \dot{y}=g(x)+\varepsilon f(x)y, \end{equation} which is called Li\'enard system of type $(m,n)$ if $g(x)$ and $f(x)$ are polynomials of degree respectively $m$ and $n$. A comprehensive study has been made in \cite{fdcr} for the cases $m +n\leq 4$, except for $(m, n)=(1,3)$. In all these cases, it has proven that at most one limit cycle can appear and for $(m, n)=(1, 3)$ the same result has been conjectured (see \cite{ccnl}). For type $(3, 2)$, there are several cases according to the portraits of the unperturbed system. Dumortier and Li \cite{ld1,ld2,ld3,ld4} have made a complete study on these cases and obtained different sharp upper bounds of the number of zeros of Abelian integrals for different cases. Li, Marde\v{s}i\'c and Roussarie \cite{ld5} investigated some Li\'enard systems of type $(3, 2)$ with symmetry and also obtained the sharp bound. Wang and Xiao \cite{wang1,wphd} investigated some Li\'enard system of type $(4,3)$ and proved that $4$ is the least upper bound and 3 is the maximum lower bound of the number of the zeros for the corresponding Abelian integral. Some other cases of type $(4,3)$ are investigated in \cite{rajaac,ssh2013}, and the least upper bound and the maximal lower one are obtained. The results of the maximum lower bound for other systems of type $(4,3)$ can be found in \cite{yhrjmaa,yhijbc,yhcma}. For the type $(5,4)$, many works concentrate on the following Li\'enard systems with symmetry \begin{equation}\label{LL} \dot{x}=y, \quad \dot{y}=\eta x (x^2-a)(x^2-b) +\varepsilon (\alpha+\beta x^2+\gamma x^4)y, \end{equation} where $\eta=\pm1$, $\alpha$, $\beta$ and $\gamma$ are real bounded number. Assume the portraits of system \eqref{LL}$_{\varepsilon=0}$ has at least one periodic annulus, there are 12 cases according to the value of $a$, $b$ and $\eta$, see Figure \ref{fig1}. \begin{figure}[ht] \begin{center} \begin{tabular}{@{}cccc@{}} % {@{}c@{}c@{}c@{}c@{}} \includegraphics[width=0.23 \textwidth]{fig1} % b1.eps &\includegraphics[width=0.23 \textwidth]{fig2} % b2.eps &\includegraphics[width=0.23 \textwidth]{fig3} % b3.eps &\includegraphics[width=0.23 \textwidth]{fig4} % b4.eps \\ \parbox{29mm}{1: $\eta=-1$, $a>0$, $b>0$, $a\neq b$} &\parbox{29mm}{2: $\eta=-1$, $ab\neq0$, $a=b$} &\parbox{29mm}{3: $\eta=-1$, $ab=0$, ${\rm sgn}(a)+{\rm sgn}(b)=1$} &\parbox{29mm}{4: $\eta=-1$, $ab<0$} \\[12pt] \includegraphics[width=0.23 \textwidth]{fig5} % b5.eps &\includegraphics[width=0.23 \textwidth]{fig6} % b6.eps &\includegraphics[width=0.23 \textwidth]{fig7} % b7.eps &\includegraphics[width=0.23 \textwidth]{fig8} % f1.eps \\ \parbox{29mm}{5: $\eta=-1$, $ab=0$, ${\rm sgn}(a)+{\rm sgn}(b)=-1$} &\parbox{29mm}{6: $\eta=-1$, $a^2+b^2=0$} &\parbox{29mm}{7: $\eta=-1$, $a<0$, $b<0$} &\parbox{29mm}{8: $\eta=1$, $0<\frac{b}{a}<\frac{1}{3}$ or $\frac{b}{a}>{3}$} \\[12pt] \includegraphics[width=0.23 \textwidth]{fig9} % f2.eps &\includegraphics[width=0.23 \textwidth]{fig10} % f3.eps &\includegraphics[width=0.23 \textwidth]{fig11} % f5.eps &\includegraphics[width=0.23 \textwidth]{fig12} % f6.eps \\ \parbox{29mm}{9: $\eta=1$, $\frac{b}{a}=\frac{1}{3}$ or $\frac{b}{a}=3$} &\parbox{29mm}{10: $\eta=1$, $\frac{1}{3}<\frac{b}{a}<{1}$ or $1<\frac{b}{a}<{3}$} &\parbox{29mm}{11: $\eta=1$, $ab=0$, ${\rm sgn}(a)+{\rm sgn}(b)=1$} &\parbox{29mm}{12: $\eta=1$, $ab<0$} \end{tabular} \end{center} \caption{Twelve cases of \eqref{LL} each having at least one annulus surrounding a center} \label{fig1} \end{figure} For case 1, Zhang et al.