Limit Cycles and Integrability of a Class of Quintic System

Abstract

In this paper, a class of quintic systems is investigated. The first 13 focal values are computed with the aid of MATHEMATICA. Then the necessary conditions of integrability and linearizability are obtained and the sufficiency of every condition is proved. Meanwhile, bifurcation of limit cycles is discussed, 13 limit cycles can be bifurcated from the origin. As far as the number of limit cycles enclosing an isolated singular point is concerned, this is so far the best result for elementary singular points.

1. Introduction

In the qualitative theory of planar vector fields, closely related to the second part of the well-known Hilbert’s 16th problem, the analysis of the existence, number, and the distribution of limit cycles has turned out to be the land of great promise for planar polynomial differential systems with the form

$$\frac{dx}{dt}=P(x,y),\frac{dy}{dt}=Q(x,y),$$

(1)

where P and Q are polynomials of degree lower than or equal to n.

Let $M\left(n\right)$ be the maximal number of small amplitude limit cycles surrounding either an elementary focus or center. On the one hand, estimating $M\left(n\right)$ is purely algebraic and algorithmically solvable. Nevertheless, it is still complicated. In the first place, let us recall some classic literature with respect to limit cycles of elementary singularities. The best known result is $M\left(2\right)=3$ obtained by [1] in 1952. Around an isolated focus, a number of good results have been obtained for $n=3$. James and Lloyd [2] described a class of cubic systems with eight limit cycles in 1991. Yu and Corless [3] employed both symbolic and numerical computations to present a cubic system which exhibits nine limit cycles in 2009. Later on, Chen et al. [4] reinvestigated this system and found all real solutions by applying modular computations based on regular chains in 2013. Lloyd and Pearson [5] presented another example of a nonlinear differential system of degree three with nine limit cycles by means of purely symbolic computation in 2012. Around an isolated center, a lot of excellent conclusions have also been gained for $n=3$. Żoła̧dek [6] proposed a rational Darboux integral and claimed the existence of 11 limit cycles in 1995, which was reconsidered and proved that this system can actually have only nine limit cycles using up to second-order Melnikov functions. More than a decade later, another two cubic-order systems admitting 11 limit cycles were constructed [7,8]. Very recently, the system considered in [8] was restudied by Yu and Tian [9] to show the existence of 12 limit cycles, which is the best result obtained so far for cubic polynomial systems with all limit cycles generated from a single singular point to the best of our knowledge. For $n\ge 4$, there are very few results due to the tedious and intricate calculations and simplifications when computing focal values. Huang et al. [10] gave an example of a quartic system with eight limit cycles bifurcated from a fine focus in 2013. Important results on the center problem for quintic systems were obtained in [11]. Next, let us come back to some classic literature in regards limit cycle bifurcation of nilpotent singularities. Liu and Li [12] proved that eight limit cycles can be created from the eight-order weakened focus of a cubic nilpotent system. Qiu and Li [13] proved that, using two different perturbations, 11 or 12 limit cycles with different distributions could be created from the third-order nilpotent critical point in a class of quartic systems. Li and Jin [14] proved that, using two different perturbations, 14 or 15 limit cycles with different distributions could be created from the third-order nilpotent critical point in a class of quintic systems. Compared with elementary singular points, more limit cycles may be produced from nilpotent singular points of vector fields with the same degree. Therefore, it is more challenging to obtain more limit cycles around an elementary singular point.

On the other hand, studying global bifurcation of limit cycles for planar differential systems is much more difficult. Let $H\left(n\right)$ denote the Hilbert number, the upper bound on the number of limit cycles that planar polynomial vector fields (1) can have. Chen and Wang [15] and Shi [16] separately proved $H\left(2\right)\ge 4$. The best results known so far for $n=3$ is $H\left(3\right)\ge 13$. Based on counting the number of zeros of some Abelian integrals, Li et al. constructed [17] a planar cubic system and demonstrated that it has at least 13 limit cycles. Motivated by [17], Yang et al. [18] considered an equivalent system and proved its possession of 13 limit cycles, which can be regarded as an improvement of [17] in terms of both simpler and more concrete condition for the existence of 13 limit cycles and simpler and more direct proof method. Liu and Li [12,19] obtained a sufficient condition for the existence of 13 limit cycles in a centrally symmetric cubic systems, having the distribution of one large limit cycle bifurcated from the equator surrounds 12 small limit cycles bifurcated from two symmetric foci. The best results reported so far for quartic-order system is $H\left(4\right)\ge 16$. Wang et al. [20] found 16 limit cycles and studied their distribution by bifurcation theory and qualitative analysis for a quartic polynomial $Z_3$-equivariant vector fields. By applying the double homoclinic loops bifurcation method and Poincaré-Bendixson theorem, Wu et al. [21] concluded that 16 limit cycles with two different configurations exist in this special planar polynomial system.

