1. Introduction
Consider the planar near-Hamiltonian system
\begin{align}
\dot{x}=H_{y}(x,y)+\varepsilon f(x,y,\delta),\quad \dot{y}=-H_{x}(x,y)+\varepsilon g(x,y,\delta),
\end{align} where the parameter 
$|\varepsilon|\ll 1$, the vector parameter 
$\delta\in D\subset\mathbf{R}^{n}$ with D compact, 
$H, f,g$ are polynomial functions in x and y. The system (1.1) describes a widely researched dynamical system that not only has numerous applications in vital areas such as celestial mechanics, molecular dynamics, statistical mechanics, and quantum mechanics [Reference Leimkuhler and Reich22, Reference Tuckerman30] but also is related to the weakened Hilbert’s 16th, as proposed by Anorld [Reference Arnold1], which is an important subject of investigation in the qualitative theory of plane differential systems.
Let 
$\omega=g(x,y,\delta)dx-f(x,y,\delta)dy$ be a 1-form. The weakened Hilbert’s 16th problem is to find an upper bound for the number of isolated zeros of the first-order Melnikov function (Abelian integral)
\begin{align*}
M(h, \delta)=\oint_{\Gamma_{h}}\omega,
\end{align*} where 
$\Gamma_{h}=\{H(x,y)=h, h\in J\}$ is defined as a family of continuous and close curves, with J being an open interval, and it is related to the lower bound of the maximum number of limit cycles for the system (1.1).
An essential approach to determining the isolated zero number of 
$M(h, \delta)$ is to investigate its expansion in h near a centre, a homoclinic loop, or a heteroclinic loop (see [Reference Euzébio, Llibre and Tonon9–Reference Gavrilov and Iliev11, Reference Han14, Reference Han, Yang, Tarţa and Gao18, Reference Han, Zang and Yang20]). The asymptotic expansion of the first-order Melnikov function 
$M(h, \delta)$ near the homoclinic or heteroclinic loop will be briefly described below, including the formulas for the coefficients of the first few terms in the expansion.
Dulac [Reference Dulac7] and Roussarie [Reference Roussarie28] gained the expression of the expansion of 
$M(h, \delta)$ near and inside the homoclinic loop L with a hyperbolic saddle,
\begin{align}
M(h, \delta)=\sum_{j \geq 0}\left(c_{2 j}(\delta)+c_{2 j+1}(\delta)\left(h-h_{s}\right) \ln \left|h-h_{s}\right|\right)\left(h-h_{s}\right)^{j}, \quad 0 \lt h_{s}-h\ll 1,
\end{align} where 
$H(S)=h_{s}$, 
$c_{0}(\delta)=\oint_{L}\omega\left.\right|_{\varepsilon=0}$, and 
$c_{1}(\delta)=-\left.\frac{1}{|\lambda|}\left(\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}\right)\right|_S$ with the eigenvalues ±λ, and the expression of the coefficient c 1 was obtained by Han and Ye [Reference Han and Ye19]. For detailed calculations of higher-order coefficients in the expansion of 
$M(h,\delta)$, refer to [Reference Han, Yang, Tarţa and Gao18, Reference Han and Ye19, Reference Tian and Han29, Reference Wei, Zhang and Zhang31].
Suppose system (1.1)
$|_{\varepsilon=0}$ has a polycycle L with m hyperbolic saddles Si for 
$i=1,\ldots, m$, encircling the family of periodic orbits 
$\Gamma_{h}$ near it, where 
$H(S_{1})=h_{s}$. Jiang and Han [Reference Jiang and Han21] proved the asymptotic expansion of 
$M(h,\delta)$ near the heteroclinic loop, and it is in the same form as (1.2). Obviously, the first coefficient 
$c_{0}(\delta)=\oint_{L} \omega\left.\right|_{\varepsilon=0}$ remains unchanged. The coefficient 
$c_{1}(\delta)=\sum_{i=1}^{m} -\left.\frac{1}{\left|\lambda_{i}\right|}\left(\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}\right)\right|_{S_{i}}$ with the eigenvalues 
$\pm \lambda_{i}$ of Si was given by Li et al. [Reference Li, Llibre and Zhang24] (see [Reference Geng and Tian13, Reference Han, Yang, Tarţa and Gao18, Reference Wei, Zhang and Zhu32] and its references for other results on the calculation of coefficients).
The coefficients 
$c_i(\delta)$ in the expansion of the first-order Melnikov function 
$M(h, \delta)$ can be utilized to acquire a lower bound on the maximum number of limit cycles: supposing there exists a constant n such that
\begin{equation*}0 \lt c_{0}(\delta)\ll -c_{1}(\delta)\ll c_{2}(\delta)\ll \cdots \ll (-1)^{n-1}c_{n-1}(\delta)\ll (-1)^{n}c_{n}(\delta),\end{equation*} this implies the integral 
$M(h, \delta)$ can possess n zeros near 
$h=h_s$. Thereby the system (1.1) has n limit cycles near the homoclinic or heteroclinic loop.
In recent years, numerous investigations have focused on the number of limit cycles inside and outside the figure-of-eight loop (double homoclinic loop) or the two-saddle loop (heteroclinic loop). This research interest not only stems from the qualitative theory of ordinary differential equations [Reference Dumortier and Li8, Reference Gavrilov and Iliev11, Reference Han, Yang and Li17, Reference Han, Yang, Tarţa and Gao18, Reference Li, Llibre and Zhang24–Reference Petrov26, Reference Tian and Han29, Reference Wei, Zhang and Zhang31, Reference Zhao and Zhang44] but also relates to various practical issues, such as the generalized Rayleigh–Liénard oscillator and the Van der Pol–Duffing oscillator [Reference Chen, Feng, Li and Wang6, Reference Euzébio, Llibre and Tonon9, Reference Gavrilov and Iliev12, Reference Wu, Han and Chen33].
In this article, we investigate the limit cycles near a double eight figure loop, which is a compound cycle with a 2-polycycle connecting two homoclinic loops, see [Reference Han15, Reference Han16, Reference Zang, Zhang and Chen38, Reference Zhao, Qi and Liu43]. Different from the articles mentioned above, the highlight of the research is that it can be considered both kinds of bifurcations (homoclinic loop and heteroclinic loop) at the same time. Furthermore, we establish a general theory for finding a lower bound of the maximal number of limit cycles near the double eight figure loop by calculating the algebraic structure (generators) of the first-order Melnikov function. The advantages of this approach are numerous. For instance, it is more convenient to gain the expressions for the coefficients of the higher-order terms in the expansion of M(h), and the results on the number of limit cycles for the system are applicable to any degree n of perturbation. An example is given as an application to illustrate these advantages.
Consider a generalized Liénard system of type (m, n) defined by
\begin{equation}
\begin{array}{l}
\dot{x}=y,\quad
\dot{y}=-g(x)+\varepsilon f(x)y,
\end{array}
\end{equation} where g(x) and f(x) are polynomials of degree m and n, respectively. Note that a necessary condition for the Hamiltonian system (1.3)
$|_{\varepsilon=0}$ to possess a double eight figure loop is 
$m\geq5$. In this article, we examine the scenario where m = 5, chosen for computational convenience, and based on the extensive research by many scholars on the problem of limit cycles of the system (1.3) when m = 5 with n fixed to a certain value [Reference Asheghi and Zangeneh2–Reference Cen, Liu, Sun and Wang4, Reference Li and Yang23, Reference Qi and Zhao27, Reference Xiong and Zhong34, Reference Xu and Li35, Reference Yang and Zhao37, Reference Zang, Zhang and Han39–Reference Zhao and Li42], and in addition it is related to the complex Ginzburg–Landau equation [Reference Chen, Chen, Jia and Tang5]. For example, Xiong and Zhong [Reference Xiong and Zhong34], Zhang et al. [Reference Zhang, Tadé and Tian40], and Yang and Zhao [Reference Yang and Zhao37] discussed the limit cycles or the zeros of first-order Melnikov function (Abelian integral) of the Liénard system
\begin{equation}
\begin{array}{l}
\dot{x}=y,\quad
\dot{y}=-x(x^{2}-2)(x^{2}-\frac{1}{2})+\varepsilon f(x)y,
\end{array}
\end{equation} with 
$0 \lt |\varepsilon|\ll 1$ for 
$f(x)=\sum _{i=0}^{2}a_{i}x^{2i}$, 
$\sum _{i=0}^{4}a_{i}x^{i}$, and 
$\sum _{i=0}^{4}a_{i}x^{2i}$, respectively. Xu and Li [Reference Xu and Li35] studied the limit cycles of the following system
\begin{equation}
\begin{array}{l}
\dot{x}=y,\quad
\dot{y}=-x(x^{2}-1)(x^{2}-9)+\varepsilon f(x)y,
\end{array}
\end{equation} where 
$0 \lt |\varepsilon|\ll 1$ and 
$f(x)=\sum _{i=0}^{5}a_{i}x^{2i}$. It is obvious that the unperturbed systems (1.4)
$|_{\varepsilon=0}$ and (1.5)
$|_{\varepsilon=0}$ both have a double eight figure loop.
