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  数学物理学报  2015, Vol. 35 Issue (2): 381-394   PDF (##pdf_size## KB)    
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万阿英
衣凤岐
郑立飞
一类扩散的Gierer-Meindardt的模型的振动模式和Hopf分支分析
万阿英1, 衣凤岐2, 郑立飞3    
1. 呼伦贝尔学院数学系 内蒙古呼伦贝尔 021008;
2. 哈尔滨工业大学应用数学系 哈尔滨 150001;
3. 西北农林科技大学应用数学研究所 陕西杨凌 712100
摘要:该文研究了带有扩散项的Gierer-Meindardt模型Hopf分支分析. 证明了该系统的Hopf分支的存在性, 同时给出了决定分支方向和分支周期解稳定性的条件. 结果表明这个著名的模型具有复杂的振动模式. 最后, 数值模拟的结果验证该理论结果的正确性.
关键词Gierer-Meindardt 模型     Hopf分支     振动模式     稳定性     分支方向    
Hopf Bifurcation Analysis and Oscillatory Patterns of A Diffusive Gierer-Meinhardt Model
Wan Aying1, Yi Fengqi2, Zheng Lifei3    
1. Department of Mathematics, Hulunbeir College, Hailar, Inner Mongolia 021008;
2. Department of Applied Mathematics, Harbin Engineering University, Harbin, Heilongjiang 150001;
3. Applied Mathematical Institute, College of Science, Northwest A&F University, Shaan'xi Yangling 712100
Abstract: In this paper, a kind of diffusive Gierer-Meindardt model is considered. We performed detailed Hopf bifurcation analysis to this reaction diffusion systems. We not only prove the existence of Hopf bifurcations, but also derived the conditions to determine the bifurcation direction and the stability of the bifurcating periodic solutions. These results suggest the complex oscillatory patterns of this famous model. Computer simulations are included to support our theoretical analysis.
Key words: Gierer-Meindardt model     Hopf bifurcation     Oscillatory patterns     Stability     Bifurcation direction    
1 引言

本文研究一个众所周知的反应扩散系统,即 Gierer-Meinhardt 模型. 该系统是由Gierer 和Meinhardt在研究图灵模式的时候提出来的$^{[1]}$. 已经有很多作者研究了这个系统,获得了很多有意义的结果,见文献[7,8,9,10]. 在文献[7,8]中,陈,史和魏研究了带有基因表达时滞和活化剂 生产饱和的Gierer-Meinhardt 模型的平衡点全局吸引性,并完成了该情况下的Hopf分支分析. 在文献[9]里,徐和魏研究了一维 Schnakenberg反应扩散模型的Hopf分支. 在文献[10]里, 刘等研究了反应扩散Schnakenberg模型的分支问题. 本文主要研究扩散的Gierer Meinhardt模型的Hopf分支,我们主要研究如下系统 λbel0{Us=D1ΔU+ρUU2V1+K2U2μUU,xΩega,s>0,[3mm]Vs=D2ΔVρVU2V1+K2U2+σV,xΩega,s>0,U(x,0)=U0(x)0,V(x,0)=V0(x)0,xΩega,νU=νV=0,xΩega,s>0,(1.1) 这里,ΩegaRn (n1) 中是有光滑边界Ωega的一个有界连通集 (反应器); 假定反应器是闭的,且符合自反的Neumann边界条件 (这里,νUu的外法向导数). U(x,s)V(x,s) 表示s>0,xΩega时的成形基因浓度. 参数 ρU,ρV,μU,σV,和 K 都是非负常数.也就是说,假设按着U2V 的形式,UV 有有效的交流作用,当U比较大的时候,这个交流作用就会饱和, 且饱和常数K不等于零. 假定U以速率ρU增加促使V以速率 ρV被消耗. U以与U成正比的速度μU移除. 对变量V, 存在一个常数生产期σV. 对于这个系统的更多更详细的论述可参见 文献 [2] 及它的参考文献.

我们引入如下新的变量 u=KU,  v=μUVσV,  u0=KU0,  v0=μUV0σV, t=μUs,  a=ρUσVKμ2U,  b=ρVK2μU,  di=DiμU, 系统 (1.1) 写成如下系统 λbel1{utd1Δu=au2v1+u2u,xΩega,t>0,[3mm]vtd2Δv=bu2v1+u2+1,xΩega,t>0,[2mm]νu=νv=0,x,Ω;t>0,u(x,0)=u0(x)0,v(x,0)=v0(x)0,  xΩega,(1.2)

本文的目的是利用半线性偏微分方程组标准的Hpof分支定理,考虑空间齐次和非齐次周期轨道对唯一正平衡点的分支的存在性. 结果表明扩散Gierer-Meinhardt模型有丰富的振动模式. 过去几年的很多研究重点在于讨论不同类型Gierer-Meinhardt模型Hpof分支, 有兴趣的读者可以阅读文献[3,4,5] 了解更多的细节.

本文由如下几个部分构成. 在第二部分,我们给出一般反应扩散 方程组的几个Hpof定理. 在第三部分,我们对反应扩散系统的Hpof分 支做了分析,并给出计算机模拟支持我们的结论.

2 Hopf分支分析

本部分对反应扩散Gierer-Meinhardt模型 (1.1) 进行Hpof分支分析.

为了方便起见,我们复制系统 (1.2) 为如下方程 λbel2{utd1Δu=au2v1+u2u,x,t>0,[3mm]vtd2Δv=bu2v1+u2+1,x,t>0,[2mm]νu=νv=0,x,t>0,u(x,0)=u0(x)0,v(x,0)=v0(x)0,  xega,(2.1)

显然,系统 (2.1) 有唯一正平衡解 (u,v):=(,(2+1)/a), 给定 a,:=a/b 相应的空间定义域为 Ωega=(0,π),R+. 系统(2.1)在平衡状态(u,v) 的线性算子为 J():=(d12x2+122+1,  a22+12(2+1),  d22x2a2+1),(2.2) 其中 b:=a/λ. 在以下的讨论中,用 λ 作为分支参数,并假设 a>0.

