## Bifurcation of Limit Cycles from a Liénard System of Degree 4

Zhu Hongying,1, Wei Minzhi,1, Yang Sumin,2, Jiang Caoqing,1

 基金资助: 国家自然科学基金.  11861009国家自然科学基金.  11761011广西自然科学基金.  2020JJB110007广西高校科研项目.  2020KY16020

 Fund supported: the NSFC.  11861009the NSFC.  11761011the NSF of Guangxi.  2020JJB110007the Middle-Aged and Young Teachers' Basic Ability Promotion Project in Guangxi and Scientific Research Project.  2020KY16020

Abstract

In this paper, we study the number of limit cycles by Poincaré bifurcation for some Liénard system of degree 4. We prove that the system can bifurcate at most 6 limit cycles from the periodic annulus, by the tools of regular chain theory in polynomial algebra and Chebyshev criteria, at least 3 limit cycles by asymptotic expansions of the related Abelian integral (first order Melnikov functions).

Keywords： Liénard system ; Chebyshev system ; Melnikov functions ; Weak Hilbert's 16th problem

Zhu Hongying, Wei Minzhi, Yang Sumin, Jiang Caoqing. Bifurcation of Limit Cycles from a Liénard System of Degree 4. Acta Mathematica Scientia[J], 2021, 41(4): 936-953 doi:

## 1 引言

$$$\dot{x} = {H}_y + \varepsilon p(x, y), \ \ \dot{y} = -{H}_x + \varepsilon q(x, y),$$$

Dumortier和Li[7-10]系统的研究了不同情况的类型为$(3, 2)$系统(1.4), 得到Abel积分零点个数并得到其上确界及分支图.

$$$\dot{x} = y, \ \dot{y} = -x(x-1)(x-\alpha)(x-\beta)+ \varepsilon f(x)y,$$$

$$$H(x, y) = \frac{1}{2}y^2+\frac{49}{32}x^2-\frac{7}{48}x^3-\frac{5}{8}x^4+\frac{1}{5}x^5.$$$

$L_h = H(x, y) = h, h\in(0, \frac{409}{480})$为(1.7)式定义的闭轨线族, 见图 2, 即对应于系统$(1.7)_{\varepsilon = 0}$的逆时针方向闭轨线族. ${H}(x, y) = \frac{409}{480}$定义通过双曲鞍点$(-1, 0)$的同宿环, 记为$L_{\frac{409}{480}}$, 幂零尖点$(\frac{7}{4}, 0)$在同宿环的外侧.

### 图 2

$$$I(h, \delta) = \oint_{L_h}(a_0+a_1x+a_2x^2+x^3)y {\rm d}x\equiv a_0I_0(h)+a_1I_1(h)+a_2I_2(h)+I_3(h),$$$

$\begin{eqnarray} I_{i}(h) = \oint_{L_h} f_i(x)y^{2s-1}{\rm d}x, \ \ i = 0, 1, \cdots, n-1, \end{eqnarray}$

(ⅰ) $W[l_0, \cdots , l_i]\neq 0$, 当$x\in (0, x_r)$和所有的$i = 0, 1, \cdots, n-2,$

(ⅱ) $W[l_0, \cdots , l_{n-1}] $$(0, x_r) 上有 k 个零点(考虑重数), (ⅲ) s> n+k-2 , 那么任何非平凡线性组合 \{I_0, I_1, \cdots, I_{n-1}\}$$ (0, h_0)$至多有$n+k-1$个零点(考虑重数). 此时称$\{I_0, I_1, \cdots, I_{n-1}\} $$(0, h_0) 上是精度为 k 的Chebyshev系统. 在研究一些系统时, s 的值不能满足条件(ⅲ), 引理2.1并不能直接应用, 此时可以使用如下引理2.2, 提高 I_i(h)$$ y$的次数, 克服上述困难, 参见文献[13]中的引理4.1.

$$${\rm rank}\frac{\partial(c_0, c_1, \cdots, c_{m-1}, b_0, b_1, \cdots, b_{k-1})}{\partial\delta}(\delta_0) = m+k,$$$

## 3 Abel积分$I(h, \delta)$的零点个数上界

$$$I_i = \oint_{L_h}x^{i}y{\rm d}x, \ i = 0, 1, 2, 3,$$$

$$$I_i(h) = \frac{1}{2h}\oint_{L_h}(2A(x)+y^2)x^{i}y{\rm d}x = \frac{1}{2h}(\oint_{L_h}2x^{i}A(x)y{\rm d}x+\oint_{L_h}x^{i}y^3{\rm d}x), \ \ i = 0, 1, 2, 3.$$$