It should be noted that better results are often acquired from $Z_n$-equivariant vector fields, see [22]. In fact, as far as an isolated focus is concerned, computing higher-order focal values with simpler expressions is not an easy task. As a result, it is hard to obtain more limit cycles by calculating the focal values of a single focus. Thus, very few results were achieved for higher-order polynomial differential systems.

In this paper, the following quintic system

$$\begin{array}{cc}\hfill \frac{dx}{dt}& =-y-B_{11}{x}^3-B_{12}{x}^4+(B_{40}{p}_1-B_{13}-B_{22}-B_{31}-B_{40})x^5\hfill \\ \hfill & -A_{11}{x}^{2}y-2x^{3}y+(A_{13}-A_{22}-3A_{31}-5A_{40}+3A_{40}{p}_1)x^{4}y\hfill \\ \hfill & -B_{11}x{y}^2-2B_{12}{x}^2{y}^2+(2B_{31}+10B_{40}-2B_{40}{p}_1-2B_{13}-2B_{22})x^3{y}^2\hfill \\ \hfill & -A_{11}{y}^3-2x{y}^3+(2A_{13}-2A_{22}-2A_{31}+10A_{40}+2A_{40}{p}_1)x^2{y}^3\hfill \\ \hfill & -B_{12}{y}^4+(3B_{31}-5B_{40}-3B_{40}{p}_1-B_{13}-B_{22})x{y}^4\hfill \\ \hfill & +(A_{13}-A_{22}+A_{31}-A_{40}-A_{40}{p}_1)y^5,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \frac{dy}{dt}& =x+A_{11}{x}^3+(1+A_{12})x^4+(A_{13}+A_{22}+A_{31}+A_{40}+A_{40}{p}_1)x^5\hfill \\ \hfill & -B_{11}{x}^{2}y+(B_{13}-B_{22}-3B_{31}-5B_{40}-3B_{40}{p}_1)x^{4}y+A_{11}x{y}^2\hfill \\ \hfill & +2A_{12}{x}^2{y}^2+(2A_{13}+2A_{22}-2A_{31}-10A_{40}-2A_{40}{p}_1)x^3{y}^2-B_{11}{y}^3\hfill \\ \hfill & +(2B_{13}-2B_{22}-2B_{31}+10B_{40}-2B_{40}{p}_1)x^2{y}^3+(A_{12}-1)y^4\hfill \\ \hfill & +(A_{13}+A_{22}-3A_{31}+5A_{40}-3A_{40}{p}_1)x{y}^4\hfill \\ \hfill & +(B_{13}-B_{22}+B_{31}-B_{40}+B_{40}{p}_1)y^5.\hfill \end{array}$$

will be studied. We suppose here $(A_{12}-\frac{1}{3})B_{31}{B}_{12}\ne 0$. Sufficient and necessary conditions for the origin of system (2) to be the center are derived. In addition, the existence of 13 small-amplitude limit cycles bifurcating from the origin is proved. We know of no example of quintic systems with more than this number bifurcating from an elementary focus. Our example appears to be the first to have been obtained without recourse to some numerical calculation.

The rest of the paper is organized as follows. In Section 2, the first 13 focal values at the origin are computed using singular point quantity method or normal form method. Furthermore, bifurcation of limit cycles will be discussed in this section, and 13 limit cycles can be obtained. Section 3 is devoted to discuss the center conditions, two center conditions are classified. In Section 4, isochronous center conditions are classified and linearization transformations are given, respectively. At the end of this paper, a conclusion is given in Section 5.

2. Focal Values and Bifurcation of Limit Cycles

Consider the following complex system,

$$\begin{array}{cc}\hfill & \frac{dz}{dT}=z+\sum _{\alpha +\beta =2}^{\infty }{a}_{\alpha \beta }{z}^{\alpha }{w}^{\beta }=Z(z,w),\hfill \\ \hfill & \frac{dw}{dT}=-w-\sum _{\alpha +\beta =2}^{\infty }{b}_{\alpha \beta }{w}^{\alpha }{z}^{\beta }=-W(z,w),\hfill \end{array}$$

(3)

where $z,w,T$ are complex variables and $a_{\alpha \beta },b_{\alpha \beta }$ are complex coefficients, $Z(z,w)$ and $W(z,w)$ are analytic functions in the neighborhood of the origin. The origin of system (3) is called a weak saddle. By transformation

$$z=x+iy,w=x-iy,T=it,i=\sqrt{-1},$$

(4)

system (3) can be brought to

$$\begin{array}{cc}\hfill & \frac{dx}{dt}=-y+\sum _{\alpha +\beta =2}^{\infty }{A}_{\alpha \beta }{x}^{\alpha }{y}^{\beta }=-y+\sum _{k=2}^{\infty }{X}_k(x,y),\hfill \\ \hfill & \frac{dy}{dt}=x+\sum _{\alpha +\beta =2}^{\infty }{B}_{\alpha \beta }{x}^{\alpha }{y}^{\beta }=x+\sum _{k=2}^{\infty }{Y}_k(x,y).\hfill \end{array}$$

(5)

In [23], the authors proved the following result.