~\cite{z1} proved that system \eqref{LL} with $a=1/2$, $b=2$ has at most $3$ zeros of the corresponding Abelian integral. For case 2, Asheghi and Zangeneh studied \eqref{LL} with $a=b=1$ and proved that the least upper bound for the number of zeros of the related Abelian integral inside the eye-figure loop is 2 in \cite{a1} and both inside and outside the eye-figure loop is 4 in \cite{a2}. For case 3, Asheghi and Zangeneh \cite{a3} studied \eqref{LL} by taking $a=0$, $b=1$ and proved that the corresponded Abelian integral has at most 2 zeros inside the double cuspidal loops. For case 3, Zhao \cite{zhaonon2014} studied system \eqref{LL} with $a=0$ and $b=1$ and obtained that $2$ is the sharp bound of the number of zeros of Abelian integral associated on the the two bounded period annuluses. For case 8, Xu and Li \cite{xl} proved that system \eqref{LL} has at least $5$ limit cycles bifurcated from $3$ annuluses of the system \eqref{LL})$_{\varepsilon=0}$ with $a=1/4$, $b=1$. For case 9, Sun \cite{sunade} proved there are at most $4$ zeros for the corresponding Abelian integral. Later, Zhao \cite{zhaodouble} proved the sharp bound of number of zeros for the corresponding Abelian integral is 2. For case 10, Qi and Zhao \cite{qz} proved that system \eqref{LL} with $a=\frac{21-\sqrt{41}}{20}$ and $b=\frac{21+\sqrt{41}}{20}$ has at most $2$ limit cycles bifurcated from each annulus. In this article, we study the cases 5, 6, 7 and 12 with some parameters. Without loss of generality we fix $\gamma=1$ in all cases. For case 12 we take $a=1$, $b=-\lambda$ without loss of generality. For convenience we assume $\lambda\geq1$, then system \eqref{LL} becomes \begin{equation} \dot{x}=y, \quad \dot{y}=\ x (x^2-1)(x^2+\lambda) +\varepsilon (\alpha+\beta x^2+ x^4)y, \label{eA} \end{equation} with the Hamiltonian function \begin{equation}\label{HL} \widetilde{ H}(x,y)=\frac{y^2}{2}+\frac{\lambda}{2} {x}^2 - \frac{\lambda-1}{4}x^4 -\frac{1}{6} {x}^6. \end{equation} The level sets (i.e. $\widetilde{H}(x, y) = h$) of Hamiltonian function \eqref{HL} are sketched in Figure \ref{fig2}. $\widetilde{H}(x, y)=h$ defines one family of ovals which correspond to a period annulus of system \eqref{eA}$_{\varepsilon=0}$ denoted by $\{\Gamma_h\}$. $H(x,y)=\frac{3\lambda+1}{12}$ defines a 2-polycycles $\Gamma^*=\{(x,y)|H(x,y)=\frac{3\lambda+1}{12}\}$ which consists of two heteroclinic orbits. $\Gamma_{0}$ is an elementary center. The Abelian integral on $\Gamma_h$ is \begin{equation}\label{Ml} I(h,\delta)=\oint_{\Gamma_h}(\alpha+\beta x^2+ x^4)y dx \equiv \alpha I_0(h)+\beta I_1(h)+ I_2(h), \end{equation} for $h\in (0,(3\lambda+1)/12)$, where $\delta=(\alpha,\beta,1)$, $I_i(h)=\oint_{\Gamma_h} x^{2i} y dx$, $i=0,1,2$. \begin{figure}[ht] \begin{center} \includegraphics[width=0.5 \textwidth]{fig13} % a.eps \end{center} \caption{The level set of $\widetilde{H}(x,y)$} \label{fig2} \end{figure} For case 7, we take $a=-\lambda_1, b=-\lambda_2$, where $\lambda_1, \lambda_2>0$, then system \eqref{LL} becomes \begin{equation} \dot{x}=y, \quad \dot{y}= -x (x^2+\lambda_1)(x^2+\lambda_2) +\varepsilon (\alpha+\beta x^2+ x^4)y. \label{eB} \end{equation} For case 6, we take $a=0, b=-\lambda_3$, where $\lambda_3>0$, then system \eqref{LL} becomes \begin{equation} \dot{x}=y, \quad \dot{y}= - x^3 (x^2+\lambda_3) +\varepsilon (\alpha+\beta x^2+ x^4)y. \label{eC} \end{equation} For case 5, we take $a=b=0$, then system \eqref{LL} becomes \begin{equation} \dot{x}=y, \quad \dot{y}= - x^5+\varepsilon (\alpha+\beta x^2+ x^4)y. \label{eD} \end{equation} The corresponding Abelian integrals of systems \eqref{eB}, \eqref{eC}, \eqref{eD} are, respectively, \begin{gather*} I^b(h,\delta)=\oint_{\Gamma_h^b}(\alpha+\beta