$k = 3 $$F(x) = 2x^{i}A(x) , 根据引理2.2, 得到 $$\oint_{L_h}2x^{i}A(x)y{\rm d}x = \oint_{L_h}G_i(x)y^3{\rm d}x,$$ 其中 G_i(x) = \frac{\rm d}{3{\rm d}x}(\frac{2x^{i}A(x)}{A'(x)}) = \frac{ x^{i} g_i(x)}{45(x+1)^{2}(4\, x-7 )^{3}} , g_i(x) = 384\, i{x}^{5}-1488\, i{x}^{4}+384\, {x}^{5}-52\, i{ x}^{3}-1248\, {x}^{4}+5250\, i{x}^{2}+592\, {x}^{3}-1715\, ix+910\, {x}^{2} -5145\, i-1960\, x-5145. 根据 (3.2) 式和 (3.3) 式以及考虑到在每条轨线 L_h$$ \frac{2A(x)+y^2}{2h} = 1$, 可得

$\begin{eqnarray} I_i(h)& = & \frac{1}{2h}\oint_{L_h}(x^{i}+G_i(x))y^3{\rm d}x = \frac{1}{4h^2}\oint_{L_h}(2A(x)+y^2)(x^{i}+G_i(x))y^3{\rm d}x {}\\ & = & \frac{1}{4h^2}\oint_{L_h}2A(x)(x^{i}+G_i(x))y^3{\rm d}x+ \frac{1}{4h^2}\oint_{L_h}(x^{i}+G_i(x))y^5{\rm d}x. \end{eqnarray}$

$k = 5 $$F(x) = 2A(x)(x^{i}+G_i(x)) , 以及由引理2.2可知 $$\oint_{L_h}2A(x)(x^{i}+G_i(x))y^3{\rm d}x = \oint_{L_h}\widetilde{G}_i(x)y^5{\rm d}x,$$ 其中, \widetilde{G}_i(x) = \frac{\rm d}{5{\rm d}x}(\frac{2A(x)(x^{i}+G_i(x))}{A'(x)}) = \frac{x^i\widetilde{g}_i(x)}{3375\, \left( x+1 \right) ^{4} \left( 4\, x-7 \right) ^{6} } , 而 (3.4) 式和 (3.5) 式以及考虑到在每条轨线 L_h$$ \frac{2A(x)+y^2}{2h} = 1$, 可知

$\begin{eqnarray} I_i(h)& = &\frac{1}{4h^2} \oint_{L_h}(x^{i}+G_i(x)+\widetilde{G}_i(x))y^5{\rm d}x{}\\ & = &\frac{1}{8h^3}\oint_{L_h}(2A(x)+y^2)(x^{i}+G_i(x)+\widetilde{G}_i(x))y^5{\rm d}x {}\\ & = &\frac{1}{8h^3} \oint_{L_h}2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x))y^5{\rm d}x+ \frac{1}{8h^3}\oint_{L_h}(x^{i}+G_i(x)+\widetilde{G}_i(x))y^7{\rm d}x.{\quad} \end{eqnarray}$

$k = 7 $$F(x) = 2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x)) , 根据引理2.2, 得到 $$\oint_{L_h}2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x))y^5{\rm d}x = \oint_{L_h}\overline{G}_i(x)y^7{\rm d}x,$$ 其中 \overline{G}_i(x) = \frac{\rm d}{7{\rm d}x}(\frac{2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x))}{A'(x)}) = \frac{x^i\overline{g}_i(x)}{354375\, \left( x+1 \right) ^{6} \left( 4\, x-7 \right) ^{9} } , 其中 根据 (3.6) 式和 (3.7) 式以及考虑到在每条轨线 L_h$$ \frac{2A(x)+y^2}{2h} = 1$, 得到

$\begin{eqnarray} I_i(h)& = & \frac{1}{8h^3} \oint_{L_h}(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x))y^7{\rm d}x {}\\ & = & \frac{1}{16h^4}\oint_{L_h}(2A(x)+y^2)(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x))y^7{\rm d}x {}\\ & = & \frac{1}{16h^4}(\oint_{L_h}2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x))y^7{\rm d}x{}\\ &&+ \oint_{L_h}(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x))y^9{\rm d}x). \end{eqnarray}$

$k = 9$, $F(x) = 2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x))$, 根据引理2.2, 得到

$$$\oint_{L_h}2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x))y^7{\rm d}x = \oint_{L_h}\widehat{G}_i(x)y^9{\rm d}x,$$$

$(3.8) $$(3.9) 式, 以及考虑到在每条轨线 L_h$$ \frac{2A(x)+y^2}{2h} = 1$, 得到

$\begin{eqnarray} I_i(h)& = & \frac{1}{16h^4}\oint_{L_h}(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x)+\widehat{G}_i(x))y^9{\rm d}x {}\\ & = & \frac{1}{32h^5}\oint_{L_h}(2A(x)+y^2)(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x)+\widehat{G}_i(x))y^9{\rm d}x {}\\ & = & \frac{1}{32h^5}(\oint_{L_h}2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x)+\widehat{G}_i(x))y^9{\rm d}x{}\\ &&+ \oint_{L_h}(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x)+\widehat{G}_i(x))y^{11}{\rm d}x). \end{eqnarray}$