Theorem 1.

For given $\left\{c_{k+1,k}\right\}$ and $\left\{d_{k+1,k}\right\}$, using a formal change of variables,

$$\xi =z+\sum _{k+j=2}^{\infty }{c}_{kj}{z}^k{w}^j,\eta =w+\sum _{k+j=2}^{\infty }{d}_{kj}{w}^k{z}^j,$$

(6)

system (3) can be uniquely reduced to the formal form of

$$\frac{d\xi }{dT}=\xi +\xi \sum _{k=1}^{\infty }{p}_k{\xi }^k{\eta }^k,\frac{d\eta }{dT}=-\eta -\eta \sum _{k=1}^{\infty }{q}_k{\xi }^k{\eta }^k.$$

(7)

Definition 1.

Denote

$${\mu }_k=p_k-q_k,{\tau }_k=p_k+q_k,k=1,2,\cdots .$$

(8)

Then ${\mu }_k$ is called the kth focal value of the origin of system (3), and ${\tau }_k$ is called the kth periodic constant of the origin of system (3).

By transformation

$$x=rcos\theta ,y=rsin\theta ,$$

(9)

system (5) is reduced to

$$\frac{dr}{dt}=R(r,\theta ),\frac{d\theta }{dt}=\Theta (r,\theta ),$$

(10)

where

$$\begin{array}{cc}\hfill & R(r,\theta )=\sum _{k=1}^{\infty }{u}_{k+2}\left(\theta \right)r^{k+1},\hfill \\ \hfill & \Theta (r,\theta )=1+\sum _{k=1}^{\infty }{v}_{k+2}\left(\theta \right)r^k.\hfill \\ \hfill & u_k\left(\theta \right)=cos\left(\theta \right)X_{k-1}(cos\left(\theta \right),sin\left(\theta \right))+sin\left(\theta \right)Y_{k-1}(cos\left(\theta \right),sin\left(\theta \right)),\hfill \\ \hfill & v_k\left(\theta \right)=cos\left(\theta \right)Y_{k-1}(cos\left(\theta \right),sin\left(\theta \right))-sin\left(\theta \right)X_{k-1}(cos\left(\theta \right),sin\left(\theta \right)).\hfill \end{array}$$

(11)

For sufficiently small h, denote $\tilde{r}(\theta ,h)$ that is the solution of system (10) satisfying the initial condition ${r|}_{\theta =0}=h$,

$$r=\tilde{r}(\theta ,h)=h+\sum _{k=2}^{\infty }{\nu }_k\left(\theta \right)h^k,$$

(12)

and

$$\begin{array}{cc}\hfill & \mathcal{T}(\vartheta ,h)={\int }_0^{\vartheta }\frac{d\theta }{\Theta (\tilde{r}(\theta ,h),\theta )}\hfill \end{array}$$

(13)

The definition of center and isochronous center were given in [24].

Definition 2.

For sufficiently small constant h, if

$$\tilde{r}(2\pi ,h)\equiv h,$$

(14)

the origin of system (5) is called to be center.

Furthermore, if the origin of system (5) is a center and

$$\mathcal{T}(2\pi ,h)\equiv 2\pi ,$$

(15)

the origin of system (5) is called to be an isochronous center.

The focal values can be computed by formal series method, for more detail, see [24].

By means of transformation (4), system (2) can be transformed to

$$\begin{array}{cc}\hfill \frac{dz}{dT}& =\frac{1}{2}z(2+2a_{11}zw+2z^{2}w+2a_{12}z{w}^2+2a_{40}{z}^4+2a_{31}{z}^{3}w+2a_{22}{z}^2{w}^2\hfill \\ \hfill & +2a_{13}z{w}^3+2p_1{b}_{40}{w}^4),\hfill \\ \hfill \frac{dw}{dT}& =-\frac{1}{2}w(2+2b_{11}wz+2w^{2}z+2b_{12}w{z}^2+2b_{40}{w}^4+2b_{31}{w}^{3}z+2b_{22}{w}^2{z}^2\hfill \\ \hfill & +2b_{13}w{z}^3+2p_1{a}_{40}{z}^4),\hfill \end{array}$$

(16)

where

$$\begin{array}{cc}\hfill & a_{11}=A_{11}+B_{11}i,a_{12}=A_{12}+B_{12}i,a_{40}=A_{40}+B_{40}i,\hfill \\ \hfill & a_{31}=A_{31}+B_{31}i,a_{22}=A_{22}+B_{22}i,a_{13}=A_{13}+B_{13}i,\hfill \\ \hfill & b_{11}=A_{11}-B_{11}i,b_{12}=A_{12}-B_{12}i,b_{40}=A_{40}-B_{40}i,\hfill \\ \hfill & b_{31}=A_{31}-B_{31}i,b_{22}=A_{22}-B_{22}i,b_{13}=A_{13}-B_{13}i,\hfill \end{array}$$

(17)

with $(a_{31}-b_{31})(-1+3b_{12})\ne 0$.