Motivated by [Reference Cen, Liu, Sun and Wang4, Reference Xiong and Zhong34, Reference Xu and Li35, Reference Yang and Zhao37, Reference Zhang, Tadé and Tian40, Reference Zhao, Qi and Liu43], in this article, we consider a more generic system of the form
\begin{equation}
\begin{array}{l}
\dot{x}=y,\quad
\dot{y}=-x(x^{2}-a)(x^{2}-b)+\varepsilon f(x)y,
\end{array}
\end{equation} where 
$0 \lt |\varepsilon|\ll 1$, 
$f(x)=\Sigma_{i=0}^{n}a_{i}x^{i}$, 
$(a_{0}, a_{1}, \ldots, a_{n})\in D\subset\mathbb{R}^{n+1}$ with D being a compact subset, and 
$a \gt b \gt 0$, which will ensure the unperturbed system (1.6)
$|_{\varepsilon=0}$ exists a double eight figure loop. Without loss of generality, we suppose a > 1 and b = 1. Otherwise, one can utilize a variable transformation
\begin{equation*}
x\rightarrow \sqrt{b}x,\quad y\rightarrow b\sqrt{b}y,\quad t\rightarrow \frac{1}{b}t,
\end{equation*}on the system (1.6) to satisfy this assumption. Compared to existing works, on the one hand, system (1.6) includes the two systems (1.4) and (1.5) described above, and on the other hand the number of limit cycles we obtain for the system holds for any value of n.
The remaining sections of this article are organized as follows. In §2, we build a general bifurcation theory to obtain the lower bound of the maximal number of limit cycles near the double eight figure loop by finding the algebraic structure of the first-order Melnikov function. As an application, we also research the system (1.6) and provide a more optimal result for the lower bound of the maximal number of the limit cycles near the double eight figure loop in §3.
2. A general bifurcation theory to a double eight figure loop
Assume that system (1.1)
$|_{\varepsilon=0}$ has a double eight figure loop (compound cycle) 
${\Gamma}\subset G$, which is defined below:
\begin{align*}
{\Gamma}=\left(\bigcup_{i=1}^{2}L_{i}\right)\bigcup\Gamma^{2},\quad \Gamma^{2}=\left(S_{1}\cup\widetilde{{L}}_{1}\right)\left(S_{2}\cup\widetilde{{L}}_{2}\right),
\end{align*} where the 2-polycycle Γ2 is composed of two hyperbolic saddles 
$S_{1}, S_{2}$ with 
$H(S_{1})=h_{s}$ and two heteroclinic orbits 
$\widetilde{{L}}_{1}, \widetilde{{L}}_{2}$ satisfying
\begin{equation*}\omega\left(\widetilde{{L}}_{1}\right)=S_{1}, \alpha\left(\widetilde{{L}}_{1}\right)=S_{2}, \omega\left(\widetilde{{L}}_{2}\right)=S_{2}, \alpha\left(\widetilde{{L}}_{2}\right)=S_{1},\end{equation*} and L 1, L 2 are two homoclinic loops outside Γ2. There are three centers 
$O(0, 0)$, C 1, C 2 surrounded by 
$\Gamma^{2}, L_{1}, L_{2}$, respectively, where 
$H(O)=0, H(C_{1})=h_{C_{1}}, H(C_{2})=h_{C_{2}}$ (see Figure 1).
Figure 1. Periodic orbits near Γ.
Near the double eight figure loop Γ, there are four families of periodic orbits: 
${\Gamma}(h)$, which is located outside Γ for 
$0 \lt h-h_{s} \lt \mu$; and 
$L_{1}(h), L_{2}(h), L_{3}(h)$, which are located inside Γ for 
$-\mu \lt h-h_{s} \lt 0$, where 
$0 \lt \mu\ll 1$. Then the four first-order Melnikov functions are as follows
\begin{align}
\begin{array}{l}
M(h)=\oint_{{\Gamma}(h)} g\mathrm{d}x-f\mathrm{d}y, \quad 0 \lt h-h_{s} \lt \mu,\\
M^{j}(h)=\oint_{L_{j}(h)} g\mathrm{d}x-f\mathrm{d}y,\quad -\mu \lt h-h_{s} \lt 0, \quad j=1,2,3.
\end{array}
\end{align}From (1.2), one has
\begin{align}
\begin{array}{l}
M(h)=\varphi(h)(h-h_{s})\ln (h-h_{s})+N(h),\quad 0 \lt h-h_{s} \lt \mu,\\
M^{j}(h)=\varphi_{j}(h)(h-h_{s})\ln|h-h_{s}|+N_{j}(h),\quad -\mu \lt h-h_{s} \lt 0, \quad j=1,2,3,
\end{array}
\end{align} where 
$\varphi_{j}(h), N_{j}(h)$, and N(h) are analytic functions defined for 
$0 \lt |h| \lt \mu$ and satisfy the following convergent expansions
\begin{align*}
\varphi(h)&=\sum_{i\geq0}c_{2i+1}(h-h_{s})^{i}, \quad\varphi_{j}(h)=\sum_{i\geq0}c_{2i+1}^{j}(h-h_{s})^{i}, \\ N_{j}(h)&=\sum_{i\geq0}c_{2i}^{j}(h-h_{s})^{i}, \quad N(h)=\sum_{i\geq0}c_{2i}(h-h_{s})^{i}.
\end{align*}From [Reference Han, Yang, Tarţa and Gao18] or Theorem 3.2.9 of the book [Reference Han16], it holds that
\begin{equation}
\varphi_{3}(h)=\varphi_{1}(h)+\varphi_{2}(h).
\end{equation} The following theorem further gives the relationships between the functions 
$\varphi_{j}(h), \varphi(h), N_{j}(h)$, and N(h).
Theorem 2.1 Assume the functions M(h) and 
$M_{j}(h)$ 
$(j=1,2,3)$ defined by (2.1) satisfy (2.2), where 
$H, f$, and g are polynomial functions on G containing double eight figure loop Γ. It follows for 
$0 \lt |h| \lt \mu$ that
\begin{equation}
\frac{1}{2}\varphi(h)=\varphi_{1}(h)+\varphi_{2}(h),
\end{equation}and
\begin{equation}
N(h)=N_{1}(h)+N_{2}(h)+N_{3}(h).
\end{equation}In fact, by applying Theorem 1.1 in [Reference Han, Yang and Li17] respectively to the right and left part of the y-axis of the phase portrait in Figure 1 one can directly obtain the conclusion of Theorem 2.1. For brevity, we omit the detailed proof here.
Depending on (2.3), (2.4), and (2.5), one has
\begin{align}
&M^{j}(h)=\sum_{i\geq0}(c_{2i}^{j}+c_{2i+1}^{j}(h-h_{s})\ln |h-h_{s}|)(h-h_{s})^{i}, \ 0 \lt h_{s}-h\ll1,\ j=1,2,\nonumber\\
&M^{3}(h)=\sum_{i\geq0}(c_{2i}^{j}+(c_{2i+1}^{1}+c_{2i+1}^{2})(h-h_{s})\ln |h-h_{s}|)(h-h_{s})^{i}, \ 0 \lt h_{s}-h\ll1,\nonumber\\
&M(h)=\sum_{i\geq0}\left(c_{2i}^{1}+c_{2i}^{2}+c_{2i}^{3}+2(c_{2i+1}^{1}+c_{2i+1}^{2})(h-h_{s})\ln |h-h_{s}| \right)(h-h_{s})^{i},\nonumber\\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\ \ 0 \lt h-h_{s}\ll1.