根据文献 [6] 可知,J(λ) 的特征值由如下算子 Jn(λ)给出 Jn(λ)=:(d1n22+1λ2λ2+1,  aλ2λ2+12λ(λ2+1),  d2n22aλλ2+1),(2.3) 其特征方程为 β2βTn(λ)+Dn(λ)=0 (n=0,1,2),(2.4) 其中, λbel3{Tn(λ):=trJn=1aλλ2λ2+1(d1+d2)n22,[3mm]Dn(λ):=detJn=d1d2n44+(ad1λλ2+1d2(1λ2)λ2+1)n22+aλλ2+1.(2.5)

下面构建Hpof分支发生的潜在临界点元素.

存在 nN{0},使得 Tn(λ)=0,Dn(λ)>0;Tj(λ)0,Dj(λ)0,  jn;

设靠近虚轴的唯一一对复特征根为 α(λ)±iΩega(λ),它穿过虚轴的横截条件为 α(λ)0.(2.63)

由式 (2.5),Tn(λ)<0 和条件 Dn(λ)>0 ,λ1, 其中后面这个条件意味着唯一正平衡解 (λ,(λ2+1)/aλ) 是局部渐近稳定的. 因此,任何一个潜在的分支点λ0一定位于区间 (0,1). 对于区间 (0,1)上的任何一个分支点 λ0, αpha(λ)±iΩega(λ) 一定是 Ln(λ) 的特征值,于是 αpha(λ)=A(λ)2(d1+d2)n222,      Ωega(λ)=Dn(λ)αpha2(λ),(2.7) 其中,A(λ)=1aλλ21+λ2. 计算表明, 对于所有的0<λ<1,下列等式总是成立的 αpha(λ)=A(λ)2=aλ24λa2(1+λ2)2<0.(2.8)

因此,横截条件总是成立的.

对于任意的 >0,λH0:=(a+a2+4)/2(0,1) 是Hpof分支点. 对应于空间齐次周期解的Hpof分支,这和已经研究过的常微分方程组的结论是一 样的. 显然, 对于任意的 >0,空间齐次周期解的Hpof分支唯一λ值记为λH0.

下面,当 n1时,寻找空间非齐次Hpof分支. 注意到 A(0)=1,A(λH0)=0,A(λ)=aλ24λa(1+λ2)2<0,   λ(0,λH0).(2.9) 定义 λbel4n=nd1+d2,  nΝ. 则对n<<n+1,及1jn,定义 λHj 为方程 A(λ)=(d1+d2)j22. 这些n 值满足 0<λHn<<λH2<λH1<λH0.

显然,对ij,Tj(λHj)=0 ,且 Ti(λHj)0. 现在,我们仅需要证明对所有的 iN0, Di(λHj,±)0,且 Dj(λHj,±)>0.

下面,推导参数满足的使得 Di(λHj,±)>0的条件. 假设 d1d2,有 λbel5Di(λHj)=d1d2i44+(ad1λHj(λHj)2+1d2(1(λHj)2)(λHj)2+1)i22+aλHj(λHj)2+1d1d2i44+d2(aλHj(λHj)2+1(1(λHj)2)(λHj)2+1)i22+aλHj(λHj)2+1=d1d2i44+aλHj(λHj)2+1>0.(2.10)

综上,得到本文的主要结果:

定理2.1假定常数 d1,d2,a>0 满足 d1d2,及n 由系统 (2.9) 定义. 则对于任意的 n<<n+1,存在 n 个点λHj(),  1jn,满足 0<λHn<<λH2<λH1<λH0. 使得系统 (1.1) 在 λ=λHjλ=λH0 具有Hpof分支. 而且,

(1)~ 在 λ=λH0 产生的Hpof周期解是空间齐次的, 这个结论和相应的常微分方程系统的周期解结论一致;

(2)~ 在 λ=λHj 产生的分支周期解是空间非齐次的.

例2.2Ωega=(0,π),d1=4,d2=1,a=2.25,通过计算可知,λH0=0.3802,n:=n(d1+d2)=5n2.2361n, Di(λHj)>0.

(1)~ 设 =3,则 (1,2](2.236,4.470]. 解方程 A(λ)=(d1+d2)n2/2,可知 λH1=0.1760. 于是Hpof分支点集为 λmbda1={λH1,λH0}={0.1760,0.3802}.

(2)~ 设 =5,则 (2,3](4.470,6.7082]. 同样, 可得 λH10.3057,λH2=0.0833. 则Hpof分支点集为 λmbda1={λH1,λH2λH0}={0.0833,0.3057,0.3802}.

3 分支方向和分支周期解的稳定性

定理3.10<a<2/3时,系统 (1.2) 在点λ=λH0的分支 是超临界的且空间齐次的分支周期解是不稳定的; 当 a>2/3 时, 在点λ=λH0的Hpof分支是次临界的,空间齐次的分支周期解是稳定的.

根据文献[6,定理2.1],为了确定分支周期解的稳定性和分支方向, 需要计算 Re(c1(λH0)), 其中c1(λH0)是按文献[6]定义的. 当 λ=λH0, 为方便起见,记 λH0=λ0. 于是 q:=(a0b0)=(11aλ20[λ201+iΩega0(1+λ20)]),(3.1) q:=(a0b0)=(1+iΩega02πiΩega0λ02π),(3.2) 其中,Ωega0=aλ0(λ0)2+1.