With the aid of computer algebraic system-MATHEMATICA, the first 13 focal values of system (16) are computed by formal series method, given by

$$\begin{array}{cc}{r}_1\hfill & =a_{11}-b_{11},\hfill \\ r_2\hfill & =a_{22}-b_{22},\hfill \\ r_3\hfill & =-a_{12}+b_{12},\hfill \\ r_4\hfill & =b_{13}{b}_{31}-a_{13}{a}_{31},\hfill \\ r_5\hfill & =-\frac{1}{2}(a_{31}-b_{31})(-1+3b_{12})(1-2b_{12}+q),\hfill \\ r_6\hfill & =(a_{40}{b}_{31}^2-a_{31}^2{b}_{40})(-1+2b_{12})(-2+3b_{12}-2p_1+b_{12}{p}_1).\hfill \end{array}$$

(18)

Case 1: $-1+2b_{12}\ne 0$.

$$\begin{array}{cc}{r}_7\hfill & =\frac{1}{2}{a}_{31}{b}_{31}(-a_{31}+b_{31})(-1+4b_{12})(-3+5b_{12})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2,\hfill \\ r_8\hfill & =-\frac{5}{64}(a_{31}-b_{31})(a_{31}+b_{31}+12a_{31}{b}_{11}{b}_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2.\hfill \end{array}$$

(19)

Subcase 1.1: $b_{12}=\frac{1}{4}$.

$$\begin{array}{cc}\hfill r_9& =\frac{(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{3072a_{31}{b}_{31}}(13a_{31}^2+26a_{31}{b}_{31}+13b_{31}^2\hfill \\ \hfill & +7488a_{31}^2{b}_{22}{b}_{31}^2+480a_{31}^3{b}_{31}^3{p}_2+672a_{31}^3{b}_{31}^3{p}_1{p}_2),\hfill \\ \hfill r_{10}& =-\frac{(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{46080a_{31}{b}_{31}}(75a_{31}^3+225a_{31}^2{b}_{31}+225a_{31}{b}_{31}^2\hfill \\ \hfill & +75b_{31}^3-43200a_{31}^3{b}_{31}^3+160a_{31}^4{b}_{31}^3{p}_2+160a_{31}^3{b}_{31}^4{p}_2\hfill \\ \hfill & +224a_{31}^4{b}_{31}^3{p}_1{p}_2+224a_{31}^3{b}_{31}^4{p}_1{p}_2),\hfill \\ \hfill r_{11}& =\frac{(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{58611990528a_{31}^3{b}_{31}^3{(a_{31}+b_{31})}^2}{f}_1,\hfill \\ \hfill r_{12}& =-\frac{5(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{54103375872a_{31}^4{b}_{31}^4(a_{31}+b_{31})}{f}_2,\hfill \\ \hfill r_{13}& =\frac{(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{48003220242432a_{31}^5{b}_{31}^5{(a_{31}+b_{31})}^3}{f}_3.\hfill \end{array}$$

(20)

Subcase 1.2: $b_{12}=\frac{3}{5}$.

$$\begin{array}{cc}\hfill r_9^{,}& =-\frac{(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{3000a_{31}{b}_{31}}(17a_{31}^2+34a_{31}{b}_{31}+17b_{31}^2\hfill \\ \hfill & +2400a_{31}^2{b}_{22}{b}_{31}^2+16a_{31}^3{b}_{31}^3{p}_2+112a_{31}^3{b}_{31}^3{p}_1{p}_2),\hfill \\ \hfill r_{10}^{,}& =-\frac{(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{22500a_{31}{b}_{31}}(-11a_{31}^3-33a_{31}^2{b}_{31}-33a_{31}{b}_{31}^2\hfill \\ \hfill & -11b_{31}^3+4320a_{31}^3{b}_{31}^3+12a_{31}^4{b}_{31}^3{p}_2+12a_{31}^3{b}_{31}^4{p}_2+84a_{31}^4{b}_{31}^3{p}_1{p}_2\hfill \\ \hfill & +84a_{31}^3{b}_{31}^4{p}_1{p}_2),\hfill \\ \hfill r_{11}^{,}& =\frac{(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{3175200000a_{31}^3{b}_{31}^3{(a_{31}+b_{31})}^2}{f}_4,\hfill \\ \hfill r_{12}^{,}& =-\frac{5(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{7620480000a_{31}^4{b}_{31}^4(a_{31}+b_{31})}{f}_5,\hfill \\ \hfill r_{13}^{,}& =\frac{(a_{31}-b_{31})(-2+3b_{12}-2p_1+b_{12}{p}_1)p_2}{32006016000000a_{31}^5{b}_{31}^5{(a_{31}+b_{31})}^3}{f}_6.\hfill \end{array}$$