\end{align}Specifically, if system (1.1) is centrally symmetric, that is,
\begin{align*}
\begin{aligned}
H(-x,-y)=H(x,y),\ f(-x, -y, \delta)=-f(x, y, \delta),\ g(-x, -y, \delta)=-g(x, y, \delta),
\end{aligned}
\end{align*} then it follows that 
$c_{2i}^{1}=c_{2i}^{2}$ and 
$c_{2i+1}^{1}=c_{2i+1}^{2}$.
At this time, (1.1) is also a 
$\mathbb{Z}_{2}$-equivariant system. Furthermore, we present the following theorem.
Theorem 2.2 Suppose that system (1.1) is centrally symmetric.
(i) If there exists a 
$\delta_{0}\in D$, such that
\begin{equation*}c_{2i}^{1}(\delta_{0})=c_{2i}^{3}(\delta_{0})=c_{2j+1}^{1}(\delta_{0})=0, \ i,j=0,1,\ldots,n-1, \end{equation*}
\begin{equation*}c_{2n}^{1}(\delta_{0}),c_{2n}^{3}(\delta_{0}) \gt 0,\end{equation*}and
\begin{align*}
\operatorname{rank} \frac{\partial\left(c_{0}^{1}, c_{0}^{3}, c_{1}^{1}, c_{2}^{1}, c_{2}^{3}, c_{3}^{1}, \ldots, c_{2n-2}^{1}, c_{2n-2}^{3}, c_{2n-1}^{1}\right)}{\partial\left(\delta_{1}, \ldots, \delta_{s}\right)}\left(\delta_{0}\right) & = 3n,
\end{align*} then system (1.1) can exist 7n limit cycles near the double eight figure loop Γ for some 
$(\varepsilon, \delta)$ near 
$(0, \delta_{0})$.
(ii) If there exists a 
$\delta_{0}\in D$, such that
\begin{equation*}c_{2i}^{1}(\delta_{0})=c_{2j}^{3}(\delta_{0})=c_{2j+1}^{1}(\delta_{0})=0,\ i=0,1,\ldots,n,\ j=0,1,\ldots,n-1,\end{equation*}
\begin{equation*}c_{2n}^{3}(\delta_{0})c_{2n+1}^{1}(\delta_{0}) \lt 0,\end{equation*}and
\begin{align*}
\operatorname{rank} \frac{\partial\left(c_{0}^{1}, c_{0}^{3}, c_{1}^{1}, c_{2}^{1}, c_{2}^{3}, c_{3}^{1}, \ldots, c_{2n-2}^{1}, c_{2n-2}^{3}, c_{2n-1}^{1}, c_{2n}^{1}\right)}{\partial\left(\delta_{1}, \ldots, \delta_{s}\right)}\left(\delta_{0}\right) & = 3n+1,
\end{align*} then system (1.1) can exist 
$7n+2$ limit cycles near the double eight figure loop Γ for some 
$(\varepsilon, \delta)$ near 
$(0, \delta_{0})$.
(iii) If there exists a 
$\delta_{0}\in D$, such that
\begin{equation*}c_{2i}^{1}(\delta_{0})=c_{2i}^{3}(\delta_{0})=c_{2j+1}^{1}(\delta_{0})=0, \ i=0,1,\ldots,n, j=0,1,\ldots,n-1, \end{equation*}
\begin{equation*}c_{2n+1}^{1}(\delta_{0})\neq0,\end{equation*}and
\begin{align*}
\operatorname{rank} \frac{\partial\left(c_{0}^{1}, c_{0}^{3}, c_{1}^{1}, c_{2}^{1}, c_{2}^{3}, c_{3}^{1}, \ldots, c_{2n}^{1}, c_{2n}^{3}\right)}{\partial\left(\delta_{1}, \ldots, \delta_{s}\right)}\left(\delta_{0}\right) & = 3n+2,
\end{align*} then system (1.1) can exist 
$7n+3$ limit cycles near the double eight figure loop Γ for some 
$(\varepsilon, \delta)$ near 
$(0, \delta_{0})$.
Proof. The proof follows a similar approach to theorem 1.2 in [Reference Han, Yang and Li17]. For clarity, we will focus on proving part (i) as outlined below. By the assumptions, there exists a 
$\delta_{0}\in D$ such that
\begin{equation*}\left|c_{0}^{1}\right| \ll\left|c_{1}^{1}\right| \ll\left|c_{2}^{1}\right| \ll\cdots \ll\left|c_{2n-2}^{1}\right|\ll\left|c_{2n-1}^{1}\right|\ll\left|c_{2n}^{1}\right|,\end{equation*}
\begin{equation*}\left|c_{0}^{3}\right|\ll\left|c_{1}^{1}\right| \ll\left|c_{2}^{3}\right|\ll\cdots \ll\left|c_{2n-2}^{3}\right|\ll\left|c_{2n-1}^{1}\right|\ll\left|c_{2n}^{3}\right|,\end{equation*} and 
$c_{4n}^{1}, c_{4n}^{3}, c_{4n+1}^{1}, c_{4n+2}^{1}, c_{4n+2}^{3}, c_{4n+3}^{1}$ with symbols 
$+,+,-,-,-,+$, respectively.
For 
$0 \lt h_s - h \ll 1$, the expansion of 
$M^{1}=M^{2}$ in (2.6) includes terms such as 
$c_{0}^{1}$, 
$c_{1}^{1}(h-h_{s})\ln |h-h_{s}|$, 
$c_{2}^{1}(h-h_{s})$, 
$c_{3}^{1} (h-h_{s})^{2} \ln|h-h_{s}|$, 
$\ldots$, which have symbols 
$+,-,+,-$, respectively, and follow this pattern repetitively. These terms satisfy 
$\left|c^{1}_{0}\right| \ll\left|c_{1}^{1}\right| \ll\left|c^{1}_{2}\right| \ll \cdots \ll\left|c_{2n-1}^{1}\right| \ll\left|c^{1}_{2 n}\right|$. As a result, both M 1 and M 2 possess at least 2n simple zeros for 
$0 \lt h_s - h \ll 1$.
Following a similar analysis as described above, we find that M 3 (resp., M) has at least 2n (resp., n) simple zeros for 
$0 \lt h_{s}-h \ll 1$ (resp., 
$0 \lt h-h_{s}\ll 1$). Consequently, system (1.1) has at least 7n limit cycles near the double figure eight loop. This concludes the proof of the theorem.
Theorem 2.3 Suppose that system (1.1) is non-centrally symmetric.
(i) If there exists a 
$\delta_{0}\in D$, such that
\begin{equation*}c_{i}^{1}(\delta_{0})=c_{i}^{2}(\delta_{0})=c_{2j}^{3}(\delta_{0})=0,\ i=0,1,\ldots,2n-1,\ j=0,1,\ldots,n-1,\end{equation*}
\begin{equation*}c_{2n}^{1}(\delta_{0}),c_{2n}^{2}(\delta_{0}),c_{2n}^{3}(\delta_{0}) \gt 0,\end{equation*}and
\begin{align*}
\operatorname{rank} \frac{\partial\left(c_{0}^{1}, c_{0}^{2}, c_{0}^{3}, c_{1}^{1}, c_{1}^{2}, \ldots, c_{2n-2}^{1}, c_{2n-2}^{2}, c_{2n-2}^{3}, c_{2n-1}^{1}, c_{2n-1}^{2}\right)}{\partial\left(\delta_{1}, \ldots, \delta_{s}\right)}\left(\delta_{0}\right) &=5n,
\end{align*} then system (1.1) can exist 7n limit cycles near the double eight figure loop Γ for some 
$(\varepsilon, \delta)$ near 
$(0, \delta_{0})$.
(ii) If there exists a 
$\delta_{0}\in D$, such that
\begin{equation*}c_{i}^{1}(\delta_{0})=c_{i}^{2}(\delta_{0})=\ c_{2j}^{3}(\delta_{0})=0,\ i=0,1,\ldots,2n,\ j=0,1,\ldots,n-1,\end{equation*}
\begin{equation*}c_{2n}^{3}(\delta_{0})c_{2n+1}^{1}(\delta_{0}) \lt 0, c_{2n}^{3}(\delta_{0})c_{2n+1}^{2}(\delta_{0}) \lt 0,\end{equation*}and
\begin{align*}
\operatorname{rank} \frac{\partial\left(c_{0}^{1}, c_{0}^{2}, c_{0}^{3}, c_{1}^{1}, c_{1}^{2}, \ldots, c_{2n-2}^{3}, c_{2n-1}^{1}, c_{2n-1}^{2}, c_{2n}^{1}, c_{2n}^{2}\right)}{\partial\left(\delta_{1}, \ldots, \delta_{s}\right)}\left(\delta_{0}\right) &=5n+2,
\end{align*} then system (1.1) can exist 
$7n+2$ limit cycles near the double eight figure loop Γ for some 
$(\varepsilon, \delta)$ near 
$(0, \delta_{0})$.