x^2+ x^4)y dx \equiv \alpha I_0^b(h)+\beta I_1^b(h)+ I_2^b(h),\\ I^c(h,\delta)=\oint_{\Gamma_h^c}(\alpha+\beta x^2+ x^4)y dx \equiv \alpha I_0^c(h)+\beta I_1^c(h)+ I_2^c(h), \\ I^d(h,\delta)=\oint_{\Gamma_h^d}(\alpha+\beta x^2+ x^4)y dx \equiv \alpha I_0^d(h)+\beta I_1^d(h)+ I_2^d(h), \end{gather*} where $I(h)$, $I^b(h)$ and $I^c(h)$ have parameters $\lambda$, $\lambda_1$, $\lambda_2$, $\lambda_3$. Using some algebraic method, some polynomial techniques and expansions of Abelian integrals, the following results are obtained. \begin{theorem} \label{thmA} For all $\alpha$ and $\beta$, each of $I(h,\delta)$, $I^b(h,\delta)$, $I^c(h,\delta)$ and $I^d(h,\delta)$ has at most $2$ zeros, counting the multiplicity. Taking $0<\alpha \ll -\beta \ll 1$, two zeros of each Abelian integral appear in some small intervals near $h=0$. Therefore, 2 is the sharp bound. \end{theorem} By the Poincar\'e-Pontryagin theorem and Theorem \ref{thmA}, each of system \eqref{eA}, \eqref{eB}, \eqref{eC}, \eqref{eD} has at most 2 limit cycles bifurcated from the corresponding period annulus, and there exist some $(\alpha,\beta)$ and $0<\varepsilon\ll 1$ such that each system has 2 limit cycles bifurcated from the corresponding period annulus. The rest of the article is organized as follows: in section 2 we will introduce some definitions and the new criteria which are used to determine the number of zeros of the Abelian integrals. In sections 3 and 4, we will prove the main results. \section{Preliminary lemmas and definitions} The method we will introduce proposes some criterion functions defined directly by Hamiltonian and integrands of Abelian integrals, through which the problem whether the basis of the vector space generated by Abelian integrals is a Chebyshev system could be reduced to the problem whether the family of criterion functions form a Chebyshev system, since the latter can be tackled by checking the nonvanishing properties of its Wronskians. For this paper to be self-contained, we list some related definitions and criterions. For more details, \cite{mp,MG} is referred. \begin{definition} \label{def2.1} \rm Suppose $f_0, f_1, f_2, \dots , f_{n-1}$ are analytic functions on an real open interval $J$. (i) The family of polynomials $\{f_0, f_1, f_2, \dots , f_{n-1}\}$ is called Chebyshev system (T-system for short) provided that any nontrivial linear combination $$ k_0f_0(x) + k_1f_1(x)+ \dots + k_{n-1}f_{n-1}(x) $$ has at most $n-1$ isolated zeros on $J$. (ii) An ordered set of $n$ functions $\{f_0, f_1, f_2, \dots , f_{n-1}\}$ is called complete Chebyshev system (CT-system for short) provided any nontrivial linear combination $k_0f_0(x) + k_1f_1(x)+ \dots + k_{i-1}f_{i-1}(x)$ has at most $i-1$ zeros for all $i = 1,2,\dots ,n$, moreover it is called extended complete Chebyshev system (ECT-system for short) if the multiplicities of zeros taken into account. (iii) The continuous Wronskian of $\{f_0, f_1, f_2, \dots , f_{n-1}\}$ at $x \in R$ is \begin{align*} W[f_0, f_1, f_2, \dots , f_{k-1}] &=\det(f_i^j)_{0\leq i,j\leq k-1}\\ &= \begin{vmatrix} f_0(x) & f_1(x) & \dots & f_{k-1} \\ f_0'(x) & f_1'(x) & \dots & f_{k-1}'(x) \\ \dots & \dots & \dots & \dots \\ f_0^{(k-1)}(x) & f_1^{(k-1)}(x) & \dots & f_{k-1}^{(k-1)}(x) \end{vmatrix}, \end{align*} where $f'(x)$ is the first order derivative of $f(x)$ and $f^{(i)}(x)$ is the $i$th order derivative of $f(x)$, $i\geq 2$. \end{definition} The above definitions imply that the function tuple $\{f_0, f_1, \dots , f_{k-1}\}$ is an ECT-system on $J$, therefore it is a CT-system on $J$, and then a T-system on $J$, however