ˆu=uuλ,ˆv=vvλ,并仍记 ˆuˆvuv,系统 (1.2) 变为 λbel9{utd1uxx=a(u+λ)2(v+(λ2+1)/aλ)1+(u+λ)2(u+λ),xΩ,t>0,[3mm]vtd2vxx=b(u+λ)2(v+(λ2+1)/aλ)1+(u+λ)2+1,xΩ,t>0,[2mm]νu=νv=0,xΩ,t>0,u(x,0)=u0(x)0,v(x,0)=v0(x)0,xΩega.(3.3)

f(λ,u,v)=a(u+λ)2(v+(λ2+1)/aλ)1+(u+λ)2(u+λ),[3mm]g(λ,u,v)=b(u+λ)2(v+(λ2+1)/aλ)1+(u+λ)2+1.(3.4)

在点(λ0,0,0),函数fg的所有偏导数为 fuu f'''_{uuu}(λ_0,0,0)=\frac{24(λ_0^2-1)}{(1+λ_0^2)^3},\;\;\;\;f'''_{uuv}(λ_0,0,0)=\frac{2a(1-3λ_0^2)}{(1+λ_0^2)^3}; g''_{uu}(λ_0,0,0)=\frac{-2+6λ_0^2}{λ_0^2(1+λ_0^2)^2},\;\;\;\;g''_{uv}(λ_0,0,0)=\frac{-2a}{(1+λ_0^2)^2}; g'''_{uuu}(λ_0,0,0)=\frac{24(1-λ_0^2)}{λ_0(1+λ_0^2)^3},\;\;\;\;g'''_{uuv}(λ_0,0,0)=\frac{-2a(1-3λ_0^2)}{λ_0(1+λ_0^2)^3}; f''_{vv}(λ_0,0,0)=f'''_{uvv}(λ_0,0,0)=f'''_{vvv}(λ_0,0,0)=0, g''_{vv}(λ_0,0,0)=g'''_{uvv}(λ_0,0,0)=g'''_{vvv}(λ_0,0,0)=0. 于是 c_0=\frac{-2+4{\rm i}Ωega_0}{λ_0(1+λ_0^2)}=-λ_0d_0,\;\;\;\; e_0=\frac{-2}{λ_0(1+λ_0^2)}=-λ_0f_0, g_0=\frac{1}{(1+λ_0^2)^3}\bigg[24(λ_0^2-1) +\frac{2}{λ_0^2}(1-3λ_0^2)[{\rm i}Ωega_0(1+λ_0^2) -3(1-λ_0^2)]\bigg]=-λ_0h_0. Q_{qq}=\left(\begin{array}{c} c_0\\ d_0\\ \end{array}\right)=\frac{2-4{\rm i}Ωega_0}{λ_0^2(1+λ_0^2)}\left(\begin{array}{c} -λ_0\\ 1\\ \end{array}\right),\;\;\;\;Q_{q\bar{q}}=\left(\begin{array}{c} e_0\\ f_0\\ \end{array}\right)=\frac{2}{λ_0^2(1+λ_0^2)}\left(\begin{array}{c} -λ_0\\ 1 \end{array}\right), C_{qq\bar{q}}=\left(\begin{array}{c} g_0\\ h_0\\ \end{array}\right)=\frac{-1}{λ_0(1+λ_0^2)^3} \bigg[24(λ_0^2-1)+\frac{2}{λ_0^2}(1-3λ_0^2) [{\rm i}Ωega_0(1+λ_0^2)-3(1-λ_0^2)]\bigg]\left(\begin{array}{c} -λ_0\\ 1 \end{array}\right), \begin{equation} \begin{array}{l} λngle q^*,Q_{qq}\rangle=\frac{-1+2{\rm i}Ωega_0}{λ_0(1+λ_0^2)},\;\; λngle q^*,Q_{q\bar{q}}\rangle=\frac{-1}{λ_0(1+λ_0^2)},\;\; λngle \bar q^*,Q_{qq}\rangle=\frac{-1+2{\rm i}Ωega_0}{λ_0(1+λ_0^2)},\\[3mm] λngle q^*,C_{qq\bar{q}}\rangle=\frac{1}{(1+λ_0^2)^3} \bigg[12(λ_0^2-1)+\frac{1}{λ_0^2}(1-3λ_0^2) [{\rm i}Ωega_0(1+λ_0^2)-3(1-λ_0^2)]\bigg]. \end{array}(3.5) \end{equation}

因此,可以直接计算得出如下结果 \begin{equation} \begin{array}{l} H_{20}=\left(\begin{array}{c} c_0\\ d_0\\ \end{array}\right)-λngle q^*,Q_{qq}\rangle\left(\begin{array}{c} a_0\\ b_0\\ \end{array}\right)-λngle \overline{q}^*,Q_{qq}\rangle\left(\begin{array}{c} \overline{a_0}\\ \overline{b_0}\\ \end{array}\right)=0,\\ [6mm] H_{11}=\left(\begin{array}{c} e_0\\ f_0\\ \end{array}\right)-λngle q^*,Q_{q\overline{q}}\rangle\left(\begin{array}{c} a_0\\ b_0\\ \end{array}\right)-λngle \overline{q}^*,Q_{q\overline{q}}\rangle\left(\begin{array}{c} \overline{a_0}\\ \overline{b_0}\\ \end{array}\right)=0, \end{array}(3.6) \end{equation} 其中 w_{20}=w_{11}=0. 于是 \begin{equation} λngle q^*,Q_{w_{11},q}\rangle=λngle q^*,Q_{w_{20},\overline{q}}\rangle=0.(3.7) \end{equation} 从而, \begin{equation} {\rm Re}(c_1(λ_0^H)) ={\rm Re}\left\{\frac{\rm i}{2Ωega_0}λngle q^*,Q_{qq}\rangle\cdotλngle q^*,Q_{q\overline{q}}\rangle+\frac{1}{2}λngle q^*,C_{q,q,\overline{q}}\rangle\right\} =\frac{2λ_0-a}{2λ_0(1+λ_0^2)^2}.(3.8) \end{equation}

注意到 λ_0:=λ_0^H=(-a+\sqrt{a^2+4})/2, 容易得,若 0<a<2/\sqrt{3}, {\rm Re}(c_1(λ_0^H))>0, 若 a>2/\sqrt{3}, {\rm Re}(c_1(λ_0^H))<0. 由文献[6,定理2.1],即可知道结论成立, 证明完毕.