(21)

Case 2: $b_{12}=\frac{1}{2}$.

$$\begin{array}{cc}\hfill {\mu }_7& =\frac{1}{8}(-a_{40}{b}_{31}-a_{40}{b}_{11}{b}_{31}^2+a_{31}{b}_{40}+a_{31}^2{b}_{11}{b}_{40})(1+3p_1),\hfill \\ \hfill {\mu }_8& =-\frac{5}{48(a_{40}{b}_{31}^2-a_{31}^2{b}_{40})}(5a_{40}^2{b}_{31}^2+12a_{40}^2{b}_{22}{b}_{31}^4+a_{31}^2{a}_{40}{b}_{40}\hfill \\ \hfill & -12a_{31}{a}_{40}{b}_{31}{b}_{40}+a_{40}{b}_{31}^2{b}_{40}-24a_{31}^2{a}_{40}{b}_{22}{b}_{31}^2{b}_{40}+5a_{31}^2{b}_{40}^2\hfill \\ \hfill & +12a_{31}^4{b}_{22}{b}_{40}^2)(1+3p_1),\hfill \\ \hfill {\mu }_9& =\frac{1}{240}{a}_{40}{b}_{40}(-a_{31}+b_{31}){(1+3p_1)}^2.\hfill \end{array}$$

(22)

The expressions of functions $f_i,i=1,\cdots ,6$ are available via the following E-mail address: lifeng@lyu.edu.cn.

Next, analyzing the 13 focal values of system (16), the following theorem can be obtained.

Theorem 2.

The origin of system (16) is a 13th order weak focus if and only if

$$\begin{array}{cc}\hfill & a_{11}=b_{11},a_{22}=b_{22},a_{12}=b_{12},a_{13}=q{b}_{31},b_{13}=q{a}_{31},\hfill \\ \hfill & q=2b_{12}-1,a_{40}=p_2{a}_{31}^2,b_{40}=p_2{b}_{31}^2,\hfill \end{array}$$

(23)

and one of the following conditions holds:

$$\begin{array}{cc}\hfill \left(1\right)& b_{12}=\frac{3}{5},b_{11}=-\frac{a_{31}+b_{31}}{12a_{31}{b}_{31}},\hfill \\ \hfill & b_{22}=\frac{-17a_{31}^2-34a_{31}{b}_{31}-17b_{31}^2-16a_{31}^3{b}_{31}^3{p}_2-112a_{31}^3{b}_{31}^3{p}_1{p}_2}{2400a_{31}^2{b}_{31}^2},\hfill \\ \hfill & p_1=\frac{1}{84a_{31}^3{b}_{31}^3(a_{31}+b_{31})p_2}(11a_{31}^3+33a_{31}^2{b}_{31}+33a_{31}{b}_{31}^2+11b_{31}^3\hfill \\ \hfill & -4320a_{31}^3{b}_{31}^3-12a_{31}^4{b}_{31}^3{p}_2-12a_{31}^3{b}_{31}^4{p}_2),\hfill \\ \hfill & f_1=f_2=0,f_3\ne 0;\hfill \\ \hfill \left(2\right)& b_{12}=\frac{1}{4},b_{11}=-\frac{a_{31}+b_{31}}{12a_{31}{b}_{31}},\hfill \\ \hfill & b_{22}=\frac{-13a_{31}^2-26a_{31}{b}_{31}-13b_{31}^2-480a_{31}^3{b}_{31}^3{p}_2-672a_{31}^3{b}_{31}^3{p}_1{p}_2}{7488a_{31}^2{b}_{31}^2},\hfill \\ \hfill & p_1=-\frac{5}{224a_{31}^3{b}_{31}^3(a_{31}+b_{31})p_2}(15a_{31}^3+45a_{31}^2{b}_{31}+45a_{31}{b}_{31}^2+15b_{31}^3\hfill \\ \hfill & -8640a_{31}^3{b}_{31}^3+32a_{31}^4{b}_{31}^3{p}_2+32a_{31}^3{b}_{31}^4{p}_2),\hfill \\ \hfill & f_4=f_5=0,f_6\ne 0.\hfill \end{array}$$

(24)

Proof. 