$k = 11$, $F(x) = 2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x)+\widehat{G}_i(x))y^9$, 根据引理2.2, 得到

$$$\oint_{L_h}2A(x)(x^{i}+G_i(x)+\widetilde{G}_i(x)+\overline{G}_i(x)+\widehat{G}_i(x))y^9{\rm d}x = \oint_{L_h}\check{G}_i(x)y^{11}{\rm d}x,$$$

(ⅱ) 为了探讨$q(x, z) $$p_2(x, z) 是否有满足条件的公共根, 我们把 q(x, z)$$ p_2(x, z)$代入到文献[37]中的附录A的程序中

> with(RegularChains):

> with(ChainTools):

> with(SemiAlgebraicSetTools):

> sys : = [p_2(x, z), q(x, z)]:

> R : = PolynomialRing([x, z]):

> dec : = Triangularize(sys, R);

[regular_chain, regular_chain, regular_chain]

> L : = map(Equations, dec, R);

> C : = Chain([L[1][2], L[1][1]], Empty(R), R);

regular_chain

> RL : = RealRootIsolate(C, R, 'abserr' = 1/10^5);

[box, box, box, box, box, box]

> map(BoxValues, RL, R);

(ⅳ) 把$q(x, z) $$p_4(x, z) 代入到上面的算法程序中, 三角分解得到正则链 其中 p^r_5(x, z) = p_{51}(z)x+p_{52}(z) , 多项式 p_{51}(z) , p_{52}(z)$$ p^r_{6}(z)$的次数分别为$559$, $561 $$680 . 对正则链 [p^r_5, p^r_6] 中的多项式实根隔离后得到如下10个区间对 由此我们知道上述10个区间对中的最后三个区间对有满足不等式(3.13)根, 假设 (x_1, z_1) , (x_2, z_2)$$ (x_3, z_3) $$p_4(x, z)$$ q(x, z)$的公共根. 把$W[l_1(x), l_2(x), l_0(x), l_3(x)]$简记为$W_4(x, z(x))$, 并对它求导得到

$$$\dot{u} = v, \ \dot{v} = \frac{1 }{9604}{u \left( 4\, u+7 \right) \left( 16\, u-49 \right) ^{2}} + \varepsilon ( \frac{4a_0}{7}+\frac{16a_1}{49} u + \frac{64a_2}{343} u^2 +\frac{256}{2401} u^3)v.$$$

$$$\begin{array}{l} \dot{u} = v, \\ { } \dot{v} = -\frac{1}{161051}u(4u-11)(16u-121)^2+\varepsilon \frac{4}{11}(a_0+a_1(\frac{4}{11}u-1) \\ {\qquad} { } +a_2(\frac{4}{11}u-1)^2+(\frac{4}{11}u-1)^3)v, \end{array}$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Arnold V I . Arnold's Problems. Berlin: Springer-Verlag, 2004

Arnold V I .

Loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields

Funct Anal Appl, 1977, 11, 85- 92

Arnold V I .

Ten problems

Adv Soviet Math, 1990, 1, 1- 8

Chen F D , Li C Z , Llibre J , Zhang Z H .

A unified proof on the weak Hilbert 16th problem for $n=2$

J Differ Equations, 2006, 221 (2): 309- 342

Chen L , Ma X Z , Zhang G M , Li C Z .

Cyclicity of several quadratic reversible systems with center of genus one

J Appl Anal Comput, 2011, 1 (4): 439- 447

Christopher C , Li C Z . Limit Cycles of Differential Equations. Basel: Birkhäuser, 2007

Dumortier F , Li C Z .

Perturbations from an elliptic Hamiltonian of degree four: (Ⅰ) Saddle loop and two saddle cycle

J Differ Equations, 2001, 176, 114- 157

Dumortier F , Li C Z .

Perturbations from an elliptic Hamiltonian of degree four: (Ⅱ) Cuspidal loop

J Differ Equations, 2001, 175, 209- 243

Dumortier F , Li C Z .

Perturbation from an elliptic Hamiltonian of degree four: Ⅲ Global center

J Differ Equations, 2003, 188, 473- 511

Dumortier F , Li C Z .

Perturbations from an elliptic Hamiltonian of degree four: (Ⅳ) Figure eight-loop

J Differ Equations, 2003, 88, 512- 514

Écalle J . Introduction Aux Fonctions Analysables et Preuve Constructive De la Conjecture De Dulac. French: Hermann, 1992

Gavrilov L .

The infinitesimal 16th Hilbert problem in the quadratic case

Invent Math, 2001, 143, 449- 497

Grau M , Mañosas F , Villadelprat J .