例3.2d_1=4,d_2=1,a=2.25>\sqrt{3}/2. 此时,λ_0^H=0.3802, 且 {\rm Re}(c_1(λ_0^H))\approx-1.4954<0, 则相应的分支周期解是稳定的. 由于 αpha'(λ_0^H)=A'(λ_0^H)/2=-1.3151<0, 我们有 \frac{1}{αpha'(λ_0^H)}{\rm Re}(c_1(λ_0^H))>0. 这样在点λ=0.3802处的分支方向是次临界的,该结果见图 1.

图(a) u(x,t); (b) v(x,t). 系统 {\rm(1.1)}在点 (u_λ,v_λ)\approx(0.3802,1.3379) 经历了一个空间齐次的Hpof分支周期轨道. 初始数据是 (u_0,v_0)=(0.40,1.35)}

定理3.3 对于系统 (1.2),如果条件\frac{1}{αpha'(λ_j^H) }{\rm Re}c_1(λ_j^H)<0 (或>0),成立,则位于点λ=λ_j^H, j\neq0 的Hpof分支是超临界的(或者次临界的),并且空间非齐次分支周期 解是不稳定的.

λ=λ_j^H (j\in {\Bbb N})时,为方便起见,记 λ_j^H=λ_j. 由此我们得到 q:=\cos\frac{j}{\ell}x(a_{j},b_{j})^T =\cos\frac{j}{\ell}x\left(1, \frac{1+λ_j^2}{aλ_j^2}({\rm i}Ωega_0-\frac{d_2j^2}{\ell^2}-\frac{aλ_j}{1+λ_j^2})\right)^T, \begin{eqnarray*} q^*&:=&\cos\frac{j}{\ell}x\left( a^*_{j},b^*_{j}\right)^T\\ &=&\cos\frac{j}{\ell}x \left( -\frac{\rm i}{\ell\piΩega_0}({\rm i}Ωega_0+\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}), \frac{λ_j(1+λ_j^2){\rm i}}{2\ell\piΩega_0} \bigg({\rm i}Ωega_0+\frac{d_2j^2}{\ell^2}+\frac{aλ_j} {1+λ_j^2}\bigg)^2\right)^T, \end{eqnarray*} 这里 Ωega_0=\left(\frac{2aλ_j}{(1+λ_j^2)^2}- \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)^2\right)^{1/2}.

经过计算,得 \left[2{\rm i}Ωega_0I-L_{2j}(λ_{j})\right]^{-1}=(αpha_1+αpha_2{\rm i})^{-1} \left(\begin{array}{cc} 2{\rm i}Ωega_0+\frac{4d_2{j}^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}&\frac{aλ_j^2}{1+λ_j^2}\\ [3mm] -\frac{2}{λ_j(1+λ_j^2)} & 2{\rm i}Ωega_0+\frac{4d_1{j}^2}{\ell^2}-\frac{1-λ_j^2}{1+λ_j^2} \end{array}\right), 其中 αpha_1:=-4Ωega_0^2+ \bigg(\frac{4d_1j^2}{\ell^2}-\frac{1-λ_j^2}{1+λ_j^2}\bigg) \bigg(\frac{4d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)+\frac{2aλ_j}{(1+λ_j^2)^2},αpha_2=\frac{6Ωega_0(d_1+d_2){j}^2}{\ell^2}; \left[2{\rm i}Ωega_0I-L_{0}(λ_{j})\right]^{-1}=(αpha_3+αpha_4{\rm i})^{-1} \left(\begin{array}{cc} 2{\rm i}Ωega_0+\frac{aλ_j}{1+λ_j^2} ~~&\frac{aλ_j^2}{1+λ_j^2}\\ [3mm] -\frac{2}{λ_j(1+λ_j^2)} ~~& 2{\rm i}Ωega_0-\frac{1-λ_j^2}{1+λ_j^2} \end{array}\right), 其中 αpha_3:=-4Ωega_0^2+\frac{aλ_j}{1+λ_j^2},\;\;αpha_4:= -\frac{2Ωega_0(d_1+d_2){j}^2}{\ell^2}. \left[L_{2j}(λ_{j})\right]^{-1}=(αpha_5)^{-1} \left(\begin{array}{cc} -\frac{4d_2j^2}{\ell^2}-\frac{aλ_j}{1+λ_j^2} ~~&-\frac{aλ_j^2}{1+λ_j^2}\\ [3mm] \frac{2}{λ_j(1+λ_j^2)} ~~& -\frac{4d_1j^2}{\ell^2}+\frac{1-λ_j^2}{1+λ_j^2} \end{array}\right), \left[L_{0}(λ_{j})\right]^{-1}=(αpha_6)^{-1} \left(\begin{array}{cc} -\frac{aλ_j}{1+λ_j^2} ~~&-\frac{aλ_j^2}{1+λ_j^2}\\[3mm] \frac{2}{λ_j(1+λ_j^2)} ~~& \frac{1-λ_j^2}{1+λ_j^2} \end{array}\right), 其中 αpha_5:=\bigg(\frac{4d_1j^2}{\ell^2}-\frac{1-λ_j^2} {1+λ_j^2}\bigg)\bigg (\frac{4d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)+\frac{2aλ_j}{(1+λ_j^2)^2},αpha_6:=\frac{aλ_j}{(1+λ_j^2)}.