(iii) If there exists a 
$\delta_{0}\in D$, such that
\begin{equation*}c_{i}^{1}(\delta_{0})=c_{i}^{2}(\delta_{0})=c_{2j}^{3}(\delta_{0})=0,\ i=0,1,\ldots,2n,\ j=0,1,\ldots,n,\end{equation*}
\begin{equation*}c_{2n+1}^{1}(\delta_{0})c_{2n+1}^{2}(\delta_{0})\neq0,\end{equation*}and
\begin{align*}
\operatorname{rank} \frac{\partial\left(c_{0}^{1}, c_{0}^{2}, c_{0}^{3}, c_{1}^{1}, c_{1}^{2},\ldots, c_{2n}^{1}, c_{2n}^{2}, c_{2n}^{3}\right)}{\partial\left(\delta_{1}, \ldots, \delta_{s}\right)}\left(\delta_{0}\right) & = 5n+3,
\end{align*} then system (1.1) can exist 
$7n+3$ limit cycles near the double eight figure loop Γ for some 
$(\varepsilon, \delta)$ near 
$(0, \delta_{0})$.
Proof. From the assumptions, there exists a 
$\delta_{0}\in D$ such that
\begin{equation*}
\begin{split}
&\left|c_{0}^{1}\right| \ll\left|c_{1}^{1}\right| \ll\left|c_{2}^{1}\right| \ll\cdots \ll\left|c_{2n-2}^{1}\right|\ll\left|c_{2n-1}^{1}\right|\ll\left|c_{2n}^{1}\right|,\\
&\left|c_{0}^{2}\right| \ll\left|c_{1}^{2}\right| \ll\left|c_{2}^{2}\right| \ll\cdots \ll\left|c_{2n-2}^{2}\right|\ll\left|c_{2n-1}^{2}\right|\ll\left|c_{2n}^{2}\right|,\\
&\left|c_{0}^{3}\right|\ll\left|c_{1}^{1}+c_{1}^{2}\right| \ll\left|c_{2}^{3}\right|\ll\cdots \ll\left|c_{2n-2}^{3}\right|\ll\left|c_{2n-1}^{1}+c_{2n-1}^{2}\right|\ll\left|c_{2n}^{3}\right|.
\end{split}
\end{equation*} And the symbols for 
$c_{4n}^{1}, c_{4n}^{2}, c_{4n}^{3}, c_{4n+1}^{1}, c_{4n+1}^{2}, c_{4n+2}^{1}, c_{4n+2}^{2}, c_{4n+2}^{3}, c_{4n+3}^{1}, c_{4n+3}^{2}$ are +, +, +, −, −, −, −, −, +, + respectively.
Similar to the proof of Theorem 2.2, we omit the details. This finishes the proof.
3. An application
Note that system (1.6) represents a Hamiltonian system with symmetry along the y-axis, defined by the Hamiltonian 
$H(x,y)=\frac{1}{2}y^{2}+\frac{1}{2}ax^{2}-\frac{1}{4}(1+a)x^{4}+\frac{1}{6}x^{6}$ for a > 1. This system possesses five equilibrium points: three elementary centres at 
$O(0,0)$, 
$C_{1}(\sqrt{a},0)$, and 
$C_{2}(-\sqrt{a},0)$; and two hyperbolic saddles at 
$S_{1}(1,0)$ and 
$S_{2}(-1,0)$. The centres C 1 and C 2 are each encircled by homoclinic loops L 1 and L 2, respectively, with 
$H(C_{1}) = H(C_{2}) = h_{c} = \frac{1}{4}a^2 - \frac{1}{12}a^3$, and 
$H(S_{1}) = H(S_{2}) = h_{s} = \frac{1}{4}a - \frac{1}{12}$. The origin 
$O(0,0)$ is surrounded by a 2-polycycle Γ2. Furthermore, for 
$0 \lt |h-h_{s}| \ll 1$, the equation 
$H(x, y) = h$ defines four families of periodic orbits: 
$L_{1}(h)$ and 
$L_{2}(h)$ for 
$h_{c} \lt h \lt h_{s}$, 
$L_{3}(h)$ for 
$0 \lt h \lt h_{s}$, and 
$L_{4}(h)$ for 
$h \gt h_{s}$. Figure 1 illustrates the phase portrait of (1.6)
$|_{\varepsilon=0}$. We have the following theorem.
Theorem 3.1 For all 
$n\geq5$, the Liénard system (1.6) can exist 
$7[\frac{n}{6}]+2[\frac{r}{2}]-[\frac{r}{4}]$ limit cycles near the double eight figure loop for some 
$(\varepsilon, \delta)$ near 
$(0, \delta_{0})$, where 
$r=\rm mod(n,6)$.
Remark 3.2. To the best of our knowledge, this number of limit cycles is maximal that we have been able to find so far near the double eight figure loop. Xu and Li [Reference Xu and Li35] (resp., Xiong and Zhong [Reference Xiong and Zhong34]) proved that the system (1.5) (resp., (1.4)), with 
$f(x)=\sum _{i=0}^{5}a_{i}x^{2i}$ (resp., 
$\sum _{i=0}^{4}a_{i}x^{2i}$), has 10 (resp., 9) limit cycles near the double eight figure loop, which is consistent with taking n = 10 (resp., n = 8) in the Theorem 3.1 for the more general system (1.6).
Let 
$I_{k}^{i}(h) = \oint_{L_{i}(h)} x^{k}ydx$ and 
$I_{k}(h) = \oint_{L(h)} x^{k}ydx$ be integrals over the curves 
$L_{i}(h)$ and L(h), respectively, as defined in §2 and illustrated in Figure 1. Based on the classification of these curves, we derive the following four first-order Melnikov functions.
Lemma 3.3. For 
$n\geq5$ and 
$0 \lt |h -h_{s}|\ll 1$.
(i) If 
$L_{i}(h)$ near the homoclinic loop Li for 
$i=1,2$, and 
$L_{i}(h)$ near the 2-polycycle Γ2 for i = 3, 
$M^{i}(h)$ can be written as
\begin{equation}
M^{i}(h)=\sum _{k=0}^{4}P_{k}(h)I_{k}^{i}(h), \quad \text{for}\quad i=1,2,3.
\end{equation}(ii) If L(h) near the double eight figure loop Γ, one has
\begin{equation}
M(h)=\sum _{k=0}^{4}P_{k}(h)I_{k}(h),
\end{equation} where 
$P_{k}(h)$ are polynomials of h, 
$\operatorname{deg}P_{i}(h)\leqslant[\frac{n-k}{6}]$ for 
$k=0,1,2,3,4$. The notation 
$[s]$ is defined as the integer part of s.
In particular, the terms 
$I_{1}(h), I_{3}(h), I_{1}^{i}(h)$, and 
$I_{3}^{i}(h)$ 
$(i=1,2,3)$ in Eqs. (3.1) and (3.2) will not appear if 
$f(-x)=f(x)$.
Proof. The idea of proof is similar to the proposition 2.4 in [Reference Zhao and Zhang44]. From the definition of 
$\{L_{1}(h), L_{2}(h), L_{3}(h), L(h)\}\triangleq\Gamma_{h}$, it can be observed that
\begin{equation}
\frac{1}{2}y^{2}+\frac{1}{2}ax^{2}-\frac{1}{4}(1+a)x^{4}+\frac{1}{6}x^{6}=h.
\end{equation} Multiplying the (3.3) by 
$x^{l}y^{m}$ and integrating it over 
$\Gamma_{h}$, we have
\begin{equation}
\frac{1}{2}I_{l,m+2}+\frac{1}{2}aI_{l+2,m}-\frac{1}{4}(1+a)I_{l+4,m}+\frac{1}{6}I_{l+6,m}=hI_{l,m},
\end{equation} where 
$I_{i,j}=\oint_{\Gamma_{h}} x^{i}y^{j}dx$.