$r_1=r_2=r_3=r_4=r_5=0$ yields that

$$a_{11}=b_{11},a_{22}=b_{22},a_{12}=b_{12},a_{13}=q{b}_{31},b_{13}=q{a}_{31},q=2b_{12}-1.$$

(25)

$r_6=0$ shows that $a_{40}=p_2{a}_{31}^2,b_{40}=p_2{b}_{31}^2$ or $b_{12}=\frac{1}{2}$. When $b_{12}=\frac{1}{2}$, the origin is a 9-th weak focus, so we take $a_{40}=p_2{a}_{31}^2,b_{40}=p_2{b}_{31}^2$. Then from $r_7=0$, we obtain $(-1+4b_{12})(-3+5b_{12})=0$, two cases will be considered. When $b_{12}=\frac{1}{4}$, $r_8=r_9=r_{10}=0$ yields that

$$\begin{array}{cc}\hfill & b_{11}=-\frac{a_{31}+b_{31}}{12a_{31}{b}_{31}},\hfill \\ \hfill & b_{22}=\frac{-17a_{31}^2-34a_{31}{b}_{31}-17b_{31}^2-16a_{31}^3{b}_{31}^3{p}_2-112a_{31}^3{b}_{31}^3{p}_1{p}_2}{2400a_{31}^2{b}_{31}^2},\hfill \\ \hfill & p_1=\frac{1}{84a_{31}^3{b}_{31}^3(a_{31}+b_{31})p_2}(11a_{31}^3+33a_{31}^2{b}_{31}+33a_{31}{b}_{31}^2+11b_{31}^3\hfill \\ \hfill & -4320a_{31}^3{b}_{31}^3-12a_{31}^4{b}_{31}^3{p}_2-12a_{31}^3{b}_{31}^4{p}_2).\hfill \end{array}$$

(26)

Furthermore, it is easy to testify that equation $f_1=f_2=f_3=0$ has no real solution which yields that the origin of system (16) is a 13th order weak focus. Similarly, we can obtain that the origin of system (16) is also a 13th order weak focus when $b_{12}=\frac{1}{4}$. □

Now, following the perturbation method in [24], we consider the perturbed system of system (16) given by

$$\begin{array}{cc}\hfill \frac{dx}{dt}& =\delta x-y-B_{11}{x}^3-B_{12}{x}^4+(B_{40}{p}_1-B_{13}-B_{22}-B_{31}-B_{40})x^5\hfill \\ \hfill & -A_{11}{x}^{2}y-2x^{3}y+(A_{13}-A_{22}-3A_{31}-5A_{40}+3A_{40}{p}_1)x^{4}y\hfill \\ \hfill & -B_{11}x{y}^2-2B_{12}{x}^2{y}^2+(2B_{31}+10B_{40}-2B_{40}{p}_1-2B_{13}-2B_{22})x^3{y}^2\hfill \\ \hfill & -A_{11}{y}^3-2x{y}^3+(2A_{13}-2A_{22}-2A_{31}+10A_{40}+2A_{40}{p}_1)x^2{y}^3\hfill \\ \hfill & -B_{12}{y}^4+(3B_{31}-5B_{40}-3B_{40}{p}_1-B_{13}-B_{22})x{y}^4\hfill \\ \hfill & +(A_{13}-A_{22}+A_{31}-A_{40}-A_{40}{p}_1)y^5,\hfill \\ \hfill \frac{dy}{dt}& =x+\delta y+A_{11}{x}^3+(1+A_{12})x^4+(A_{13}+A_{22}+A_{31}+A_{40}+A_{40}{p}_1)x^5\hfill \\ \hfill & -B_{11}{x}^{2}y+(B_{13}-B_{22}-3B_{31}-5B_{40}-3B_{40}{p}_1)x^{4}y+A_{11}x{y}^2\hfill \\ \hfill & +2A_{12}{x}^2{y}^2+(2A_{13}+2A_{22}-2A_{31}-10A_{40}-2A_{40}{p}_1)x^3{y}^2-B_{11}{y}^3\hfill \\ \hfill & +(2B_{13}-2B_{22}-2B_{31}+10B_{40}-2B_{40}{p}_1)x^2{y}^3+(A_{12}-1)y^4\hfill \\ \hfill & +(A_{13}+A_{22}-3A_{31}+5A_{40}-3A_{40}{p}_1)x{y}^4\hfill \\ \hfill & +(B_{13}-B_{22}+B_{31}-B_{40}+B_{40}{p}_1)y^5.\hfill \end{array}$$

(27)

The following consequence can be derived directly from Theorem 2.

Theorem 3.