通过计算,可得函数fg 在点(λ_j,0,0)的所有偏导数 f''_{uu}=\frac{2-6λ_j^2}{λ_j(1+λ_j^2)^2},\;\;\;\;f''_{uv}=\frac{2aλ_j}{(1+λ_j^2)^2}; f'''_{uuu}=\frac{24(λ_j^2-1)}{(1+λ_j^2)^3},\;\;\;\;f'''_{uuv}=\frac{2a(1-3λ_j^2)}{(1+λ_j^2)^3}; g''_{uu}=\frac{-2+6λ_j^2}{λ_j^2(1+λ_j^2)^2},\;\;\;\;g''_{uv}=\frac{-2a}{(1+λ_j^2)^2}; g'''_{uuu}=\frac{24(1-λ_j^2)}{λ_j(1+λ_j^2)^3},\;\;\;\;g'''_{uuv}=\frac{-2a(1-3λ_j^2)}{λ_j(1+λ_j^2)^3}; f''_{vv}=f'''_{uvv}=f'''_{vvv}=0,~~g''_{vv}=g'''_{uvv}=g'''_{vvv}=0. c_j=f_{uu}+2f_{uv} \bigg({\rm i}Ωega_0-\frac{d_2j^2}{\ell^2}-\frac{aλ_j}{1+λ_j^2} \bigg)=-λ_jd_j, e_j=f_{uu}-2f_{uv}\frac{1+λ_j^2}{aλ_j^2} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)=-λ_jf_j, g_j=f_{uuu}+2f_{uuv}\frac{1+λ_j^2}{aλ_j^2} \bigg({\rm i}Ωega_0-\frac{3d_2j^2}{\ell^2}-\frac{3aλ_j} {1+λ_j^2}\bigg)=-λ_jh_j. 由文献 [6] Q_{qq}=\cos^2\frac{j}{\ell}x\left(\begin{array}{c} c_j\\ d_j \end{array}\right) ,\;\;\;\; Q_{q\bar{q}}=\cos^2\frac{j}{\ell}x\left(\begin{array}{c} e_j\\ f_j \end{array}\right) ,\;\;\;\; C_{qq\bar{q}}=\cos^3\frac{j}{\ell}x\left(\begin{array}{c} g_j\\ h_j \end{array}\right), λngle q^*,Q_{qq}\rangle=λngle q^*,Q_{q\overline{q}}\rangle=λngle \overline{q}^*,Q_{qq}\rangle=λngle \overline{q}^*,\;Q_{\overline{q}\overline{q}}\rangle=0. 这样, H_{20}=Q_{qq}=\cos^2\frac{j}{\ell}x\left(\begin{array}{c} c_{j}\\ d_{j}\\ \end{array}\right)=\left(\frac{1}{2}\cos\frac{2j}{\ell}x+\frac{1}{2}\right)\left(\begin{array}{c} c_{j}\\ d_{j}\\ \end{array}\right), H_{11}=Q_{q\bar{q}}=\cos^2\frac{j}{\ell}x\left(\begin{array}{c} e_{j}\\ f_{j}\\ \end{array}\right)=\left(\frac{1}{2}\cos\frac{2j}{\ell}x+\frac{1}{2}\right)\left(\begin{array}{c} e_{j}\\ f_{j}\\ \end{array}\right), \begin{eqnarray*} w_{20}&=&[2{\rm i}Ωega_0I-L(λ_j)]^{-1}H_{20}\\ &=&\left[\frac{[2{\rm i}Ωega_0I-L_{2j}(λ_j)]^{-1}}{2}\cos\frac{2j}{\ell}x+\frac{[2{\rm i}Ωega_0I-L_{0}(λ_j)]^{-1}}{2}\right]\cdot\left(\begin{array}{c} c_j\\ d_j\\ \end{array}\right)\\ &=&\left(\begin{array}{c} \xi\\ \eta\\ \end{array}\right)\cos\frac{2jx}{\ell} +\left(\begin{array}{c} \tau\\ \chi \end{array}\right), \end{eqnarray*} \begin{eqnarray*} w_{11}&=&-[L(λ_j)]^{-1}H_{11}\\ &=&-\left[\frac{[L_{2j}(λ_j)]^{-1}}{2}\cos\frac{2j}{\ell}x+\frac{[L_{0}(λ_j)^{-1}]}{2}\right]\cdot\left(\begin{array}{c} e_j\\ f_j\\ \end{array}\right)\\ &=&\left(\begin{array}{c} \tilde{\xi}\\ \tilde{\eta}\\ \end{array}\right)\cos\frac{2jx}{\ell} +\left(\begin{array}{c} \tilde{\tau}\\ \tilde{\chi} \end{array}\right), \end{eqnarray*} \xi=\frac{1}{2}(αpha_1+{\rm i}αpha_2)^{-1}\left[\left(2{\rm i}Ωega_0+\frac{4d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\right)c_j+\frac{aλ_j^2}{1+λ_j^2}d_j\right], \eta=\frac{1}{2}(αpha_1+{\rm i}αpha_2)^{-1}\left[-\frac{2}{λ_j(1+λ_j^2)}c_j+\left(2{\rm i}Ωega_0+\frac{4d_2j^2}{\ell^2}-\frac{1-λ_j^2}{1+λ_j^2}\right)d_j\right], \tau=\frac{1}{2}(αpha_3+{\rm i}αpha_4)^{-1}\left[\left(2{\rm i}Ωega_0+\frac{aλ_j}{1+λ_j^2}\right)c_j+\frac{aλ_j^2}{1+λ_j^2}d_j\right], \chi=\frac{1}{2}(αpha_3+{\rm i}αpha_4)^{-1}\left[-\frac{2}{λ_j(1+λ_j^2)}c_j+\left(2{\rm i}Ωega_0-\frac{1-λ_j^2}{1+λ_j^2}\right)d_j\right], \widetilde{\xi}=\frac{1}{2αpha_5}\left[\left(\frac{4d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\right)e_j+\frac{aλ_j^2}{1+λ_j^2}f_j\right], \widetilde{\eta}=\frac{1}{2αpha_5}\left[-\frac{2}{λ_j(1+λ_j^2)}e_j+\left(\frac{4d_1j^2}{\ell^2}-\frac{1-λ_j^2}{1+λ_j^2}\right)f_j\right], \widetilde{\tau}=\frac{1}{2αpha_6}\left(\frac{aλ_j}{1+λ_j^2}e_j+\frac{aλ_j^2}{1+λ_j^2}f_j\right), \widetilde{\chi}=\frac{1}{2αpha_6}\left(-\frac{2}{λ_j(1+λ_j^2)}e_j-\frac{1-λ_j^2}{1+λ_j^2}f_j\right). 