It follows from (3.3) that
\begin{equation}
y\frac{\partial y}{\partial x}+ax-(1+a)x^{3}+x^{5}=0.
\end{equation} Multiplying (3.5) by 
$x^{l-5}y^{m}$ 
$(l\geq5)$ and integrating it over 
$\Gamma_{h}$ by parts, one has
\begin{equation}
-\frac{l-5}{m+2}I_{l-6,m+2}+aI_{l-4,m}-(1+a)I_{l-2,m}+I_{l,m}=0.
\end{equation}Hence by (3.4) and (3.6), we obtain
\begin{equation}
I_{l+6,m}=\frac{3(2m+l+5)}{2(3m+l+7)}(1+a)I_{l+4,m}-\frac{3(m+l+3)}{3m+l+7}aI_{l+2,m}+\frac{6(l+1)}{3m+l+7}hI_{l,m}.
\end{equation}If l = 5 in (3.6), it is direct that
\begin{equation*}I_{5,m}=(1+a)I_{3,m}-aI_{1,m}.\end{equation*} Therefore, it follows from taking m = 1 in (3.7) that the generators of M(h) (resp., 
$M^{i}(h)$) are 
$I_{0}(h), I_{1}(h)$, 
$I_{2}(h)$, 
$I_{3}(h)$, and 
$I_{4}(h)$ (resp., 
$I_{0}^{i}(h), I_{1}^{i}(h)$, 
$I_{2}^{i}(h)$, 
$I_{3}^{i}(h)$, and 
$I_{4}^{i}(h)$). By induction in n, it is easy to see the dimensions of 
$P_{0}(h)$, 
$P_{1}(h)$, 
$P_{2}(h)$, 
$P_{3}(h)$, and 
$P_{4}(h)$.
It can be proved in a similar way if 
$f(-x)=f(x)$. This finishes the proof.
Proof of Theorem 3.1
From Lemma 3.3, we suppose for 
$k=0,1,2,3,4$ that
\begin{align}
P_{k}(h)=m_{0}^{k}+m_{1}^{k}(h-h_{s})+m_{2}^{k}(h-h_{s})^{2}+\cdots+m_{[\frac{n-k}{6}]}^{k}(h-h_{s})^{[\frac{n-k}{6}]}.
\end{align} Setting 
$f(x)=\Sigma_{0}^{n}a_{i}x^{i}$, it is easy to verify that
\begin{align*}
\left|\frac{\partial\left(m_{0}^{0}, m_{0}^{1}, m_{0}^{2}, m_{0}^{3}, m_{0}^{4},\ldots, m_{[\frac{n}{6}]}^{0}, m_{[\frac{n-1}{6}]}^{1}, m_{[\frac{n-2}{6}]}^{2}, m_{[\frac{n-3}{6}]}^{3}, m_{[\frac{n-4}{6}]}^{4}\right)}{\partial\left(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{6}, a_{7}, a_{8}, a_{9}, a_{10},\ldots, a_{n}\right)}\right|\neq 0,
\end{align*} where 
$5\neq mod(n,6)$. If 
$f(-x)=f(x)$, one has
\begin{align*}
\left|\frac{\partial\left(m_{0}^{0}, m_{0}^{2}, m_{0}^{4}, \ldots, m_{[\frac{n}{6}]}^{0}, m_{[\frac{n-2}{6}]}^{2}, m_{[\frac{n-4}{6}]}^{4}\right)}{\partial\left(a_{0}, a_{2}, a_{4}, a_{6}, a_{8}, a_{10},\ldots, a_{n}\right)}\right|\neq 0,
\end{align*}where n is even. And hence, the coefficients in (3.8) are independent.
For 
$0 \lt |h-h_{s}|\ll 1$, assume that 
$I_{k}(h)$ and 
$I_{k}^{i}(h)$ for 
$i=1,2,3$ in Lemma 3.3 can be represented as
\begin{align}
\begin{split}
I_{k}^{i}(h)&=a_{k,0}^{i}+\widetilde{a}_{k,1}^{i}(h-h_{s})\ln |h-h_{s}|+\cdots\\
&+\widetilde{a}_{k,n}^{i}(h-h_{s})^{n}\ln |h-h_{s}|+a_{k,n}^{i}(h-h_{s})^{n}+O((h-h_{s})^{n}),\\
I_{k}(h)&=a_{k,0}^{4}+\widetilde{a}_{k,1}^{4}(h-h_{s})\ln |h-h_{s}|+\cdots\\
&+\widetilde{a}_{k,n}^{4}(h-h_{s})^{n}\ln |h-h_{s}|+a_{k,n}^{4}(h-h_{s})^{n}+O((h-h_{s})^{n}).
\end{split}
\end{align} (i) Centrally symmetric. If 
$[\frac{n}{6}]=[\frac{n-2}{6}]=[\frac{n-4}{6}]=s$ in (3.8), we set
\begin{align*}
\pmb\delta^{1}&\triangleq(m_{0}^{0},m_{0}^{2},m_{0}^{4}, m_{1}^{0},m_{1}^{2},m_{1}^{4}, \ldots, m_{s}^{0},m_{s}^{2},m_{s}^{4}),\\
\pmb E_{1}&\triangleq\frac{\partial\left(c_{0}^{1}, c_{0}^{3}, c_{1}^{1}, c_{2}^{1}, c_{2}^{3}, c_{3}^{1}, \ldots, c_{2s}^{1}, c_{2s}^{3}, c^{1}_{2s+1}\right)}{\partial\pmb\delta^{1}},
\end{align*} where the coefficients 
$c_{i}^{j}$ appear in (2.6).
Substituting (3.8) and (3.9) into (3.1), it follows from (2.6) that
\begin{equation*}\tiny \pmb E_{1}= \left(\begin{array}{llllllllllllll}
a_{0,0}^{1} & a_{2,0}^{1} & a_{4,0}^{1} & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\
a_{0,0}^{3} & a_{2,0}^{3} & a_{4,0}^{3} & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\
\widetilde{a}_{0,1}^{1} & \widetilde{a}_{2,1}^{1} & \widetilde{a}_{4,1}^{1} & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\
a_{0,1}^{1} & a_{2,1}^{1} & a_{4,1}^{1} & a_{0,0}^{1} & a_{2,0}^{1} & a_{4,0}^{1} & \cdots & 0 & 0 & 0 \\
a_{0,1}^{3} & a_{2,1}^{3} & a_{4,1}^{3} & a_{0,0}^{3} & a_{2,0}^{3} & a_{4,0}^{3} & \cdots & 0 & 0 & 0 \\
\widetilde{a}_{0,2}^{1} & \widetilde{a}_{2,2}^{1} & \widetilde{a}_{4,2}^{1} & \widetilde{a}_{0,1}^{1} & \widetilde{a}_{2,1}^{1} & \widetilde{a}_{4,1}^{1} & \cdots & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
a_{0,s}^{1} & a_{2,s}^{1} & a_{4,s}^{1} & a_{0,s-1}^{1} & a_{2,s-1}^{1} & a_{4,s-1}^{1} & \cdots & a_{0,0}^{1} & a_{2,0}^{1} & a_{4,0}^{1} \\
a_{0,s}^{3} & a_{2,s}^{3} & a_{4,s}^{3} & a_{0,s-1}^{3} & a_{2,s-1}^{3} & a_{4,s-1}^{3} & \cdots & a_{0,0}^{3} & a_{2,0}^{3} & a_{4,0}^{3} \\
\widetilde{a}_{0,s+1}^{1} & \widetilde{a}_{2,s+1}^{1} & \widetilde{a}_{4,s+1}^{1} & \widetilde{a}_{0,s}^{1} & \widetilde{a}_{2,s}^{1} & \widetilde{a}_{4,s}^{1} & \cdots & \widetilde{a}_{0,1}^{1} & \widetilde{a}_{2,1}^{1} & \widetilde{a}_{4,1}^{1}
\end{array}\right).
