造血系统的下临界Hopf分岔

Subcritical Hopf Bifurcation of a Hematopological System

摘要: 本文讨论了造血干细胞引发周期噬中性粒子血液病的两房模型。数学上，该模型是一多时滞的时滞微分系统的复杂非线性模型。变化时滞可带来系统的Hopf分岔行为，通过计算范式，分析得出该分岔点处发生了下临界Hopf分岔。数值模拟与理论分析结果一致。

Abstract: The periodic cyclical neutrophil oscillating model of hematopoietic is discussed. Mathematically, the model is a multi-delay DDEs with high nonlinearity. Varying time delay brings forth Hopf bifurcation in system dynamic behavior. By calculating coefficients of normal form, it is analyzed that Hopf bifurcation is sub-critical. Numerical simulation coincides with the theoretical analyze result.

1. 引言

在系统生态学动力学行为研究中，造血干细胞模型一直是人们非常关注的一个话题。在数学上，人们致力于应用不同的数学模型来描述干细胞增殖分化的时空动力系统。造血干细胞经分化得到人类必须的三类血液细胞：白细胞、红细胞和血小板。造血功能失调会带来多种血液学疾病，典型的例子有长期发烧、“打摆子”，和慢性的嗜中性粒子白血球减小症。病人呈现出周期性的嗜中性粒子低水平振荡现象，振荡周期经生物学实验证实为11天~19天。应用非线性动力学模型来刻画血液细胞的动力学振荡现象可以帮助理解其复杂动力学机制。

近十几年来，Mackey等人给出了建立血液学疾病机理的基本数学框架 [1] - [7] 。从建模起，描述血液学疾病机理的数学模型是一含有多个滞量的时滞微分系统(DDEs)。造血系统由激发分化和增殖的机制以及成熟机制组成，由于细胞的分化和成熟都需要时间，因此在系统中含有多个时滞。一般地，细胞需要经历的成熟期为2到6天，变化不同参数组合，得到振荡的周期解 [8] 。医学上，人们应用集落刺激因子管理嗜中性粒子减少血液病 [3] [4] [5] 。集落刺激因子管理有助于降低血液细胞的程序性凋亡率和增加增殖分化所需时间。国内Lei等人发展了基于集落刺激因子管理嗜中性粒子的血液病振荡数学模型，一般地，他们给出的数学模型如下 [9] [10] ，

$\begin{array}{l}\frac{\text{d}Q}{\text{d}t}=-\left(\beta \left(Q\right)+k_N\left(N\right)+k_{\delta }\right)Q+2{\text{e}}^{-{\gamma }_s{\tau }_s}\beta \left(Q_{{\tau }_s}\right)Q_{{\tau }_s}\\ \frac{\text{d}N}{\text{d}t}=-{\gamma }_{N}N+{\text{e}}^{{\eta }_{NP}{\tau }_{NP}-{\gamma }_0{\tau }_{NM}}{k}_N\left(N_{{\tau }_N}\right)Q_{{\tau }_N}\end{array}$ (1)

其中Holling函数

$\begin{array}{l}{k}_N\left(N\right)=\frac{f_0{\theta }_1^m}{{\theta }_1^m+N^m},\beta \left(Q\right)=\frac{k_0{\theta }_2^n}{{\theta }_2^n+Q^n}\\ k_N\left(N_{{\tau }_N}\right)=\frac{f_0{\theta }_1^m}{{\theta }_1^m+N{\left(t-{\tau }_N\right)}^m},\beta \left(Q\right)=\frac{k_0{\theta }_2^n}{{\theta }_2^n+Q{\left(t-{\tau }_s\right)}^n}\end{array}$

可以看出，模型(1)中干细胞以速率 $\beta$ 进入其增殖相，并以速率 $k_N$ 进入嗜中性粒子房。其它参数的生理学意义如下，

(1) 造血过程由激发分化和成熟机制组成，具体地，造血干细胞经由细胞分裂分化成为全部血液细胞包括白细胞、红细胞和血小板，并能自我更新， ${\gamma }_s$ 表示凋亡率，增殖相需要的天数是。

(2) 与嗜中性粒细胞不同的是，血液干细胞分化以速率 $k_{\delta }$ 分化成为其它血液细胞；

(3) 嗜中性粒细胞耦合时滞为 ${\tau }_N$ ，由两部分和组成，分别是其前驱细胞的增殖相需要的天数 ${\tau }_{NP}$ 和成熟相需要的天数 ${\tau }_{NM}$ 。 ${\eta }_{NP}$ 是增殖速率， ${\gamma }_0$ 是成熟相中的死亡率。因而，嗜中性粒细胞的组合相中的放大系数为