the inverse implications are not true at all. Recall that the authors of \cite{MG} studied the number of isolated zero of Abelian integrals using a purely algebraic criteria which is developed from the idea introduced in \cite{lizhang}. Let $H(x, y) = A(x) + \frac{1}{2}y^2$ be an analytic function in some open subset of the plane which has a local minimum at $(0,0)$. Then there exists a punctured neighborhood $P$ of the origin foliated by ovals $L_h:{H(x, y) =h}$ which correspond to the clockwise closed orbits of $(\ref{eq11a})$. The set of ovals $L_h$ inside the period annulus, is parameterized by the energy levels $h \in(0, h_1)=J$ for some $h_1 \in (0,+\infty]$. The projection of $P$ on the $x$-axis is an interval $(x_l, x_r)$ with $x_l <0 < x_r$. Under the above assumptions it is easy to verify that $x A'(x) > 0$ for all $x\in(x_l, x_r)\setminus \{0\}$, $A(x)$ has a zero of even multiplicity at $x = 0$ and there exists an analytic involution $z(x)$ such that $$ A(x) = A(z(x)) $$ for all $x\in(x_l, x_r)$. It is obvious that $z(x)=-x$ if $A(x)$ is a even function. For the number of isolated zeros of nontrivial linear combination of some integrals of special form, the algebraic criterion in \cite[Theorem B]{MG} can be stated as follows: \begin{lemma} \label{lem2.1} Assume that the function $f_i(x)$ is analytic on the interval $(x_l,x_r)$ for $i = 0, 1, \dots ,n-1$, and consider $$ A_{i}(h)=\int_{L_h} f_i(x)y^{2s-1}dx, \quad i=0, 1,\dots , n-1, $$ where for each $h \in(0,h_0)$, $L_h$ is the oval surrounding the origin inside an level curve $\{A(x)+\frac{1}{2}y^2=h\}$. We define $$ l_i(x):=\frac{f_i(x)}{A'(x)}- \frac{f_i(z(x))}{A'(z(x))}. $$ Then, $\{A_0, A_1,\dots , A_{n-1}\}$ is an ECT-system on $(0,h_1)$ if $\{l_0, l_1,\dots , l_{n-1}\}$ is a CT-system on $(x_l,0)$ or $(0, x_r )$ and $s > n-2$. And $\{l_0, l_1,\dots , l_{n-1}\}$ is an ECT-system on $(x_0,x_r)$ or $(x_l,x_0)$ if and only if the continuous Wronskian of $\{l_0,l_1,\dots ,l_{k-1}\}$ does not vanish for $ \forall x \in (0, x_r)$ or for all $z\in (x_l,0)$ and $k = 1,\dots ,n$. \end{lemma} Usually $s$ is not big enough, Lemma \ref{lem2.1} can not be applied directly. To overcome this problem the next result (see \cite[Lemma 4.1]{MG}) is useful to increase the power of $y$ in $A_i(h)$. \begin{lemma} \label{lem2.2} Let $L_h$ be an oval inside the level curve ${A(x) + \frac{1}{2}(x)y^2 = h}$ and consider a function $F(x)$ satisfying $\frac{F(x)}{A'(x)}$ is analytic at $x = 0$. Then, for any $k \in N$, $$ \oint_{L_h} F(x)y^{k-2} dx = \oint_{L_h}G(x)y^k dx $$ where $G(x) = \frac{1}{k}(\frac{F}{A'})'(x)$. \end{lemma} \section{Proof of main result} For briefness we prove only case 12, other cases can be proved similarly. In what follows, we proved that the following generating elements of $I(h,\delta)$, $$ I_i(h)=\int_{\Gamma_h}x^{2i}ydx,\quad i=0, 1, 2 $$ have the Chebyshev property for $h\in (0,\frac{3\lambda+1}{12})$. By Lemma \ref{lem2.1}, $A(x)=\widetilde{H}(x,0)=-\frac{3}{2} {x}^2+{x}^4-\frac{1}{6} {x}^6$ and $s = 1$, $n = 3$ for system \eqref{eA}. The period annulus is foliated by the ovals $\Gamma_h$, and the projection of the period annulus on the plan is an open interval $(-1, 1)$. Noting that $xA'(x) > 0$ for all $x \in (-1, 1)\setminus \{0\}$, therefore there exists an analytic involution $z(x)$ such that $$ A(x)=A(z(x)). $$ Our goal is to prove that the vector space generated by Abelian integral $I_i(h)$ has the Chebyshev property for $x\in (0, 1)$ by Lemma \ref{lem2.1}. However, for $s = 1$ and $n = 3$ it does not satisfy the hypothesis $s> n - 2$ in Lemma \ref{lem2.1}. Thus