\end{equation*} Let 
$$F_{0}=\left(\begin{array}{lll}
a_{0,0}^{1} & a_{2,0}^{1} & a_{4,0}^{1}\\
a_{0,0}^{3} & a_{2,0}^{3} & a_{4,0}^{3} \\\
\widetilde{a}_{0,1}^{1} & \widetilde{a}_{2,1}^{1} & \widetilde{a}_{4,1}^{1}
\end{array}\right)$$, 
$A(x)=\sqrt {\frac{1}{2}a-\frac{1}{6}-a{x}^{2}+\frac{1}{2}(1+a)x^{4}-\frac{1}{3}x^{6}}$ and 
$\gamma_{1}=\arcsin\left(\frac{\sqrt{2}}{\sqrt {3a-1}}\right)$, where 
$\widetilde{a}_{0,1}^{1}=\widetilde{a}_{2,1}^{1}=\widetilde{a}_{4,1}^{1}=-\frac{1}{\sqrt{2(a-1)}}$,
\begin{align*}
\begin{split}
a_{0,0}^{1}=& 2\,\int_{1}^{\frac{1}{2}\sqrt{6a-2}}A(x){\rm d}x\\
=&\frac{3\sqrt{2}}{16}\sqrt{a-1}(a+1)-\frac{\sqrt{3}}{16}(3a-1)(a-3)\left(\gamma_{1}-\frac{\pi}{2}\right),\\
a_{2,0}^{1}=&2\int_{1}^{\frac{1}{2}\sqrt{6a-2}}x^{2}A(x){\rm d}x\\
=&\frac{\sqrt{2}}{64}\sqrt{a-1}(9a^{2}-14a+9)-\frac {\sqrt {3}}{
192}\left( 3\,a-5 \right) ( 3a-1 ) ^{2}\left(\gamma_{1}-\frac{\pi}{2}\right),\\
a_{4,0}^{1}=&2\,\int_{1}^{\frac{1}{2}\sqrt{6a-2}}x^{4}A(x){\rm d}x\\
=&{\frac {3\sqrt {2}}{1024}}\sqrt {a-1}\left( 45\,{a}^{3}-73\,{a}^{2}+23\,a+
13 \right)-\frac{\sqrt{3}}{1024}(5a-7)(3a-1)^{3}\left(\gamma_{1}-\frac{\pi}{2}\right),\\
a_{0,0}^{3}=&2\,\int_{-1}^{1}A(x){\rm d}x\\
=&\frac{3\sqrt{2}}{8}\sqrt{a-1}(a+1)-\frac{\sqrt{3}}{8}(3a-1)(a-3)\gamma_{1},\\
a_{2,0}^{3}=&2\,\int_{-1}^{1}x^{2}A(x){\rm d}x\\
=&\frac {\sqrt {2} }{32}\sqrt {a-1}\left( 9\,{a}^{2}-14\,a+9 \right)-{\frac {\sqrt {3}}{96}} \left( 3\,a-5 \right) \left( 3\,a-1 \right) ^{2}\gamma_{1},\\
a_{4,0}^{3}=&2\,\int_{-1}^{1}x^{4}A(x){\rm d}x\\
=&\frac {3\sqrt {2}}{512}\sqrt {a-1} \left( 45\,{a}^{3}-73\,{a}^{2}+23\,a+
13 \right) \sqrt {3}-\frac{\sqrt {3}}{512}( 5\,a-7 ) ( 3\,a-1 )^{3}\gamma_{1}.
\end{split}
\end{align*} From 
$|\pmb F_{0}|=-{\frac {3\,\sqrt {3}}{128}\pi \, \left( 3\,a-1 \right) \left( a-1 \right)
^{3}}\neq0,$ it follows that 
$|\pmb E_{1}|=|\pmb F_{0}|^{s+1}\neq0$. By Theorem 2.2 (iii), the system (1.6) has at least 
$7[\frac{n}{6}]+3$ limit cycles near 
$h=h_{s}$.
For 
$[\frac{n}{6}]=[\frac{n-2}{6}]=[\frac{n-4}{6}]+1=s$, there are 
$3s+2$ free coefficients 
$m_{0}^{0}$, 
$m_{0}^{2}$, 
$m_{0}^{4}$, 
$m_{1}^{0}$, 
$m_{1}^{2}$, 
$m_{1}^{4}$, 
$\ldots$, 
$m_{s-1}^{0}$, 
$m_{s-1}^{2}, m_{s-1}^{4}, m_{s}^{0}, m_{s}^{2}$. Let
\begin{align*}
\pmb\delta^{2}&\triangleq(m_{0}^{0},m_{0}^{2},m_{0}^{4}, m_{1}^{0},m_{1}^{2},m_{1}^{4}, \ldots, m_{s-1}^{0},m_{s-1}^{2},m_{s-1}^{4}, m_{s}^{0}),\\
\pmb E_{2}&\triangleq\frac{\partial\left(c_{0}^{1}, c_{0}^{3}, c_{1}^{1}, c_{2}^{1}, c_{2}^{3}, c_{3}^{1}, \ldots, c_{2s-2}^{1}, c_{2s-2}^{3}, c_{2s-1}^{1}, c_{2s}^{1}\right)}{\partial\pmb\delta^{2}}.
\end{align*} It can be seen that 
$|\pmb E_{2}|=a_{0,0}^{1}\cdot|\pmb F_{0}|^{s}\neq0$. Note that 
$c_{2s}^{3}c_{2s+1}^{1}=(a_{2,0}^{3}m_{s}^{2})(\widetilde{a}_{2,1}^{1}m_{s}^{2}) \lt 0$ when 
$\pmb\delta^{2}=\pmb0$. Applying Theorem 2.2 (ii), the system has at least 
$7[\frac{n}{6}]+2$ limit cycles near 
$h=h_{s}$.
If 
$[\frac{n}{6}]=[\frac{n-2}{6}]+1=[\frac{n-4}{6}]+1=s$, it implies that the 
$3s+1$ coefficients 
$m_{0}^{0}, m_{0}^{2}$, 
$m_{0}^{4}$, 
$m_{1}^{0}, m_{1}^{2}$, 
$m_{1}^{4}, \ldots, m_{s-1}^{0}, m_{s-1}^{2}, m_{s-1}^{4}, m_{s}^{0}$ are free. Let
\begin{align*}
\pmb\delta^{3}&\triangleq(m_{0}^{0},m_{0}^{2},m_{0}^{4}, m_{1}^{0},m_{1}^{2},m_{1}^{4}, \ldots, m_{s-1}^{0},m_{s-1}^{2},m_{s-1}^{4}),\\
\pmb E_{3}&\triangleq\frac{\partial\left(c_{0}^{1}, c_{0}^{3}, c_{1}^{1}, c_{2}^{1}, c_{2}^{3}, c_{3}^{1}, \ldots, c_{2s-2}^{1}, c_{2s-2}^{3}, c_{2s-1}^{1}\right)}{\partial\pmb\delta^{3}},
\end{align*} then we obtain 
$|\pmb E_{3}|=|\pmb F_{0}|^{s}\neq0$. In addition, it is not hard to verify that 
$c_{2s}^{1}=a_{0,0}^{1}m_{s}^{0}$, 
$c_{2s}^{3}=a_{0,0}^{3}m_{s}^{0}$, 
$c_{2s}^{4}=c_{2s}^{1}+c_{2s}^{3}=(a_{0,0}^{1}+a_{0,0}^{3})m_{s}^{0}$ when 
$\pmb\delta^{3}=\pmb0$. Since the parameter 
$m_{s}^{0}$ is free and 
$a_{0,0}^{1}, a_{0,0}^{3} \gt 0$, we suppose 
$c_{2s}^{1}, c_{2s}^{3}, c_{2s}^{4} \gt 0$. Utilizing Theorem 2.2 (i), the system has at least 
$7[\frac{n}{6}]$ limit cycles near 
$h=h_{s}$.
(ii) Non-centrally symmetric. Set 
$[\frac{n}{6}]=[\frac{n-1}{6}]=[\frac{n-2}{6}]=[\frac{n-3}{6}]=[\frac{n-4}{6}]=s$ in (3.8) and
\begin{align*}
\widehat{\pmb\delta}^{1}&\triangleq(m_{0}^{0},m_{0}^{1}, m_{0}^{2},m_{0}^{3}, m_{0}^{4}, \ldots, m_{s}^{0},m_{s}^{1}, m_{s}^{2},m_{s}^{3}, m_{s}^{4}),\\
\widehat{\pmb E}_{1}&\triangleq\frac{\partial\left(c_{0}^{1}, c_{0}^{2}, c_{0}^{3}, c_{1}^{1}, c_{1}^{2}, \ldots, c_{2s}^{1}, c_{2s}^{2}, c_{2s}^{3}, c_{2s+1}^{1}, c_{2s+1}^{2}\right)}{\partial\widehat{\pmb\delta}^{1}}.