When $(A_{12}-\frac{1}{3})B_{31}{B}_{12}\ne 0$, if the origin of system (16) is a 13th order weak focus, there exist 13 limit cycles in the neighborhood of the origin of system (27).

3. Center Conditions

Now, we study the center conditions of system (16). The necessary and sufficient conditions for the origin of system (16) to be a center is that all focal values are zero. However, it is impossible to present all focal values, we can obtain some necessary conditions from some focal values, then prove the sufficiency of every condition one by one. The following proposition can be obtained directly.

Proposition 1.

The first 13 focal values of system (16) are zero if and only if one of the following conditions holds:

$$\begin{array}{cc}\hfill C_1:& a_{11}=b_{11},a_{22}=b_{22},a_{12}=b_{12},a_{13}=q{b}_{31},b_{13}=q{a}_{31},\hfill \\ \hfill & q=2b_{12}-1,b_{12}=\frac{2(1+p_1)}{3+p_1};\hfill \\ \hfill C_2:& a_{11}=b_{11},a_{22}=b_{22},a_{12}=b_{12},a_{13}=q{b}_{31},b_{13}=q{a}_{31},\hfill \\ \hfill & q=2b_{12}-1,a_{40}=b_{40}=0.\hfill \end{array}$$

(28)

According to Theorem in [24], the origin of system is a center if and only if there exists an analytic integral factor in a neighborhood of the origin. So we have the following result.

Theorem 4.

When $(a_{31}-b_{31})(-1+3b_{12})\ne 0$, the origin of system (16) is a center if and only if one of the conditions in Proposition 1 holds.

Proof. 

When condition $C_1$ holds, system (16) can be rewritten as

$$\begin{array}{cc}\hfill \frac{dz}{dT}& =\frac{z}{3+p_1}[3+p_1+b_{11}(3+p_1)zw+(3+p_1)z^{2}w+2(1+p_1)z{w}^2\hfill \\ \hfill & +a_{40}(3+p_1)z^4+a_{31}(3+p_1)z^{3}w+b_{22}(3+p_1)z^2{w}^2\hfill \\ \hfill & +b_{31}(1+3p_1)z{w}^3+b_{40}{p}_1(3+p_1)w^4],\hfill \\ \hfill \frac{dw}{dT}& =-\frac{w}{3+p_1}[3+p_1+b_{11}(3+p_1)wz+(3+p_1)w^{2}z+2(1+p_1)w{z}^2\hfill \\ \hfill & +b_{40}(3+p_1)w^4+b_{31}(3+p_1)w^{3}z+b_{22}(3+p_1)w^2{z}^2\hfill \\ \hfill & +a_{31}(1+3p_1)w{z}^3+a_{40}{p}_1(3+p_1)z^4],\hfill \end{array}$$

(29)

which has the analytic first integral

$$\left\{\begin{array}{c}zw,p_1=1,\\ {\left(zw\right)}^{\frac{4}{p_1-1}}{g}_1,p_1\ne 1,\end{array}\right.$$

(30)

where

$$\begin{array}{cc}\hfill g_1& =(1+p_1)(3+p_1)(1+a_{40}{z}^4)+4(1+p_1)zw(b_{11}+z+a_{31}{z}^2)\hfill \\ \hfill & +2z{w}^2[2(1+p_1)+b_{22}(3+p_1)z]+4b_{31}(1+p_1)z{w}^3\hfill \\ \hfill & +b_{40}(1+p_1)(3+p_1)w^4.\hfill \end{array}$$

(31)

Therefore, the origin is a center.

When condition $C_2$ holds, system (16) can be transformed to

$$\begin{array}{cc}\hfill \frac{dz}{dT}& =z[1+b_{11}zw+z^{2}w+b_{12}z{w}^2+a_{31}{z}^{3}w+b_{22}{z}^2{w}^2+b_{31}(2b_{12}-1)z{w}^3],\hfill \\ \hfill \frac{dw}{dT}& =-w[1+b_{11}wz+w^{2}z+b_{12}w{z}^2+b_{31}{w}^{3}z+b_{22}{w}^2{z}^2+a_{31}(2b_{12}-1)w{z}^3],\hfill \end{array}$$

(32)

which has the analytic first integral

$$\left\{\begin{array}{c}zw,b_{12}=1,\\ zw{g}_2^{\frac{1}{2}},b_{12}=2,\\ {\left(zw\right)}^{\frac{2-b_{12}}{b_{12}-1}}{g}_3,b_{12}\ne 1,2,\end{array}\right.$$

(33)

where

$$\begin{array}{cc}\hfill g_2& =e^{-1+2b_{11}zw+2z^{2}w+2z{w}^2+2a_{31}{z}^{3}w+b_{22}{z}^2{w}^2+2b_{31}z{w}^3},\hfill \\ \hfill g_3& =-b_{12}+(-2+b_{12})b_{12}zw(b_{11}+z+a_{31}{z}^2)\hfill \\ \hfill & +(-2+b_{12})z{w}^2(b_{12}+b_{22}z)+b_{31}(-2+b_{12})b_{12}z{w}^3.\hfill \end{array}$$

(34)

Therefore, the origin is a center. □

4. Isochronous Center Conditions

In this section, we will study the isochronous center conditions when condition $C_1$ or $C_2$ holds. The main method to study isochronous center is periodic constants, which can be computed using the method of time-angle difference or formal series method; for more detail, see [24]. We compute periodic constants for each case first and obtai then isochronous center condition. Then, the sufficiencies will be proved one by one.