the power $s$ of $y$ in the integrand of $I_i(h)$ should be increased such that the condition $s > n -2$ holds. \begin{lemma} \label{lem3.1} For $i=0,1,2$, we have $$ 2h I_i(h)=\int_{\Gamma_h}{f}_i(x)y^3dx, $$ where $f_i(x)= \frac {{x}^{2 i} \widetilde{f}_i(x)}{18(x-1)^2(x+1)^2 (x^2+\lambda)^2}$ with \begin{align*} \widetilde{f}_i(x) &= 20{x}^8+39{x}^6\lambda+21{\lambda}^2{x}^4-39{\lambda}^ {2}{x}^2+24{\lambda}^2+4 i{x}^8-10 i{x}^6+6 i{x}^4\\ &\quad +12i{\lambda}^2+10 i{x}^6\lambda -28 i{x}^4\lambda +18i\lambda{x}^2+6 i{\lambda}^2{x}^4-18 i{\lambda}^2{x}^2\\ &\quad +21{x}^4- 70{x}^4\lambda+39\lambda{x}^2-39{x}^6. \end{align*} \end{lemma} \begin{proof} It is clear that on every periodic orbits $\Gamma_h:\{\widetilde{H}(x,y)=h\}$, $\frac{2A(x)+y^2}{2h}=1$ holds. Therefore, \begin{equation}\label{51} I_i(h)=\frac{1}{2h}\int_{\Gamma_h}(2A(x)+ y^2)x^{2i}ydx =\frac{1}{2h}\int_{\Gamma_h}2x^{2i}A(x)ydx + \frac{1}{2h}\int_{\Gamma_h}x^{2i}y^3dx, \end{equation} for $i=0,1,2$. Noting that the functions $\frac{2x^{2i}A(x)}{A'(x)}$ are analytic on $x=1$, by Lemma \ref{lem2.2}, we have \begin{equation}\label{32} \int_{\Gamma_h}2x^{2i}A(x)ydx=\int_{\Gamma_h}G_i(x)y^3dx, \end{equation} where $$ G_i(x)= \frac{ x^{2i} g_i(x)}{( x-1)^2 (x+1)^2 (x^2+\lambda)^2} $$ with \begin{align*} g_i(x)& = 2{x}^8+3{x}^6\lambda+3{\lambda}^2{x}^4-3{\lambda}^2{ x}^2+6{\lambda}^2+4 i{x}^8-10 i{x}^6+6 i{x}^4\\ &\quad +12 i{\lambda}^2+10 i{x}^6\lambda -28 i{x}^4\lambda+18 i\lambda{x}^2 +6 i{\lambda}^2{x}^4\\ &\quad -18 i{\lambda}^2{x}^2+3{x}^4+2{x}^4\lambda+3\lambda{x}^2-3{x}^6. \end{align*} Combine \eqref{51} and \eqref{32}, so Lemma \ref{lem3.1} is proved. \end{proof} Let $$ \widetilde{I}_i(h)=\int_{\Gamma_h}{f}_i(x)y^3dx. $$ Then $\{I_0, I_1,I_2\}$ is an ECT-system on $(0,\frac{3\lambda+1}{12})$ if and only if $\{\widetilde{I}_0, \widetilde{I}_1,\widetilde{I}_2\}$ is as well. Since $s = 2$, $n = 3$ and the condition $s > n-2$ holds, lemma \ref{lem2.1} can be used to study if $\{\widetilde{I}_0, \widetilde{I}_1,\widetilde{I}_2\}$ is an ECT-system on $(0,\frac{3\lambda+1}{12})$. Thus, setting the criteria functions \begin{equation}\label{lx} l_i(x)=(\frac{f_i}{A'})(x)-(\frac{f_i}{A'})(z(x)), \ 0 < x < 1, i= 0, 1, 2, \end{equation} where $z(x)$ is the analytic involution $z(x)$ defined by $A(x)=A(z)$. By symmetry of system \eqref{eA}, it is obvious $z(x)=-x$. Inserting $z(x)=-x$ in \eqref{lx} gives $$ l_i(x)=- {\frac {( {x}^{2 i}+ (-x) ^{2 i})\widetilde{l_i}(x)}{18 (x-1) ^3 (x+1)^3 ({x}^2+\lambda)^3x}} $$ with \begin{align*} \widetilde{l_i}(x) &= 20{x}^8+21{\lambda}^2{x}^4-39{ \lambda}^2{x}^2+24{\lambda}^2+4 i{x}^8-10 i{x}^6+6 i{x}^4\\ &\quad +12 i{\lambda}^2+39{x}^6\lambda+10 i{x}^6\lambda -28 i{x}^4\lambda+18 i\lambda{x}^2+6 i{\lambda}^2{x}^4\\ &\quad -18 i{\lambda}^2{x}^2+21{x}^4-70{x}^4\lambda+39\lambda{x}^{2}-39{x}^6. \end{align*} Next, we check that the ordered set of criterion functions $\{l_1(x), l_2(x), l_0(x)\}$ is an ECT-system for $x \in(0, 1)$ by verifying the non-vanishing property of continuous Wronskians $W[l_1]$, $W[l_1, l_2]$, $W[ l_1, l_2, l_0]$. \begin{lemma} \label{lem3.2} The function tuple $\{l_1(x), l_2(x), l_0(x)\}$ is an ECT-system for $x \in (0, 1)$. \end{lemma} \begin{proof} By the Definition \ref{def2.1} (iii) about the continuous Wronskian, with the aid of Maple 13, we have \begin{gather*} W[l_1(x)]= \frac{- x w_1(x,\lambda)}{9(x-1)^3 (x+1)^3 (x^2+\lambda)^3}, \\ W[l_1(x),l_2(x)] = \frac{2x^3 w_2(x,\lambda)}{81(x-1)^5 (x+1)^5 (x^2+\lambda)^5}, \\ W[l_1(x),l_2(x),l_0(x)] = \frac{- 16 w_3(x,\lambda)}{243 (x-1)^7 (x+1)^7 (x^2+\lambda)^7}, \end{gather*} where \begin{gather*} \begin{aligned} w_1(x,\lambda) & = 24{x}^8+27{\lambda}^2{x}^4-57{\lambda}^2{x}^2+36{ \lambda}^2-49{x}^6+27{x}^4+49{x}^6\lambda\\ &\quad -98{x}^4\lambda+57\lambda{x}^2, \end{aligned} \\ \begin{aligned} w_2(x,\lambda) & = 672{x}^{12}-2072{x}^{10}+2072{x}^{10}\lambda-6174{x}^8 \lambda+2295{x}^8+2295{x}^8{\lambda}^2\\ &\quad -891{x}^6+6993{x}^6\lambda -6993{\lambda}^2{x}^6+891{x}^6{\lambda}^3 +8382 {\lambda}^2{x}^4-2871{\lambda}^3{x}^4\\ &\quad -2871{x}^4\lambda -3672{\lambda}^2{x}^2+3672{\lambda}^3{x}^2 -1728{\lambda}^3, \end{aligned}\\ \begin{aligned} w_3(x,\lambda) & = 13824{\lambda}^4+6237{x}^8+40392{\lambda}^2{x}^4+22275\, {x}^6\lambda+6237{\lambda}^4{x}^8-22275{\lambda}^4{x}^6\\ &\quad +40392{\lambda}^4{x}^4 -37152{\lambda}^4{x}^2+4480{x}^{16} -17248{x}^{14}+27636{x}^{12}\\ &\quad -20979{x}^{10}-71280{x}^8{ \lambda}^3+27636{x}^{12}{\lambda}^2 -92073{x}^{10}{\lambda}^2 +17248{x}^{14}\lambda\\ &\quad -59304{x}^{12}\lambda +20979{x}^{10}{\lambda}^3+37152{\lambda}^3{x}^2 -71280{x}^8\lambda-122265{\lambda}^2{x}^6\\ &\quad -106128{\lambda}^3{x}^4 +92073{x}^{10}\lambda+149094{x}^8{\lambda}^2 +122265{x}^6{\lambda}^3, \end{aligned} \end{gather*} of degree $8$, $12$ and $16$, respectively. To check if three Wronskians vanish for $x \in (0,1)$, we only need check if three two-variable polynomials $w_1(x,\lambda)$, $w_2(x,\lambda)$ and $w_3(x,\lambda)$ vanish for $x \in (0,1)$. In order to avoid complicated symbolic computation, such as regular chains with parameter, and real roots isolation, we introduce some transforms. First, let $\alpha>0$ and introduce $x=\frac{1}{1+\alpha}$, which satisfies $00, \lambda\geq 1\}$ since its coefficients are all positive. Therefore, $W[l_1(x)]$ has not root for $x\in(0,1)$ obviously. Second, let $\alpha>0$, $\beta\geq 0$ and introduce $$ x=\frac{1}{1+\alpha}, \quad \lambda=1+\beta. $$ Then $w_2(x,\lambda)=-p_2(\alpha,\beta)/(1+\alpha)^{12}$, where $p_2(\alpha,\beta)$ is a polynomial with positive coefficients and has no root on $\{(\alpha,\lambda):\alpha>0, \beta\geq 0\}$ (see Appendix A). Hence, $W[l_1(x),l_2(x)]$ has no roots for $x\in(0,1)$. Last, taking $\alpha>0$ and substituting $x=1/(1+\alpha)$ into $w_3(x,\lambda)$ yields $w_3(x,\lambda)= \frac{p_3(\alpha,\lambda)}{(1+\alpha)^{16}}$. The polynomial $p_3(\alpha,\lambda)$ has positive coefficients (see Appendix A), and it has no root on $\{(\alpha,\lambda)|\alpha>0, \lambda\geq 1\}$. Hence, $W[l_1(x),l_2(x),l_0(x)]$ has no root for $x\in(0,1)$. Lemma \ref{lem3.2} is proved. \end{proof} By Lemmas \ref{lem2.1} and \ref{lem3.2}, $\{\widetilde{I}_1(h), \widetilde{I}_2(h),\widetilde{I}_0(h)\}$ is an ECT-system on $(0,(3\lambda+1)/12)$, and so $\{ {I}_1, {I}_2, {I}_0\}$ is as well. Therefore, $I(h,\delta)$ has at most $2$ zeros. \begin{remark} \rm With the same methods and techniques, it is not difficult to prove each of $I^b(h,\delta)$, $I^c(h,\delta)$ and $I^d(h,\delta)$ has at most $2$ zeros, we omit the proofs here for brevity. \end{remark} \section{Finding zeros in small intervals} Usually, it is difficult to find zeros of $A_n(h)$. One popular method is to detect the expansions of $A_n(h)$ near a center, homoclinic loop and heteroclinic loop of system \eqref{eq11a}, see \cite{hanexp}. When the annulus $\{L_h\}$ of system \eqref{eq11a} has a homoclinic loop, a heteroclinic loop as the outer boundary, the expansions of $A_n(h)$ near these outer boundaries was studied in \cite{jdde2008} and the expression of coefficients are also given. When the inner boundary of $\{L_h\}$ is a center, the expansion of $A_n(h)$ near an elementary