\end{align*} Combining with (2.6) and substituting (3.8) and (3.9) into (3.1), one determines that 
$\widehat{\pmb E}_{1}$ can be expressed as
\begin{equation*}\tiny
\left(\begin{array}{lllllllllllllll}
a_{0,0}^{1} & a_{1,0}^{1} & a_{2,0}^{1} & a_{3,0}^{1} & a_{4,0}^{1} & \cdots & 0 & 0 & 0 & 0 & 0 \\
a_{0,0}^{1} & -a_{1,0}^{1} & a_{2,0}^{1} &-a_{3,0}^{1} & a_{4,0}^{1} & \cdots & 0 & 0 & 0 & 0 & 0 \\
a_{0,0}^{3} & 0 & a_{2,0}^{3} & 0 & a_{4,0}^{3} & \cdots & 0 & 0 & 0 & 0 & 0 \\
\widetilde{a}_{0,1}^{1} & \widetilde{a}_{1,1}^{1} & \widetilde{a}_{2,1}^{1} & \widetilde{a}_{3,1}^{1} & \widetilde{a}_{4,1}^{1} & \cdots & 0 & 0 & 0 & 0 & 0 \\
\widetilde{a}_{0,1}^{1} & -\widetilde{a}_{1,1}^{1} & \widetilde{a}_{2,1}^{1} & -\widetilde{a}_{3,1}^{1} & \widetilde{a}_{4,1}^{1} & \cdots & 0 & 0 & 0 & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots \\
a_{0,s}^{1} & a_{1,s}^{1} & a_{2,s}^{1} & a_{3,s}^{1} & a_{4,s}^{1} & \cdots & a_{0,0}^{1} & a_{1,0}^{1} & a_{2,0}^{1} & a_{3,0}^{1} & a_{4,0}^{1} \\
a_{0,s}^{1} & -a_{1,s}^{1} & a_{2,s}^{1} & -a_{3,s}^{1} & a_{4,s}^{1} & \cdots & a_{0,0}^{1} & -a_{1,0}^{1} & a_{2,0}^{1} & -a_{3,0}^{1} & a_{4,0}^{1} \\
a_{0,s}^{3} & 0 & a_{2,s}^{3} & 0 & a_{4,s}^{3} & \cdots & a_{0,0}^{3} & 0 & a_{2,0}^{3} & 0 & a_{4,0}^{3}\\
\widetilde{a}_{0,s+1}^{1} & \widetilde{a}_{1,s+1}^{1} & \widetilde{a}_{2,s+1}^{1} & \widetilde{a}_{3,s+1}^{1} & \widetilde{a}_{4,s+1}^{1} & \cdots & \widetilde{a}_{0,1}^{1} & \widetilde{a}_{1,1}^{1} & \widetilde{a}_{2,1}^{1} & \widetilde{a}_{3,1}^{1} & \widetilde{a}_{4,1}^{1}\\
\widetilde{a}_{0,s+1}^{1} & -\widetilde{a}_{1,s+1}^{1} & \widetilde{a}_{2,s+1}^{1} & -\widetilde{a}_{3,s+1}^{1} & \widetilde{a}_{4,s+1}^{1} & \cdots & \widetilde{a}_{0,1}^{1} & -\widetilde{a}_{1,1}^{1} & \widetilde{a}_{2,1}^{1} & -\widetilde{a}_{3,1}^{1} & \widetilde{a}_{4,1}^{1}\\
\end{array}\right).
\end{equation*} Let 
$$\widehat{F}_{0}=\left(\begin{array}{lllll}
a_{0,0}^{1} & a_{1,0}^{1} & a_{2,0}^{1} & a_{3,0}^{1} & a_{4,0}^{1} \\
a_{0,0}^{1} & -a_{1,0}^{1} & a_{2,0}^{1} & -a_{3,0}^{1} & a_{4,0}^{1} \\
a_{0,0}^{3} & 0 & a_{2,0}^{3} & 0 & a_{4,0}^{3}\\
\widetilde{a}_{0,1}^{1} & \widetilde{a}_{1,1}^{1} & \widetilde{a}_{2,1}^{1} & \widetilde{a}_{3,1}^{1} & \widetilde{a}_{4,1}^{1} \\
\widetilde{a}_{0,1}^{1} & -\widetilde{a}_{1,1}^{1} & \widetilde{a}_{2,1}^{1} & -\widetilde{a}_{3,1}^{1} & \widetilde{a}_{4,1}^{1}
\end{array}\right)$$, in which all coefficients of 
$\widehat{\pmb F}_{0}$, except for the following six, have been determined under the condition of central symmetry,
\begin{align*}
a_{1,0}^{1}=& 2\,\int_{1}^{\frac{1}{2}\sqrt{6a-2}}xA(x){\rm d}x
=\frac{3\sqrt{2}}{10}\sqrt{a-1}(a-1)^{2},\\
a_{3,0}^{1}=&2\,\int_{1}^{\frac{1}{2}\sqrt{6a-2}}x^{3}A(x){\rm d}x
=\frac{3\sqrt{2}}{70}\sqrt{a-1}(6a+1)(a-1)^{2},\\
a_{1,0}^{3}=&2\,\int_{-1}^{1}xA(x){\rm d}x=0,\
a_{3,0}^{3}=2\,\int_{-1}^{1}x^{3}A(x){\rm d}x=0,\
\widetilde{a}_{1,1}^{1}=\widetilde{a}_{3,1}^{1}=-\frac{1}{\sqrt{2(a-1)}}.
\end{align*} It follows that 
$|\widehat{\pmb F}_{0}|=\frac{27\sqrt{3}}{1120}\pi(3a-1)(a-1)^{6}\neq0,$ which gives 
$|\widehat{\pmb E}_{1}|=|\widehat{\pmb F}_{0}|^{s+1}\neq0$. By Theorem 2.3 (iii), the system (1.6) has at least 
$7[\frac{n}{6}]+3$ limit cycles near 
$h=h_{s}$.
If 
$[\frac{n}{6}]=[\frac{n-1}{6}]=[\frac{n-2}{6}]=[\frac{n-3}{6}]=[\frac{n-4}{6}]+1=s$, the 
$5s+4$ coefficients 
$m_{0}^{0}$, 
$m_{0}^{1}$, 
$m_{0}^{2}$, 
$m_{0}^{3}$, 
$m_{0}^{4}$, 
$\ldots$, 
$m_{s-1}^{0}$, 
$m_{s-1}^{1}, m_{s-1}^{2},m_{s-1}^{3}, m_{s-1}^{4}, m_{s}^{0}, m_{s}^{1}, m_{s}^{2}, m_{s}^{3}$ are independent. Let
\begin{align*}
\widehat{\pmb \delta}^{2}&\triangleq(m_{0}^{0},m_{0}^{1}, m_{0}^{2},m_{0}^{3}, m_{0}^{4}, \ldots, m_{s-1}^{0},m_{s-1}^{1}, m_{s-1}^{2},m_{s-1}^{3}, m_{s-1}^{4}, m_{s}^{0},m_{s}^{1}),\\
\widehat{\pmb E}_{2}&\triangleq\frac{\partial\left(c_{0}^{1}, c_{0}^{2}, c_{0}^{3}, c_{1}^{1}, c_{1}^{2}, \ldots, c_{2s-2}^{1}, c_{2s-2}^{2}, c_{2s-2}^{3}, c_{2s-1}^{1}, c_{2s-1}^{2}, c_{2s}^{1}, c_{2s}^{2}\right)}{\partial\widehat{\pmb \delta}^{2}}.
\end{align*}It is not difficult to verify that
\begin{equation*}|\widehat{\pmb E}_{2}|=\left|\begin{array}{ll}
a_{0,0}^{1} & a_{1,0}^{1} \\
a_{0,0}^{1} & -a_{1,0}^{1}
\end{array}\right|\cdot|\widehat{\pmb F}_{0}|^{s}\neq0.\end{equation*} For 
$m_{s}^{2}$ and 
$m_{s}^{3}$, one has
\begin{align*}
c_{2s}^{3}=a_{2,0}^{3}m_{s}^{2},\quad c_{2s+1}^{1}=\widetilde{a}_{2,1}^{1}m_{s}^{2}+\widetilde{a}_{3,1}^{1}m_{s}^{3},\quad c_{2s+1}^{2}=\widetilde{a}_{2,1}^{1}m_{s}^{2}-\widetilde{a}_{3,1}^{1}m_{s}^{3},
\end{align*} when 
$\widehat{\pmb\delta}^{2}=\pmb0$. Setting 
$|m_{s}^{3}|\ll |m_{s}^{2}|$, one has 
$c_{2s}^{3}c_{2s+1}^{1} \lt 0, c_{2s}^{3}c_{2s+1}^{2} \lt 0$ from 
$a_{2,0}^{3}\widetilde{a}_{2,1}^{1} \lt 0$. Taking Theorem 2.3 (ii) into account, the system (1.6) has at least 
$7[\frac{n}{6}]+2$ limit cycles near 
$h=h_{s}$.