$A_N={\text{e}}^{{\eta }_{NP}{\tau }_{NP}-{\gamma }_0{\tau }_{NM}}$

注意到速率 ${\gamma }_s,{\eta }_{NP},{\gamma }_0$ 和成熟时间 ${\tau }_{NM}$ 依赖于粒细胞集落刺激因子的管理。考虑到粒细胞集落刺激因子在医学上的成功使用，学者们在周期噬中性粒子的演化模型中，在增殖系数中用时滞来减弱系数。其估计的参数值列出如下 $k_0=2.0$ ， $f_0=0.4$ ， $k_{\delta }=0.01$ ， ${\theta }_1=0.36\times {10}^8细胞/千克$ ，， ${\gamma }_N=2.4$ ， ${\tau }_{NP}=5$ ， ${\eta }_{NP}=2.5379$ ， ${\gamma }_0=0.27$ ， ${\tau }_{NM}=6.6909$ ， ${\tau }_N=11.6909$ 。选取 ${\gamma }_s$ 和 ${\tau }_s$ 为分岔参数。文 [11] 分析了系统(1)的长周期解，以及倍周期分岔和多周期解共存等复杂非线性现象，并证实了Haurie生理学实验的结论 [1] [2] [3] [4] [5] ，即造血模型的周期解存在着一个很广泛的周期，例从11天到89天等。

本文首先分析了系统的Hopf分岔，其次通过范式分析，证实了Hopf分岔现象是下临界的Hopf分岔现象，并给出了下临界Hopf分岔的数值模拟。

2. Hopf分岔分析

设 $\left(Q_0,N_0\right)$ 是系统(1)的平衡点，计算方程在平衡点的特征值，如果方程(1)的特征方程有一对纯虚根 $\pm i{\omega }_0$ ，并满足横截条件，则方程(1)发生了Hopf分岔，且经由Hopf分岔，系统(1)出现稳定或不稳定的周期解。作坐标平移，系统的三次Taylor截断展开式为

$\begin{array}{l}{x}^{\prime }=a_{11}x+a_{12}y+b_{1}x\left(t-{\tau }_s\right)+f\left(x,x\left(t-{\tau }_s\right),y\right)\\ y^{\prime }=-{\gamma }_{N}y+b_{2}x\left(t-{\tau }_N\right)+b_{3}y\left(t-{\tau }_N\right)+g\left(x\left(t-{\tau }_N\right),y,y\left(t-{\tau }_N\right)\right)\end{array}$

其中

$\begin{array}{l}{a}_{11}=-\beta \left(Q^{*}\right)-k_N\left(N^{*}\right)-k_{\delta }-{\beta }^{\prime }\left(Q^{*}\right)Q^{*},a_{12}=-{k^{\prime }}_N\left(N^{*}\right)Q^{*},\\ b_1=2{\text{e}}^{-r_s{\tau }_s}\left(\beta \left(Q^{*}\right)+{\beta }^{\prime }\left(Q^{*}\right)Q^{*}\right),b_2={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}k\left(N^{*}\right),\\ b_3={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k}^{\prime }\left(N^{*}\right)Q^{*},\end{array}$

和

$\begin{array}{c}f\left(x,x\left(t-{\tau }_s\right),y\right)=-{\beta }^{\prime }\left(Q^{*}\right)x^2-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)Q^{*}{x}^2-{k^{\prime }}_N\left(N^{*}\right)xy-\frac{1}{2}{k^{″}}_N\left(N^{*}\right)Q^{*}{y}^2\\ +2{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{\prime }\left(Q^{*}\right)x^2\left(t-{\tau }_s\right)+{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)Q^{*}{x}^2\left(t-{\tau }_s\right)\\ +{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)x^3\left(t-{\tau }_s\right)-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)x^3-\frac{1}{6}{\beta }^{‴}\left(Q^{*}\right)Q^{*}{x}^3-\frac{1}{2}{k^{″}}_N\left(N^{*}\right)x{y}^2\\ -\frac{1}{6}{k^{‴}}_N\left(N^{*}\right)Q^{*}{y}^3+{\text{e}}^{-r_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)x^3\left(t-{\tau }_s\right)+\frac{1}{3}{\text{e}}^{-r_s{\tau }_s}{\beta }^{‴}\left(Q^{*}\right)Q^{*}{x}^3\left(t-{\tau }_s\right)\\ +o\left(\Vert x^3\Vert ,\Vert x^3(t-{\tau }_s)\Vert ,\Vert y^3\Vert \right)\end{array}$