center is investigated in \cite{hanijbc2009}. and the expansion of $A_n(h)$ near a nilpotent center is investigated in \cite{yanghamca}. By the results of \cite{jdde2008,hanijbc2009,yanghamca}, the expansions of $I(h,\delta)$, $I^b(h,\delta)$, $I^c(h,\delta)$ and $I^d(h,\delta)$ near the centers are as follows \begin{gather*} I(h,\delta) =b_0(\delta)h+b_1(\delta)h^2+b_2(\delta)h^3+h.o.t., \quad h\in (0,\varepsilon_1), \\ I^b(h,\delta) =\widetilde{b}_0(\delta)h+\widetilde{b}_1(\delta)h^2 +\widetilde{b}_2(\delta)h^3+h.o.t., \quad h\in (0,\varepsilon_2), \\ I^c(h,\delta) =\overline{b}_0(\delta)h^{\frac{3}{4}} +\overline{b}_1(\delta)h^{\frac{5}{4}}+\overline{b}_2(\delta)h^{\frac{7}{4}}+h.o.t., \ \ \ h\in(0,\varepsilon_3), \\ I^d(h,\delta) =\widehat{b}_0(\delta)h^{\frac{4}{6}}+\widehat{b}_1(\delta) h^{\frac{8}{6}}+\widehat{b}_2(\delta)h^{\frac{10}{6}}+h.o.t., \quad h\in(0,\varepsilon_4), \end{gather*} where $0<\varepsilon_1, \varepsilon_2, \varepsilon_3, \varepsilon_4 \ll 1$ and \begin{gather*} b_0(\delta)=2\pi \alpha, \quad b_1(\delta)=\frac{3\pi}{4}(\lambda-1)\alpha+\pi \beta, \\ b_2(\delta)=\frac{5\pi}{96}(21\lambda^2-42\lambda+37)\alpha +\frac{5\pi}{4}(\lambda-1)\beta+\pi, \\ \widetilde{b}_0(\delta)=2\pi \alpha, \quad \widetilde{b}_1(\delta)= \frac{3\pi}{4}(\lambda_1-\lambda_2)\alpha+ \pi \beta, \\ \widetilde{b}_2(\delta)= \frac{105\pi}{96}( \lambda_1^2+2 \lambda_1\lambda_2+ \lambda_2^2+\frac{80}{105})\alpha+ \frac{5\pi}{4}(\lambda_1+\lambda_2) \beta+ \pi, \\ \overline{b}_0(\delta)=\frac{4 \pi^{\frac{3}{2}} \sqrt{2} \alpha}{3 \Gamma^2 (\frac{3}{4}) \sqrt[4]{\lambda_3}}, \quad \overline{b}_1(\delta)=\frac{8\sqrt{2} \Gamma^2(\frac{3}{4})(2\lambda_3\beta-\alpha)}{5 {\lambda_3}^{\frac{7}{4}} \sqrt{\pi}}, \\ \overline{b}_2(\delta)=\frac{2\sqrt{2}\pi^{\frac{3}{2}} (15 \alpha-20 \lambda_3 \beta+24 \lambda_3^2)}{63\Gamma^2(\frac{3}{4}){\lambda_3}^{\frac{13}{4}}}, \quad \widehat{b}_0(\delta)=\frac {\sqrt{2}{6}^{\frac{1}{6}}{\pi } ^{\frac{3}{2}} \alpha} {\Gamma(\frac{5}{6})\Gamma(\frac{2}{3})}, \\ \widehat{b}_1(\delta ) = \frac{2\sqrt{3} \pi \beta}{3}, \quad \widehat{b}_2(\delta ) = \frac{3\times 6^{\frac{4}{3}} \Gamma(\frac{5}{6})\Gamma(\frac{2}{3}) }{8 \sqrt{\pi}}, \end{gather*} Taking $\alpha=\beta=0$, then $b_0=b_1=\widetilde{b}_0=\widetilde{b}_1=\overline{b}_0=\overline{b}_1 =\widehat{b}_0=\widehat{b}_1=0$ and $b_2=\widetilde{b}_2=\overline{b}_2=\widehat{b}_2=\pi$. It is easy to find that $$ \det \frac{\partial (b_0, b_1)}{\partial (a_0, a_1)} = \det \frac{\partial (\widetilde{b}_0, \widetilde{b}_1)}{\partial (a_0, a_1)} = \det \frac{\partial (\overline{b}_0, \overline{b}_1)}{\partial (a_0, a_1)} = \det \frac{\partial (\widehat{b}_0, \widehat{b}_1)}{\partial (a_0, a_1)}= 2. $$ Let us take $\alpha\ll -\beta \ll 1$, then $b_0\ll -b_1 \ll b_2$, $\widetilde{b}_0\ll -\widetilde{b}_1 \ll \widetilde{b}_2$, $\overline{b}_0\ll -\overline{b}_1 \ll \overline{b}_2$ and $\widehat{b}_0\ll -\widehat{b}_1 \ll \widehat{b}_2$, which imply there exist $2$ zeros for each Abelian integral $I(h,\delta)$, $I^b(h,\delta)$, $I^c(h,\delta)$ and $I^d(h,\delta)$. \subsection{Conclusion} The number of zeros of Abelian integral for system \eqref{LL} has been studied for all 12 cases except for case $4$. Up to now, the sharp bounds of the numbers of zeros for the corresponding Abelian integrals defined on all period annuluses for one case of system \eqref{LL} are obtained for case $5$, $6$, $7$, $12$ and case $9$, the sharp bound for other cases are our further research. \subsection*{Appendix A} This section shows two polynomials with positive coefficients that have no