When 
$[\frac{n}{6}]=[\frac{n-1}{6}]=[\frac{n-2}{6}]=[\frac{n-3}{6}]+1=[\frac{n-4}{6}]+1=s$, we have 
$5s+3$ free coefficients 
$m_{0}^{0},m_{0}^{1}, m_{0}^{2},m_{0}^{3}, m_{0}^{4}$, 
$\ldots$, 
$m_{s-1}^{0},m_{s-1}^{1}$, 
$m_{s-1}^{2},m_{s-1}^{3}, m_{s-1}^{4}, m_{s}^{0},m_{s}^{1}, m_{s}^{2}$. Owing to 
$|\widehat{\pmb E}_{2}|\neq0$, we will consider 
$m_{s}^{2}$ below. If 
$\widehat{\pmb\delta}^{2}=\pmb0$, it holds that
\begin{align*}
c_{2s}^{3}=a_{2,0}^{3}m_{s}^{2},\quad c_{2s+1}^{1}=\widetilde{a}_{2,1}^{1}m_{s}^{2},\quad c_{2s+1}^{2}=\widetilde{a}_{2,1}^{1}m_{s}^{2},
\end{align*} which imply 
$c_{2s}^{3}c_{2s+1}^{1} \lt 0, c_{2s}^{3}c_{2s+1}^{2} \lt 0$. From Theorem 2.3 (ii), the system (1.6) has at least 
$7[\frac{n}{6}]+2$ limit cycles near 
$h=h_{s}$.
For 
$[\frac{n}{6}]=[\frac{n-1}{6}]=[\frac{n-2}{6}]+1=[\frac{n-3}{6}]+1=[\frac{n-4}{6}]+1=s$, there are 
$5s+2$ free coefficients 
$m_{0}^{0}, m_{0}^{1}, m_{0}^{2}$, 
$m_{0}^{3}, m_{0}^{4}$, 
$\ldots$, 
$m_{s-1}^{0}, m_{s-1}^{1}, m_{s-1}^{2}, m_{s-1}^{3}, m_{s-1}^{4}, m_{s}^{0}, m_{s}^{1}$. Let
\begin{align*}
\widehat{\pmb\delta}^{3}&\triangleq(m_{0}^{0},m_{0}^{1}, m_{0}^{2},m_{0}^{3}, m_{0}^{4}, \ldots, m_{s-1}^{0},m_{s-1}^{1}, m_{s-1}^{2},m_{s-1}^{3}, m_{s-1}^{4}),\\
\widehat{\pmb E}_{3}&\triangleq\frac{\partial\left(c_{0}^{1}, c_{0}^{2}, c_{0}^{3}, c_{1}^{1}, c_{1}^{2}, \ldots, c_{2s-2}^{1}, c_{2s-2}^{2}, c_{2s-2}^{3}, c_{2s-1}^{1}, c_{2s-1}^{2}\right)}{\partial\widehat{\pmb\delta}^{3}},
\end{align*} and hence 
$|\widehat{\pmb E}_{3}|= |\widehat{\pmb F}_{0}|^{s}\neq0$. It follows that
\begin{align*}
c_{2s}^{1}&=a_{0,0}^{1}m_{s}^{0}+a_{1,0}^{1}m_{s}^{1}, \quad c_{2s}^{2}=a_{0,0}^{1}m_{s}^{0}-a_{1,0}^{1}m_{s}^{1}, \\
c_{2s}^{3}&=a_{0,0}^{3}m_{s}^{0}, \quad c_{2s}^{4}=c_{2s}^{1}+c_{2s}^{2}+c_{2s}^{3}=(2a_{0,0}^{1}+a_{0,0}^{3})m_{s}^{0},
\end{align*} when 
$\widehat{\pmb\delta}^{3}=\pmb0$. By virtue of the fact that the coefficients 
$m_{s}^{0}$ and 
$m_{s}^{1}$ are independent, we assume 
$|m_{s}^{1}|\ll |m_{s}^{0}|$. Naturally, we arrive at 
$c_{2s}^{1}, c_{2s}^{2}, c_{2s}^{3}, c_{2s}^{4} \gt 0$ from 
$a_{0,0}^{1}, a_{0,0}^{3} \gt 0$. By Theorem 2.3 (i), the system (1.6) has at least 
$7[\frac{n}{6}]$ limit cycles near 
$h=h_{s}$.
If 
$[\frac{n}{6}]=[\frac{n-1}{6}]+1=[\frac{n-2}{6}]+1=[\frac{n-3}{6}]+1=[\frac{n-4}{6}]+1=s$, we gain 
$5s+1$ free coefficients 
$m_{0}^{0}, m_{0}^{1}, m_{0}^{2}$, 
$m_{0}^{3}, m_{0}^{4}$, 
$\ldots$, 
$m_{s-1}^{0}, m_{s-1}^{1}, m_{s-1}^{2}, m_{s-1}^{3}, m_{s-1}^{4}, m_{s}^{0}$. By 
$|\widehat{\pmb E}_{3}|= |\widehat{\pmb F}_{0}|^{s}\neq0$, we first consider the coefficient 
$m_{s}^{0}$. It follows from 
$\widehat{\pmb\delta}^{3}=\pmb0$ that
\begin{align*}
c_{2s}^{1}&=a_{0,0}^{1}m_{s}^{0},\quad c_{2s}^{2}=a_{0,0}^{1}m_{s}^{0},\\
c_{2s}^{3}&=a_{0,0}^{3}m_{s}^{0},\quad c_{2s}^{4}=c_{2s}^{1}+c_{2s}^{2}+c_{2s}^{3}=(2a_{0,0}^{1}+a_{0,0}^{3})m_{s}^{0}.
\end{align*} Combining 
$m_{s}^{0}$ is independent and 
$a_{0,0}^{1}, a_{0,0}^{3} \gt 0$, one gets 
$c_{2s}^{1}, c_{2s}^{2}, c_{2s}^{3}, c_{2s}^{4} \gt 0$. As a result of Theorem 2.3 (i), the system (1.6) has at least 
$7[\frac{n}{6}]$ limit cycles near 
$h=h_{s}$.
4. Conclusions
A double eight figure loop is one of the common topological structures in differential system. Moreover, one can utilize it to investigate the simultaneous existence of two (homoclinic loop and heteroclinic loop) bifurcations. This article we establish a general theory to find the lower bound of the maximal number of limit cycles near the double eight figure loop with hyperbolic saddles. In addition, the new approach facilitates the computation of expressions for higher-order coefficients in the expansion of the first-order Melnikov function (Abelian integral) M(h) near the double eight figure loop by finding the algebraic structure (generators) of M(h), and the conclusion gained for the limit cycles can be valid for the perturbation with any degree n.
As an application of our theory and inspired by [Reference Cen, Liu, Sun and Wang4, Reference Xiong and Zhong34, Reference Xu and Li35, Reference Yang and Zhao37, Reference Zhang, Tadé and Tian40, Reference Zhao, Qi and Liu43], we study the number of limit cycles in an 
$(n+1)$th-order generalized Liénard differential system, whose unperturbed system is a Hamiltonian with double eight figure loop passing two hyperbolic saddles. Besides the fact that system (1.6) can contain the two systems (1.4) and (1.5), and the result for the limit cycles holds for any 
$n\geq5$.
After receiving notification of the acceptance of the article, we noticed Yang and Han’s recent work [Reference Yang and Han36], which also studied the problem of limit cycles near a double eight figure loop.
4. Declaration of competing interest
There is no competing interest.
Acknowledgements
I would like to thank Professor Changjian Liu and Professor Yuzhen Bai for their invaluable suggestions, which greatly improved the quality of this article.