When condition $C_1$ holds, the first four periodic constants are

$$\begin{array}{cc}\hfill T_1& =b_{11},T_2=b_{22},T_3=\frac{(1+3p_1)(5+3p_1)}{3+p_1},\hfill \\ \hfill T_4& =-\frac{20}{9}(9a_{31}{b}_{31}+a_{40}{b}_{40}).\hfill \end{array}$$

(35)

Furthermore, we have the following result.

Proposition 2.

When condition $C_1$ holds, the origin of system (16) is an isochronous center if and only if

$$\begin{array}{cc}\hfill I_1:& a_{11}=b_{11}=a_{22}=b_{22}=a_{13}=b_{31}=b_{13}=a_{31}=a_{40}=b_{40}=0,\hfill \\ \hfill & b_{12}=\frac{2(1+p_1)}{3+p_1},(1+3p_1)(5+3p_1)=0.\hfill \end{array}$$

(36)

Proof. 

When condition in Proposition 2 holds, it can be divided into two cases $p_1=-\frac{1}{3}$ and $p_1=-\frac{5}{3}$. If $p_1=-\frac{1}{3}$, system (16) can be simplified to

$$\begin{array}{cc}\hfill \frac{dz}{dT}& =\frac{1}{2}z(2+2z^{2}w+z{w}^2),\hfill \\ \hfill \frac{dw}{dT}& =-\frac{1}{2}w(2+z^{2}w+2z{w}^2).\hfill \end{array}$$

(37)

By the linearization transformation

$$\begin{array}{cc}\hfill u& =\frac{1+3z^{2}w+3z{w}^2}{z{w}^2},\hfill \\ \hfill v& =\frac{1+3z^{2}w+3z{w}^2}{z^{2}v},\hfill \end{array}$$

system (37) can be changed to

$$\begin{array}{cc}\hfill \frac{du}{dT}& =\frac{du}{dz}\frac{dz}{dT}+\frac{du}{dw}\frac{dw}{dT}=u,\hfill \\ \hfill \frac{dv}{dT}& =\frac{dv}{dz}\frac{dz}{dT}+\frac{dv}{dw}\frac{dw}{dT}=-v.\hfill \end{array}$$

If $p_1=-\frac{5}{3}$, system (16) can be rewritten as

$$\begin{array}{cc}\hfill \frac{dz}{dT}& =z(1+z^{2}w-z{w}^2),\hfill \\ \hfill \frac{dw}{dT}& =-w(1-z^{2}w+z{w}^2),\hfill \end{array}$$

(38)

then by the linearization transformation

$$\begin{array}{cc}\hfill u& =\frac{2+3z{w}^2}{2z{w}^2},\hfill \\ \hfill v& =\frac{8z^4{w}^5}{(2+3z^{2}w){(2+3z{w}^2)}^2},\hfill \end{array}$$

system (37) can be changed to

$$\begin{array}{cc}\hfill \frac{du}{dT}& =u,\hfill \\ \hfill \frac{dv}{dT}& =-v.\hfill \end{array}$$

□

When condition $C_2$ holds, the first four periodic constants are

$$\begin{array}{cc}\hfill T_1& =b_{11},T_2=b_{22},T_3=-2(1+b_{12})(-1+2b_{12})\hfill \\ \hfill T_4& =-4a_{31}{b}_{31}{b}_{12}(-2+3b_{12}).\hfill \end{array}$$

(39)

The first four periodic constants are all zero yield the same isochronous conditions as $C_1$. Namely, the origin of system (16) is an isochronous center if and only if the conditions in Proposition 2 hold.

5. Conclusions

Compared with elementary singular points, many results show that more limit cycles can be produced from nilpotent singular points of vector fields with the same degree. It is more difficult to obtain more limit cycles around an elementary focus. In this paper, we constructed an example which has 13 limit cycles around an elementary focus. Meanwhile, center and isochronous center conditions are obtained, and the sufficiency of every condition are also proved. Even if one parameter is added, the calculation of focal values and periodic constants may become very complicated so there are still great difficulties to obtain more limit cycles.