$\begin{array}{l}g\left(x\left(t-{\tau }_N\right),y,y\left(t-{\tau }_N\right)\right)\\ ={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right)x\left(t-{\tau }_N\right)y\left(t-{\tau }_N\right)\\ +\frac{1}{2}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)Q^{*}{y}^2\left(t-{\tau }_N\right)+\frac{1}{2}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)x\left(t-{\tau }_N\right)y^2\left(t-{\tau }_N\right)\\ +\frac{1}{6}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{‴}}_N\left(N^{*}\right)Q^{*}{y}^3\left(t-{\tau }_N\right)+\frac{1}{6}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{‴}}_N\left(N^{*}\right)x\left(t-{\tau }_N\right)y^3\left(t-{\tau }_N\right)\end{array}$

我们引入向量 $m={\left(x,y\right)}^{\text{T}}$ ，设参数向量 $\mu =\left({\gamma }_s,{\tau }_s\right)$ ，那么方程(1)可以写成其向量形式

$\begin{array}{c}{m}^{\prime }\left(t\right)=L\left(\mu \right)m\left(t\right)+R_1\left(\mu \right)m\left(t-{\tau }_s\right)+R_2\left(\mu \right)m\left(t-{\tau }_N\right)\\ +F\left(m\left(t\right),m\left(t-{\tau }_s\right),m\left(t-{\tau }_N\right),\mu \right)\end{array}$ (2)

其中

$L=\left(\begin{array}{cc}a{}_{11}& a_{12}\\ 0& -{\gamma }_N\end{array}\right)$ , $R_1=\left(\begin{array}{cc}{b}_1& 0\\ 0& 0\end{array}\right)$ , $R_2=\left(\begin{array}{cc}0& 0\\ b_2& b_3\end{array}\right)$

和

$\begin{array}{l}F\left(m\left(t\right),m\left(t-{\tau }_s\right),m\left(t-{\tau }_N\right),\mu \right)\\ =\frac{1}{2}B\left(m\left(t\right),m\left(t\right)\right)+\frac{1}{2}\tilde{B}\left(m\left(t-{\tau }_s\right),m\left(t-{\tau }_s\right)\right)\\ +\frac{1}{2}\hat{B}\left(m\left(t-{\tau }_N\right),m\left(t-{\tau }_N\right)\right)+\frac{1}{6}C\left(m\left(t\right),m\left(t\right),m\left(t\right)\right)\\ +\frac{1}{6}\tilde{C}\left(m\left(t-{\tau }_s\right),m\left(t-{\tau }_s\right),m\left(t-{\tau }_s\right)\right)+\frac{1}{6}\hat{C}\left(m\left(t-{\tau }_N\right),m\left(t-{\tau }_N\right),m\left(t-{\tau }_N\right)\right)\end{array}$ (3)

设 $C$ 表示连续可微函数 $u:\left[-{\tau }_N,0\right]\to R^2$ 构成的巴拿赫空间，定义时间平移 $u_t\left(\theta \right)=m\left(t+\theta \right)$ , $\theta \in \left[-{\tau }_N,0\right]$ ，那么方程(2)的算子微分方程形式为

${m^{\prime }}_t=A{m}_t+F\left(m_t\right)$ (4)

这里，设在Hopf分岔临界值处的线性算子 $A$ 为

$Au\left(\theta \right)=\{\begin{array}{l}\frac{\text{d}}{\text{d}\theta }u\left(\theta \right),\theta \in \left[-{\tau }_N,0\right),\\ Lu\left(0\right)+R_{1}u\left(-{\tau }_s\right)+R_{2}u\left(-{\tau }_N\right),\theta =0,\end{array}$ (5)

非线性算子 $H$ 可被写作(6)

$H\left(u\right)\left(\theta \right)=\{\begin{array}{l}0,\theta \in \left[-{\tau }_N,0\right),\\ F\left(u\left(0\right),u\left(-{\tau }_s\right),u\left(-{\tau }_N\right)\right),\theta =0,\end{array}$ (6)