则有 Q_{w_{20},\overline{q}} =\left(\begin{array}{c} f_{uu}\xi+f_{uv}\eta+f_{uv}\overline{b_j}\xi\\ g_{uu}\xi+g_{uv}\eta+g_{uv}\overline{b_j}\xi\\ \end{array}\right)\cos\frac{jx}{\ell}\cos\frac{2jx}{\ell} +\left(\begin{array}{c} f_{uu}\tau+f_{uv}\chi+f_{uv}\overline{b_j}\tau\\ g_{uu}\tau+g_{uv}\chi+g_{uv}\overline{b_j}\tau\\ \end{array}\right)\cos\frac{jx}{\ell}, Q_{w_{11},q} =\left(\begin{array}{c} f_{uu}\widetilde{\xi}+f_{uv}\widetilde{\eta}+f_{uv}b_j\widetilde{\xi}\\ g_{uu}\widetilde{\xi}+g_{uv}\widetilde{\eta}+g_{uv}b_j\widetilde{\xi}\\ \end{array}\right)\cos\frac{2jx}{\ell}\cos\frac{jx}{\ell} +\left(\begin{array}{c} f_{uu}\widetilde{\tau}+f_{uv}\widetilde{\chi}+f_{uv}b_j\widetilde{\tau}\\ g_{uu}\widetilde{\tau}+g_{uv}\widetilde{\chi}+f_{uv}b_j\widetilde{\tau}\\ \end{array}\right)\cos\frac{jx}{\ell}, 注意对任意的 j\in {\Bbb N}, \int_0^{\ell\pi}\cos^2\frac{jx}{\ell}{\rm d}x=\frac{1}{2}\ell\pi,\int_0^{\ell\pi}\cos\frac{2jx}{\ell}\cos^2\frac{jx}{\ell}{\rm d}x=\frac{1}{4}\ell\pi, \int_0^{\ell\pi}\cos^4\frac{{j}x}{\ell}{\rm d}x=\frac{3}{8}\ell\pi,\begin{eqnarray*} λngle q^*,Q_{w_{20},\overline{q}}\rangle &=&\frac{\ell\pi}{4}\left\{\overline{a_j^*}(f_{uu}\xi+f_{uv}\eta+f_{uv}\xi\overline{b_j})+\overline{b_j^*}(g_{uu}\xi+g_{uv}\eta+g_{uv}\xi\overline{b_j})\right\}\\ &&+\frac{\ell\pi}{2}\left\{\overline{a_j^*}(f_{uu}\tau+f_{uv}\chi+f_{uv}\tau\overline{b_j})+\overline{b_j^*}(g_{uu}\tau+g_{uv}\chi+g_{uv}\tau\overline{b_j})\right\}, \end{eqnarray*} \begin{eqnarray*} λngle q^*,Q_{w_{11},q}\rangle &=&\frac{\ell\pi}{4}\left\{\overline{a_j^*}(f_{uu}\widetilde{\xi}+f_{uv}\widetilde{\eta}+f_{uv}\widetilde{\xi}b_j) +\overline{b_j^*}(g_{uu}\widetilde{\xi}+g_{uv}\widetilde{\eta}+g_{uv}\widetilde{\xi}b_j)\right\}\\ &&+\frac{\ell\pi}{2}\left\{\overline{a_j^*}(f_{uu}\widetilde{\tau}+f_{uv}\widetilde{\chi}+f_{uv}\widetilde{\tau}b_j) +\overline{b_j^*}(g_{uu}\widetilde{\tau}+g_{uv}\widetilde{\chi}+g_{uv}\widetilde{\tau}b_j)\right\}, \end{eqnarray*} λngle q^*,C_{q,q,\overline{q}}\rangle =\frac{3\ell\pi}{4}(\overline{a_j^*}g_j +\overline{b_j^*}h_j). \begin{eqnarray*} &&{\rm Re}λngle q^*,Q_{w_{20},\bar{q}}\rangle \\ &=&\frac{1}{4}\bigg\{f _{uu}(\xi_R+2\tau_R) +f_{uv}(\eta_R+2\chi_R) -f_{uv}\frac{1+λ_j^2}{aλ_j^2} \bigg[(\xi_R+2\tau_R) \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \\ && -(\xi_I+2\tau_I)Ωega_0\bigg]\bigg\} -\frac{1}{4Ωega_0} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)\\ &&\cdot\bigg\{f_{uu}(\xi_I+2\tau_I)+f_{uv}(\eta_I+2\chi_I) -f_{uv}\frac{1+λ_j^2}{aλ_j^2}\bigg[(\xi_I+2\tau_I) \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \\ && +(\xi_R+2\tau_R)Ωega_0\bigg]\bigg\} -\frac{λ_j(1+λ_j^2)}{4} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)\\ &&\cdot\bigg\{g_{uu}(\xi_R+2\tau_R)+g_{uv}(\eta_R+2\chi_R) -g_{uv}\frac{1+λ_j^2}{aλ_j^2}\bigg[(\xi_R+2\tau_R) \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)\\ &&- (\xi_I+2\tau_I)Ωega_0\bigg]\bigg\} +\frac{λ_j(1+λ_j^2)}{8Ωega_0} \bigg[\bigg(\frac{d_2j^2}{\ell^2}+ \frac{aλ_j}{1+λ_j^2}\bigg)^2-Ωega_0^2\bigg]\\ &&\cdot\bigg\{g_{uu}(\xi_I+2\tau_I) +g_{uv}(\eta_I+2\chi_I)\\ &&-g_{uv}\frac{1+λ_j^2}{aλ_j^2} \bigg[(\xi_I+2\tau_I)\bigg(\frac{d_2j^2}{\ell^2}+ \frac{aλ_j}{1+λ_j^2}\bigg) +(\xi_R+2\tau_R)Ωega_0\bigg]\bigg\}, \end{eqnarray*} \begin{eqnarray*} &&{\rm Re}λngle q^*,Q_{w_{11},q}\rangle \\ &=&\frac{1}{4}\bigg[f _{uu}(\tilde{\xi}+2\tilde{\tau})+f_{uv}(\tilde{\eta}+2\tilde{\chi}) -f_{uv}(\tilde{\xi}+2\tilde{\tau}) \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg(\frac{1+λ_j^2}{aλ_j^2}+1\bigg) \bigg]\\ &&-\frac{λ_j(1+λ_j^2)}{4}\bigg[g _{uu}(\tilde{\xi}+2\tilde{\tau})+g_{uv} (\tilde{\eta}+2\tilde{\chi})\bigg] \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)\\ &&+\frac{λ_j(1+λ_j^2)}{4}g_{uv}(\tilde{\xi}+2\tilde{\tau}) \bigg[\bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)^2 \bigg(\frac{1+λ_j^2}{aλ_j^2}+1 \bigg)-\frac{aλ_j} {(1+λ_j^2)^2} \bigg], \end{eqnarray*} \begin{eqnarray*} &&{\rm Re}λngle q^*,C_{q,q,\overline{q}}\rangle\\ &=&\frac{3}{8}\bigg[f _{uuu}-6f_{uuv}\frac{1+λ_j^2}{aλ_j^2} \bigg( \frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)\bigg] \bigg[1+(1+λ_j^2)\bigg( \frac{d_2j^2}{\ell^2}+\frac{2aλ_j}{1+λ_j^2}\bigg)\bigg]\\ &&-f_{uuv}\frac{3(1+λ_j^2)}{4aλ_j^2} \bigg[\frac{d_2j^2}{\ell^2}+\frac{2aλ_j}{1+λ_j^2} -Ωega_0^2(1+λ_j^2)\bigg]. \end{eqnarray*}