A Chebyshev criterion for Abelian integrals

Trans Amer Math Soc, 2011, 363, 109- 129

Han M A .

Asymptotic expansions of Melnikov functions and limit cycle bifurcations

Int J Bifur Chaos, 2012, 22, 1250296

Han M A .

Existence of at most 1, 2, or 3 zeros of a melnikov function and limit cycles

J Differ Equations, 2001, 170 (2): 325- 343

Han M A .

On Hopf cyclicity of planar systems

J Math Anal Appl, 2000, 245 (2): 404- 422

Han M A , Wang Z , Zang H .

Limit cycles by Hopf and homoclinic bifurcations for near-Hamiltonian systems

Chinese Ann Math Series A, 2007, 62 (5): 3214- 3234

Han M A , Yang J M , Tarta A A , Gao Y .

Limit cycles near homoclinic and heteroclinic loops

J Dynam Differ Equat, 2008, 20, 923- 944

Han M A , Yang J M , Yu P .

Hopf bifurcations for near-Hamiltonian systems

Inter J Bifur Chaos, 2009, 19 (12): 4117- 4130

Hilbert D .

Mathematical problems

Bull Amer Math Soc, 1902, 8, 437- 479

Horozov E , Iliev I D .

On the number of limit cycles in perturbations of quadratic Hamiltonian systems

Proc London Math Soc, 1994, 69, 198- 224

Ilyashenko Y S . Finiteness Theorems for Limit Cycles. Providence RI: Amer Math Soc, 1991

Li C Z .

Abelian integrals and limit cycles

Qual Theory Dyn Syst, 2012, 11, 111- 128

Li C Z , Li W G .

Weak Hilbert's 16th problem and the relative research

Adv Math(CHINA), 2010, 39 (5): 513- 526

Li C Z , Zhang Z F .

A criterion for determining the monotonicity of the ratio of two abelian integrals

J Differ Equations, 1996, 124, 407- 424

Li C Z , Zhang Z H .

Remarks on 16th weak Hilbert problem for $n = 2$

Nonlinearity, 2002, 15, 1975- 1992

Li J B .

Hilberts 16th problem and bifurcations of planar polynomial vector fields

Inter J Bifur Chaos, 2003, 13, 47- 106

Liu C J .

The cyclicity of period annuli of a class of quadratic reversible systems with two centers

J Differ Equations, 2012, 252 (10): 5260- 5273

Liu C J , Xiao D M .

The monotonicity of the ratio of two Abelian integrals

Tran the Ame Math Soc, 2013, 365 (10): 5525- 5544

Manosas F , Villadelprat J .

Bounding the number of zeros of certain Abelian integrals

J Differ Equations, 2011, 251 (6): 1656- 1669

Petrov G S .

Number of zeros of complete elliptic integrals

Funct Anal App, 1984, 18 (2): 148- 149

Pontryagin L S .

On dynamical systems close to hamiltonian ones

Zh Exp Theor Phys, 1934, 4, 234- 238

Smale S .

Mathematical problems for the next century

Math Intelligencer, 1998, 2, 7- 15

Sun X B , Huang W T .

Bounding the number of limit cycles for a polynomial Liénard system by using regular chains

J Symb Comput, 2017, 79, 197- 210

Sun X B , Su J , Han M A .

On the number of zeros of Abelian integral for some Liénard system of type (4, 3)

Chaos Soliton Fract, 2013, 51, 1- 12

Sun X B , Wu K L .

The sharp bound on the number of zeros of Abelian integral for a perturbation of hyper-elliptic Hamiltonian system

Science in China, 2015, 45 (6): 751- 764

Sun X B , Xi H J , Zangeneh H R Z , Kazemi R .

Bifurcation of limit cycles in small perturbation of a class of Liénard systems

Int J Bifur Chaos, 2014, 24 (01): 1450004

Tian Y , Han M A .

Hopf and homoclinic bifurcations for near-Hamiltonian systems

J Differ Equat, 2017, 262 (4): 3214- 3234

Wang J H .

Estimate of the number of zeros of Abelian integrals for a perturbation of hyperelliptic Hamiltonian system with nilpotent center

Chaos Soliton Fract, 2012, 45, 1140- 1146

Wang J H , Xiao D M .

On the number of limit cycles in small perturbations of a class of hyperelliptic Hamiltonian systems with one nilpotent saddle

J Differ Equations, 2011, 250, 2227- 2243

Xiao D M .

Bifurcations on a five-parameter family of planar vector field

J Dyn Diff Equat, 2008, 20, 961- 980

Zhang Z F , Li C Z .

On the number of limit cycles of a class of quadratic Hamiltonian systems under quadratic perturbations

Adv in Math, 1997, 26, 445- 460

/

 〈 〉