其中 $u\in \sum$ 。

设连续可微函数 $v:\left[0,{\tau }_N\right]\to R^2$ 构成空间 $C^{*}$ ，在上定义线性算子 $A$ 的伴随线性算子为

$A^{*}v\left(s\right)=\{\begin{array}{l}-\frac{\text{d}}{\text{d}s}v\left(s\right),s\in \left(0,{\tau }_N\right)\\ L^{\text{T}}v\left(0\right)+R_1^{\text{T}}v\left({\tau }_s\right)+R_2^{\text{T}}v\left({\tau }_N\right),s=0\end{array}$

定义双线性形式 $(,):C^{*}\times C\to R$ 为

$\langle v,u\rangle ={\bar{v}}^{\text{T}}\left(0\right)u\left(0\right)+{\displaystyle {\int }_{-{\tau }_s}^0{\bar{v}}^{\text{T}}\left(\xi +{\tau }_s\right)R_{1}u\left(\xi \right)\text{d}\xi }+{\displaystyle {\int }_{-{\tau }_N}^0{\bar{v}}^{\text{T}}\left(\xi +{\tau }_N\right)R_{2}u\left(\xi \right)\text{d}\xi }$ (7)

容易证明，有

$\langle v,Au\rangle =\langle A^{*}v,u\rangle$

成立。

在临界Hopf分岔值 $u_0=\left({\gamma }_s,{\tau }_s\right)$ 处，有一对纯虚特征值 $\Lambda =\left\{-i{\omega }_0,i{\omega }_0\right\}$ ，且其余特征值都具有负实部。因而 $C$ 可被分解为

$C=P_{\Lambda }\oplus Q_{\Lambda }$ ，其中 $P_{\Lambda }$ 为相应于 $\Lambda$ 的特征空间，而 $Q_{\Lambda }$ 是其补空间。计算得 $A\left(0\right)$ 相应于特征值 $i{\omega }_0$ 的特征向量为 $q\left(\theta \right)=\left(\begin{array}{c}{\gamma }_N+i{\omega }_0-b_3{\text{e}}^{-i{\omega }_0{\tau }_N}\\ b_2{\text{e}}^{-i{\omega }_0{\tau }_N}\end{array}\right){\text{e}}^{i{\omega }_0\theta }$ ， $-{\tau }_N\le \theta \le 0$ ， $A^{*}\left(0\right)$ 相应于 $-i{\omega }_0$ 的特征向量为 $q^{*}\left(s\right)=l\left(\begin{array}{c}-b_2{\text{e}}^{i{\omega }_0{\tau }_N}\\ a_{11}+i{\omega }_0+b_1{\text{e}}^{i{\omega }_0{\tau }_s}\end{array}\right){\text{e}}^{i{\omega }_{0}s}$ ， $0\le s\le {\tau }_N$ ,满足 $\langle q^{*},q\rangle =1$ ，进一步可证明 $\langle q^{*},\bar{q}\rangle =0$ 。

注意到

$\begin{array}{l}Aq\left(\theta \right)=i{\omega }_{0}q\left(\theta \right),-{\tau }_N\le \theta \le 0,\\ A^{*}{q}^{*}\left(s\right)=-i{\omega }_0{q}^{*}\left(s\right),0\le s\le {\tau }_N\end{array}$ (8)

每个 $u\in C$ 可以被写作

$u_t=zq+\bar{z}\bar{q}+w$ (9)

使得

$z=\langle q^{*},m_t\rangle ,w\in Q_{\Lambda }$ (10)

对(9)两边微分，注意到(8)式成立，因而可得

$\begin{array}{c}{z}^{\prime }\left(t\right)=\langle q^{*},{m^{\prime }}_t\rangle =\langle q^{*},Azq\rangle +\langle q^{*},H\left(zq+\bar{z}\bar{q}+w\left(t\right)\right)\rangle \\ =i{\omega }_{0}z+{\bar{q}}^{*}\left(0\right)F^{*}\left(z,\bar{z}\right)\end{array}$ (11)

其中

$F^{*}\left(z,\bar{z}\right)=F\left(2\mathrm{Re}\left\{zq\left(\theta \right)\right\}+w\left(t\right)\right)$ (12)

3. 二维中心流形

将 $w=v\left(z,\bar{z}\right)$ 展开成级数

$w=v\left(z,\bar{z}\right)=\left(\begin{array}{c}{\displaystyle \underset{2\le i+j}{\sum }\frac{v_{ij}^{\left(1\right)}}{i!j!}\left(z,\bar{z}\right)}\\ {\displaystyle \underset{2\le i+j}{\sum }\frac{v_{ij}^{\left(2\right)}}{i!j!}\left(z,\bar{z}\right)}\end{array}\right)$ (13)