\Gamma=\xi,\eta,\tau,\chi,定义 \Gamma_R:={\rm Re} \Gamma\Gamma_I:={\rm Im} \Gamma. 进一步, \begin{eqnarray*} \xi_R&=&\frac{1}{2(αpha_1^2+αpha_2^2)} \bigg\{2\bigg[f_{uu}+2f_{uv} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg]\bigg(αpha_2Ωega_0+\frac{2αpha_1d_2j^2}{\ell^2}\bigg) \\ && +4f_{uv}\bigg(\frac{2αpha_2d_2j^2}{\ell^2}-αpha_1Ωega_0^2 \bigg)\bigg\}, \end{eqnarray*} \begin{eqnarray*} \xi_I&=&\frac{1}{2(αpha_1^2+αpha_2^2)} \bigg\{2\bigg[f_{uu}+2f_{uv}\bigg(\frac{d_2j^2}{\ell^2} +\frac{aλ_j}{1+λ_j^2}\bigg)\bigg] \bigg(αpha_1Ωega_0-\frac{2αpha_2d_2j^2}{\ell^2}\bigg) \\ &&+4Ωega_0f_{uv}\bigg(\frac{2αpha_1d_2j^2} {\ell^2}-αpha_2Ωega_0\bigg)\bigg\}, \end{eqnarray*} \begin{eqnarray*} \eta_R&=&-\frac{1}{2λ_j(αpha_1^2+αpha_2^2)} \bigg\{\bigg[f_{uu}+2f_{uv} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg] \bigg[2αpha_2Ωega_0+αpha_1 \bigg(1+\frac{4d_2j^2}{\ell^2}\bigg)\bigg]\bigg\}\\ &&-\frac{Ωega_0f_{uv}}{λ_j(αpha_1^2+αpha_2^2)} \bigg[αpha_2\bigg(1+\frac{4d_2j^2}{\ell^2}\bigg) -2αpha_1Ωega_0\bigg], \end{eqnarray*} \begin{eqnarray*} \eta_I&=&-\frac{1}{2λ_j(αpha_1^2+αpha_2^2)} \bigg\{\bigg[f_{uu}+2f_{uv}\bigg(\frac{d_2j^2} {\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)\bigg] \bigg[2αpha_1Ωega_0-αpha_2 \bigg(1+\frac{4d_2j^2}{\ell^2}\bigg)\bigg]\bigg\}\\ &&-\frac{Ωega_0f_{uv}}{λ_j(αpha_1^2+αpha_2^2)} \bigg[αpha_1\bigg(1+\frac{4d_2j^2}{\ell^2}\bigg) -2αpha_2Ωega_0\bigg], \end{eqnarray*} \begin{eqnarray*} \tau_R=-\frac{Ωega_0}{(αpha_3^2+αpha_4^2)} \bigg\{2Ωega_0αpha_3f_{uv}-αpha_4\bigg[f_{uu}+2f_{uv} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg]\bigg\}, \end{eqnarray*} \begin{eqnarray*} \tau_I=-\frac{Ωega_0}{(αpha_3^2+αpha_4^2)} \bigg\{2Ωega_0αpha_4f_{uv}-αpha_3\bigg[f_{uu}+2f_{uv} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg]\bigg\}, \end{eqnarray*} \begin{eqnarray*} \chi_R&=&-\frac{1}{2λ_j(αpha_3^2+αpha_4^2)} \bigg\{(2Ωega_0αpha_4+αpha_3)\bigg[f_{uu}+2f_{uv} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg] \\ &&+2Ωega_0f_{uv}(αpha_4-2Ωega_0αpha_3)\bigg\}, \end{eqnarray*} \begin{eqnarray*} \chi_I&=&-\frac{1}{2λ_j(αpha_3^2+αpha_4^2)} \bigg\{(2Ωega_0αpha_3-αpha_4) \bigg[f_{uu}+2f_{uv} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2} \bigg)\bigg] \\ &&+2Ωega_0f_{uv}(αpha_3+2Ωega_0αpha_4)\bigg\}, \end{eqnarray*} 由此,得到 \begin{eqnarray*} &&{\rm Re}(c_1(λ_{j}))\\ &=&{\rm Re}λngle q^*,Q_{w_{11},q}\rangle +\frac{1}{2}{\rm Re}λngle q^*,Q_{w_{20},\overline{q}}\rangle+\frac{1}{2}{\rm Re}λngle q^*,C_{q,q,\overline{q}}\rangle\\ &=&\frac{1}{4}\bigg[f _{uu}(\tilde{\xi}+2\tilde{\tau})+f_{uv}(\tilde{\eta}+2\tilde{\chi}) -f_{uv}(\tilde{\xi}+2\tilde{\tau}) \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg(\frac{1+λ_j^2}{aλ_j^2}+1\bigg) \bigg]\\ &&-\frac{λ_j(1+λ_j^2)}{4} \bigg[g _{uu}(\tilde{\xi}+2\tilde{\tau}) +g_{uv}(\tilde{\eta}+2\tilde{\chi})\bigg] \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)\\ &&+\frac{λ_j(1+λ_j^2)}{4}g_{uv}(\tilde{\xi}+2\tilde{\tau}) \bigg[\bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)^2 \bigg(\frac{1+λ_j^2}{aλ_j^2}+1\bigg)-\frac{aλ_j} {(1+λ_j^2)^2}\bigg]\\ &&+\frac{1}{8}\bigg\{f _{uu}(\xi_R+2\tau_R)+f_{uv}(\eta_R+2\chi_R) \\ && -f_{uv}\frac{1+λ_j^2}{aλ_j^2}\ \bigg[(\xi_R+2\tau_R) \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) -(\xi_I+2\tau_I)Ωega_0\bigg]\bigg\}\\ &&-\frac{1}{8Ωega_0} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg\{f_{uu}(\xi_I+2\tau_I)+f_{uv}(\eta_I+2\chi_I) \\ &&-f_{uv}\frac{1+λ_j^2}{aλ_j^2} \bigg[(\xi_I+2\tau_I) \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) +(\xi_R+2\tau_R)Ωega_0\bigg]\bigg\}\\ &&-\frac{λ_j(1+λ_j^2)}{8} \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg\{g_{uu}(\xi_R+2\tau_R)+g_{uv}(\eta_R+2\chi_R) \\ &&-g_{uv}\frac{1+λ_j^2}{aλ_j^2} \bigg[(\xi_R+2\tau_R) \bigg(\frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg)- (\xi_I+2\tau_I)Ωega_0\bigg]\bigg\} \\ &&+\frac{λ_j(1+λ_j^2)}{16Ωega_0} \bigg[\bigg(\frac{d_2j^2}{\ell^2}+ \frac{aλ_j}{1+λ_j^2}\bigg)^2-Ωega_0^2\bigg] \bigg\{g_{uu}(\xi_I+2\tau_I) +g_{uv}(\eta_I+2\chi_I)\\ &&-g_{uv}\frac{1+λ_j^2}{aλ_j^2} \bigg[(\xi_I+2\tau_I)\bigg(\frac{d_2j^2}{\ell^2} +\frac{aλ_j}{1+λ_j^2}\bigg) +(\xi_R+2\tau_R)Ωega_0\bigg]\bigg\}\\ &&+\frac{3}{16}\bigg[f _{uuu}-6f_{uuv}\frac{1+λ_j^2}{aλ_j^2}\bigg( \frac{d_2j^2}{\ell^2}+\frac{aλ_j}{1+λ_j^2}\bigg) \bigg]\bigg[1+(1+λ_j^2)\bigg( \frac{d_2j^2}{\ell^2}+\frac{2aλ_j}{1+λ_j^2}\bigg) \bigg]\\ &&-f_{uuv}\frac{3(1+λ_j^2)}{8aλ_j^2} \bigg[\frac{d_2j^2}{\ell^2}+\frac{2aλ_j}{1+λ_j^2} -Ωega_0^2(1+λ_j^2)\bigg], \end{eqnarray*} 于是,如果 \frac{1}{αpha'(λ_{j,\pm}^H)}{\rm Re}(c_1(λ_{j,\pm}^H))<0 (>0),分支的周期解是超临界的 (或次临界的). 证明完毕.

例3.4Ωega=\left(0,5\pi\right),d_1=4,d_2=1,a=2.25.则 根据文献 [7]中的例子 λmbda_1=\{ λ_2^H,λ_1^H, λ_0^H\}=\{0.0833,\;0.3057,\;0.3802\}. {\rm Re}(c_1(λ_1^H))\approx-36.9473<0,~~ {\rm Re}(c_1(λ_2^H))\approx-263.3673<0, αpha'(λ_1^H)=A'(λ_1^H)/2\approx-1.3643<0,~~ αpha'(λ_2^H)=A'(λ_1^H)/2\approx-1.2662<0. 这样,在点λ=λ_1^Hλ=λ_2^H,分支方向是次临界的.

参考文献
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一类扩散的Gierer-Meindardt的模型的振动模式和Hopf分支分析
万阿英, 衣凤岐, 郑立飞