root on $\{(\alpha,\lambda):\alpha>0, \lambda\geq 1\}$. \begin{align*} &p_2(\alpha,\beta) \\ &=1728{\alpha}^{12}+380160{\alpha}^9+114048{\alpha}^{10}+20736\, {\alpha}^{11}+238656{\alpha}^3+672144{\alpha}^4\\ &\quad +1220736{\alpha}^5 +1522752{\alpha}^6 +1347456{\alpha}^7+852720{ \alpha}^8+49632{\alpha}^2+5952\alpha\\ &\quad +64\beta+36{\beta}^3+20736{\beta}^3{\alpha}^{11} +1728{\beta}^3{\alpha}^{12}+3622848\beta{\alpha}^7 +2395560\beta{\alpha}^8\\ &\quad +3226296{\beta}^2{\alpha}^7+2235831{\beta}^2{\alpha}^8 +950904{\beta}^3{\alpha}^7 +692991{\beta}^3{\alpha}^8 +1103760\beta{\alpha}^9 \\ &\quad +338472\beta{\alpha}^{10} +1067040{\beta}^2{\alpha}^ {9}+334800{\beta}^2{\alpha}^{10} +343440{\beta}^3{\alpha}^9 +110376{\beta}^3{\alpha}^{10}\\ &\quad +62208\beta{\alpha}^{11}+5184 \beta{\alpha}^{12} +62208{\beta}^2{\alpha}^{11}+5184{\beta}^2{\alpha}^{12} +11360\beta\alpha +101296\beta{\alpha}^2 \\ &\quad +496896\beta{\alpha}^3+1491744\beta{\alpha}^4+96{\beta}^2 +7356{\beta}^2\alpha+69162{\beta}^2{\alpha}^2+349356{\beta}^2{\alpha}^3\\ &\quad +1102515{\beta}^2{\alpha}^4 +2910624\beta{\alpha}^5 +3875376\beta{\alpha}^6 +2293896{\beta}^2{\alpha}^5 +3258564{\beta}^2{\alpha}^6\\ &\quad +1638{\beta}^3\alpha +15831{ \beta}^3{\alpha}^2 +82476{\beta}^3{\alpha}^3 +271845{\beta}^3{\alpha}^4+598662{\beta}^3{\alpha}^5\\ &\quad +905049{\beta}^3{\alpha}^6. \end{align*} \begin{align*} &p_3(\alpha,\lambda)\\ &= 126+1012\lambda+1026{\lambda}^4+2988{\lambda}^3+2784{\lambda}^2 +40236{\alpha}^3+149541{\alpha}^4+223398{\alpha}^5\\ &\quad +153657{\alpha}^6+49896{\alpha}^7+6237{\alpha}^8 +8519{\alpha}^2+12912{\lambda}^2\alpha+123300{\lambda}^2{\alpha}^2\\ &\quad +832788{\lambda}^2{\alpha}^3+3401511{\lambda}^2{\alpha}^4 +8976510{\lambda}^2{\alpha}^5 +15729117{\lambda}^2{\alpha}^6\\ &\quad +18511416{\lambda}^2{\alpha}^7 +14641209{\lambda}^2{\alpha}^8 +2228\lambda\alpha+49054\lambda{\alpha}^2+285564\lambda{\alpha}^3\\ &\quad +1009941\lambda{\alpha}^4 +2174058\lambda{\alpha}^5 +2773983\lambda{\alpha}^6+70\alpha +484704{\lambda}^2{\alpha}^{11}\\ &\quad+40392{\lambda}^2{\alpha}^{12} +222750\lambda{\alpha}^9 +22275\lambda{\alpha}^{10} +57553848{\lambda}^3{\alpha}^7\\ &\quad +64464741{\lambda}^3{\alpha}^8 +52252794{\lambda}^3{\alpha}^9 +30306969{\lambda}^3{\alpha}^{10}+12249792{\lambda}^3{\alpha}^{11}\\ &\quad +3274704{\lambda}^3{\alpha}^{12} +520128{\lambda}^3{\alpha}^{13} +37152{\lambda}^3{\alpha}^{ 14}+2102760\lambda{\alpha}^7 +931095\lambda{\alpha}^8\\ &\quad+ 7663590{\lambda}^2{\alpha}^9 +2543607{\lambda}^2{\alpha}^{10} +12906{\lambda}^4\alpha +116181{\lambda}^4{\alpha}^2+780624{\lambda}^4{\alpha}^3\\ &\quad +3723408{\lambda}^4{\alpha}^4 +12731364 {\lambda}^4{\alpha}^5 +31954230{\lambda}^4{\alpha}^6+60008256{\lambda}^4{\alpha}^7\\ &\quad +85345326{\lambda}^4{\alpha}^8+92431746{\lambda}^4{\alpha}^9 +76157037{\lambda}^4{\alpha}^{10}+47344608{\lambda}^4{\alpha}^{11}\\ &\quad +21819240{\lambda}^4{\alpha}^{12}+7221312{\lambda}^4{\alpha}^{13} +1621728{\lambda}^{4}{\alpha}^{14}+221184{\lambda}^4{\alpha}^{15}\\ &\quad +13824{\lambda}^4 {\alpha}^{16}+24876{\lambda}^3\alpha +197154{\lambda}^3{\alpha}^2+1274868{\lambda}^3{\alpha}^3 +5656527{\lambda}^3{\alpha}^4 \\ &\quad +17269902{\lambda}^3{\alpha}^5 +37205973{\lambda}^3{\alpha}^6. \end{align*} \subsection*{Acknowledgements} This work was supported by the National Natural Science Foundations of China (No. 11101118, 11261013), Natural Science Foundation of Hebei Province (A2012205074) and Research project of Guangxi Universities (2013YB216). \begin{thebibliography}{00} \bibitem{a1} R. 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