其中系数 $\left\{v_{ij}\right\}$ 依赖于 $\theta$ ，令 $G\left(z,\bar{z}\right)={\bar{q}}^{*}\left(0\right)F^{*}\left(z,\bar{z}\right)$ ，把方程(11)写作

$z^{\prime }\left(t\right)=i{\omega }_{0}z+G\left(z,\bar{z}\right)$ (13)

其中 $G\left(z,\bar{z}\right)$ 有展式

$G\left(z,\bar{z}\right)={\displaystyle \underset{2\le i+j}{\sum }\frac{G_{ij}}{i!j!}\left(z,\bar{z}\right)}$ (14)

其中 $G_{ij}$ 为常数，计算见附录。

从方程(9)和(11)可得

$\frac{\text{d}v}{\text{d}t}=\{\begin{array}{l}Av-2\mathrm{Re}\left\{{\bar{q}}^{*}\left(0\right)\hat{F}q\left(\theta \right)\right\},-{\tau }_N\le \theta <0,\\ Av-2\mathrm{Re}\left\{{\bar{q}}^{*}\left(0\right)\hat{F}q\left(\theta \right)\right\}+\hat{F},\theta =0,\end{array}$ (15)

方程(14)的第一个方程可以写成

$\frac{\text{d}v}{\text{d}t}=Av\left(z,\bar{z}\right)+R\left(z,\bar{z}\right)$ (16)

其中

$R\left(z,\bar{z}\right)=-2\mathrm{Re}\left\{{\bar{q}}^{*}\left(0\right)\hat{F}q\left(\theta \right)\right\}=\left(\begin{array}{c}\frac{R_{20}^{\left(1\right)}}{2}{z}^2+R_{11}^{\left(1\right)}z\bar{z}+\frac{R_{02}^{\left(1\right)}}{2}{\bar{z}}^2+\cdots \\ \frac{R_{20}^{\left(2\right)}}{2}{z}^2+R_{11}^{\left(2\right)}z\bar{z}+\frac{R_{02}^{\left(2\right)}}{2}{\bar{z}}^2+\cdots \end{array}\right)$

其中系数 $R_{ij}^{\left(1,2\right)}=R_{ij}^{\left(1,2\right)}\left(\theta \right)$ 。

由于

$\begin{array}{l}-2\mathrm{Re}\left\{{\bar{q}}^{*}\left(0\right)\hat{F}q\left(\theta \right)\right\}\\ =-G\left(z,\bar{z}\right)q\left(\theta \right)-\bar{G}\left(z,\bar{z}\right)\bar{q}\left(\theta \right)\\ =-\left(\frac{G_{20}}{2}{z}^2+G_{11}z\bar{z}+\frac{G_{02}}{2}{\bar{z}}^2\right)q\left(\theta \right)-\left(\frac{{\bar{G}}_{20}}{2}{\bar{z}}^2+{\bar{G}}_{11}z\bar{z}+\frac{{\bar{G}}_{02}}{2}{z}^2\right)\bar{q}\left(\theta \right)\end{array}$

所以

$R_{20}^{\left(1\right)}=-G_{20}{q}_1{\text{e}}^{i{\omega }_0\theta }-{\bar{G}}_{02}{\bar{q}}_1{\text{e}}^{-i{\omega }_0\theta }$ , $R_{20}^{\left(2\right)}=-G_{20}{q}_2{\text{e}}^{i{\omega }_0\theta }-{\bar{G}}_{02}{\bar{q}}_2{\text{e}}^{-i{\omega }_0\theta }$

$R_{11}^{\left(1\right)}=-G_{11}{q}_1{\text{e}}^{i{\omega }_0\theta }-{\bar{G}}_{11}{\bar{q}}_1{\text{e}}^{-i{\omega }_0\theta }$ , $R_{11}^{\left(2\right)}=-G_{11}{q}_2{\text{e}}^{i{\omega }_0\theta }-{\bar{G}}_{11}{\bar{q}}_2{\text{e}}^{-i{\omega }_0\theta }$

$R_{02}^{\left(1\right)}=-G_{02}{q}_1{\text{e}}^{i{\omega }_0\theta }-{\bar{G}}_{20}{\bar{q}}_1{\text{e}}^{-i{\omega }_0\theta }$ , $R_{02}^{\left(2\right)}=-G_{02}{q}_2{\text{e}}^{i{\omega }_0\theta }-{\bar{G}}_{20}{\bar{q}}_2{\text{e}}^{-i{\omega }_0\theta }$

为了决定分岔解的行为，通过下式计算 $w=v\left(z,\bar{z}\right)$ 。

$\begin{array}{l}A{v}_{20}^{\left(1,2\right)}=2i{\omega }_0{v}_{20}^{\left(1,2\right)}-R_{20}^{\left(1,2\right)},\\ A{v}_{11}^{\left(1,2\right)}=-R_{11}^{\left(1,2\right)},\\ A{v}_{02}^{\left(1,2\right)}=-2i{\omega }_0{v}_{02}^{\left(1,2\right)}-R_{02}^{\left(1,2\right)}\end{array}$ (17)

注意到算子 $A$ 的定义，(17)式即

$\begin{array}{l}\{\begin{array}{l}\frac{\text{d}{v}_{20}^{\left(1\right)}}{\text{d}t}=2i{\omega }_0{v}_{20}^{\left(1\right)}+G_{20}{q}_1{\text{e}}^{i{\omega }_0\theta }+{\bar{G}}_{02}{\bar{q}}_1{\text{e}}^{-i{\omega }_0\theta }\\ \frac{\text{d}{v}_{20}^{\left(2\right)}}{\text{d}t}=2i{\omega }_0{v}_{20}^{\left(2\right)}+G_{20}{q}_2{\text{e}}^{i{\omega }_0\theta }+{\bar{G}}_{02}{\bar{q}}_2{\text{e}}^{-i{\omega }_0\theta }\end{array}\\ \{\begin{array}{l}\frac{\text{d}{v}_{11}^{\left(1\right)}}{\text{d}t}=G_{11}{q}_1{\text{e}}^{i{\omega }_0\theta }+{\bar{G}}_{11}{\bar{q}}_1{\text{e}}^{-i{\omega }_0\theta }\\ \frac{\text{d}{v}_{11}^{\left(2\right)}}{\text{d}t}=G_{11}{q}_2{\text{e}}^{i{\omega }_0\theta }+{\bar{G}}_{11}{\bar{q}}_2{\text{e}}^{-i{\omega }_0\theta }\end{array}\\ \{\begin{array}{l}\frac{\text{d}{v}_{02}^{\left(1\right)}}{\text{d}t}=-2i{\omega }_0{v}_{02}^{\left(1\right)}+G_{02}{q}_1{\text{e}}^{i{\omega }_0\theta }+{\bar{G}}_{20}{\bar{\sout{q}}}_1{\text{e}}^{-i{\omega }_0\theta }\\ \frac{\text{d}{v}_{02}^{\left(2\right)}}{\text{d}t}=-2i{\omega }_0{v}_{02}^{\left(1\right)}+G_{02}{q}_2{\text{e}}^{i{\omega }_0\theta }+{\bar{G}}_{20}{\bar{q}}_2{\text{e}}^{-i{\omega }_0\theta }\end{array}\end{array}$ ， $-{\tau }_N\le \theta <0$ (18)

相似地，由方程(15)的第二个方程得到初值条件

$\{\begin{array}{l}{a}_{11}{v_{20}^{\left(1\right)}|}_{\theta =0}+a_{12}{v_{20}^{\left(2\right)}|}_{\theta =0}-2i{\omega }_0{v_{20}^{\left(1\right)}|}_{\theta =0}+b_1{v_{20}^{\left(1\right)}|}_{\theta =-{\tau }_s}+R_{20}^{\left(1\right)}+2f_0=0\\ -{\gamma }_N{v_{20}^{\left(2\right)}|}_{\theta =0}-2i{\omega }_0{v_{20}^{\left(2\right)}|}_{\theta =0}+b_2{v_{20}^{\left(1\right)}|}_{\theta =-{\tau }_N}+b_3{v_{20}^{\left(2\right)}|}_{\theta =-{\tau }_N}+R_{20}^{\left(2\right)}+2g_0=0\end{array}$

$\{\begin{array}{l}{a}_{11}{v_{11}^{\left(1\right)}|}_{\theta =0}+a_{12}{v_{11}^{(2)}|}_{\theta =0}+b_1{v_{11}^{\left(1\right)}|}_{\theta =-{\tau }_s}+R_{11}^{\left(1\right)}+f_1=0\\ -{\gamma }_N{v_{11}^{\left(2\right)}|}_{\theta =0}+b_2{v_{11}^{\left(1\right)}|}_{\theta =-{\tau }_N}+b_3{v_{11}^{\left(2\right)}|}_{\theta =-{\tau }_N}+R_{11}^{\left(2\right)}+g_1=0\end{array}$ (19)

$\{\begin{array}{l}{a}_{11}{v_{02}^{\left(1\right)}|}_{\theta =0}+a_{12}{v_{02}^{\left(2\right)}|}_{\theta =0}+2i{\omega }_0{v_{02}^{\left(1\right)}|}_{\theta =0}+b_1{v_{02}^{\left(1\right)}|}_{\theta =-{\tau }_s}+R_{02}^{\left(1\right)}+2f_2=0\\ -{\gamma }_N{v_{02}^{\left(2\right)}|}_{\theta =0}+2i{\omega }_0{v_{02}^{\left(2\right)}|}_{\theta =0}+b_2{v_{02}^{\left(1\right)}|}_{\theta =-{\tau }_N}+b_3{v_{02}^{\left(2\right)}|}_{\theta =-{\tau }_N}+R_{02}^{\left(2\right)}+2g_2=0\end{array}$

对方程(18)积分在初值条件(19)下积分，可得出

$\{\begin{array}{l}{v}_{20}^{\left(1\right)}=E_1{\text{e}}^{2i{\omega }_0\theta }+\frac{i{G}_{20}{q}_1}{{\omega }_0}{\text{e}}^{i{\omega }_0\theta }+\frac{i{\bar{G}}_{02}{\bar{q}}_1}{3{\omega }_0}{\text{e}}^{-i{\omega }_0\theta }\\ v_{20}^{\left(2\right)}=E_2{\text{e}}^{2i{\omega }_0\theta }+\frac{i{G}_{20}{q}_2}{{\omega }_0}{\text{e}}^{i{\omega }_0\theta }+\frac{i{\bar{G}}_{02}{\bar{q}}_2}{3{\omega }_0}{\text{e}}^{-i{\omega }_0\theta }\end{array}$ (20.1)

$\{\begin{array}{l}{v}_{11}^{\left(1\right)}=E_3-\frac{i{G}_{11}{q}_1}{{\omega }_0}{\text{e}}^{i\omega {}_0\theta }+\frac{i{\bar{G}}_{11}{\bar{q}}_1}{{\omega }_0}{\text{e}}^{-i{\omega }_0\theta }\\ v_{11}^{\left(2\right)}=E_4-\frac{i{G}_{11}{q}_2}{{\omega }_0}{\text{e}}^{i\omega {}_0\theta }+\frac{i{\bar{G}}_{11}{\bar{q}}_2}{{\omega }_0}{\text{e}}^{-i{\omega }_0\theta }\end{array}$ (20.2)

(20.3)

其中系数 $E_1,E_2,E_3,E_4,E_5,E_6$ 由方程初值条件(19)决定。

4. 下临界Hopf分岔

当特征值穿越虚轴时，系统(1)的平衡点处发生了Hopf分岔，一般的Hopf分岔分为超临界的Hopf分岔和下临界的Hopf分岔。若当参数值小于临界值时分岔出不稳定的周期解，Hopf分岔被称为是下临界的；若当参数值大于临界值时分岔出的周期解是稳定的，Hopf分岔被称为是超临界的。对平衡点作分析，方程(1)的线性化方程的特征方程为

$\Delta \left(\lambda ,{\gamma }_s,{\tau }_s\right)=\left|\begin{array}{cc}{a}_{11}-\lambda +b_1{\text{e}}^{-\lambda {\tau }_s}& a_{12}\\ b_2{\text{e}}^{-\lambda {\tau }_N}& -{\gamma }_N-\lambda +b_3{\text{e}}^{-\lambda {\tau }_N}\end{array}\right|=0$ (21)

当有一对共轭复特征值穿越虚轴时，方程(1)的平衡点的稳定性发生改变，这时方程(21)确定了一对

Figure 1. Periodic solutions near the subcritical Hopf point, wherein stable periodic solutions is denoted by blue lines, whilst unstable perioidc solutions represented by red lines

图1. 下临界hopf点附近的周期解。红线表示不稳定周期解，蓝线表示稳定的周期解

Hopf分岔的临界值 ${\mu }_0=\left({\gamma }_s,{\tau }_s\right)=\left(0.0969,4.9789\right)$ 。并由方程(21)确定 $\mathrm{Re}\left\{{\frac{\text{d}\lambda }{\text{d}\tau }|}_{\tau ={\tau }_s}\right\}\ne 0$ ，即系统在Hopf临界值处满足横截性。

由特征方程(21)可以计算 $\frac{\text{d}\lambda }{\text{d}{\tau }_s}>0$ ，以上计算得到大于零的第一Lyapunov数，又由于 $\mu =-\frac{l_1\left(0\right)}{\mathrm{Re}\left\{{\frac{\text{d}\lambda }{\text{d}\tau }|}_{\tau ={\tau }_s}\right\}}<0$ ，从而Hopf分岔是下临界的。数值模拟得到该Hopf分岔点附近的周期解，如图1所示。

5. 总结

本文应用Hopf分岔原理研究造血系统中噬中性粒子数周期性振荡现象发生的机制。模型中引入集落刺激因子管理，有助于降低血液细胞的程序性凋亡率和增加增殖分化所需时间，在数学上，使得造血系统成为含时滞依赖物理参数的多时滞系统。变化时滞得到系统的Hopf分岔点，应用范式理论进一步得到该分岔是下临界Hopf分岔现象。连续延拓得到系统共存的稳定周期解和不稳定周期解。

附录

计算 $A\left(0\right)$ 相应于特征值 $i{\omega }_0$ 的特征向量为

$q\left(\theta \right)=\left(\begin{array}{c}{\gamma }_N+i{\omega }_0-b_3{\text{e}}^{-i{\omega }_0{\tau }_N}\\ b_2{\text{e}}^{-i{\omega }_0{\tau }_N}\end{array}\right){\text{e}}^{i{\omega }_0\theta }=\left(\begin{array}{c}{q}_1\\ q_2\end{array}\right){\text{e}}^{i{\omega }_0\theta }$ ， $-{\tau }_N\le \theta \le 0$ ，

$A^{*}\left(0\right)$ 相应于 $-i{\omega }_0$ 的特征向量为

$q^{*}\left(s\right)=l\left(\begin{array}{c}-b_2{\text{e}}^{i{\omega }_0{\tau }_N}\\ a_{11}+i{\omega }_0+b_1{\text{e}}^{i{\omega }_0{\tau }_s}\end{array}\right){\text{e}}^{i{\omega }_{0}s}=l\left(\begin{array}{c}{q}_1^{*}\\ q_2^{*}\end{array}\right){\text{e}}^{i{\omega }_{0}s}$ ， $0\le s\le {\tau }_N$

由定义直接计算得

$\varphi \left(0\right)=z\left(\begin{array}{c}{q}_1\\ q_2\end{array}\right)+\bar{z}\left(\begin{array}{c}{\bar{q}}_1\\ {\bar{q}}_2\end{array}\right)+{v\left(z,\bar{z}\right)|}_{\theta =0}$

$\begin{array}{c}\varphi \left(-{\tau }_s\right)=2\mathrm{Re}\left\{zq\left(-{\tau }_s\right)\right\}+{v\left(z,\bar{z}\right)|}_{\theta =-{\tau }_s}\\ =z\left(\begin{array}{c}{q}_1\\ q_2\end{array}\right){\text{e}}^{-i{w}_0{\tau }_s}+\bar{z}\left(\begin{array}{c}{\bar{q}}_1\\ {\bar{q}}_2\end{array}\right){\text{e}}^{-i{w}_0{\tau }_s}+{v\left(z,\bar{z}\right)|}_{\theta =-{\tau }_s}\end{array}$

$\begin{array}{c}\varphi \left(-{\tau }_N\right)=2\mathrm{Re}\left\{zq\left(-{\tau }_N\right)\right\}+{v\left(z,\bar{z}\right)|}_{\theta =-{\tau }_N}\\ =z\left(\begin{array}{c}{q}_1\\ q_2\end{array}\right){\text{e}}^{-i{w}_0{\tau }_N}+\bar{z}\left(\begin{array}{c}{\bar{q}}_1\\ {\bar{q}}_2\end{array}\right){\text{e}}^{-i{w}_0{\tau }_N}+{v\left(z,\bar{z}\right)|}_{\theta =-{\tau }_N}\end{array}$

而且

$\langle q^{*}\left(0\right),\hat{F}\left(z,\bar{z}\right)\rangle =\langle q^{*}\left(0\right),(2\mathrm{Re}\left\{zq\left(\theta \right)\right\}+w\left(t\right)\rangle$

比较 $z^i{\bar{z}}^j$ 在方程两边的系数得

$\begin{array}{l}{z}^2:\frac{1}{2}{G}_{20}=\langle q^{*}\left(0\right),\frac{1}{2}B\left(q\left(0\right),q\left(0\right)\right)+\frac{1}{2}\tilde{B}\left(q\left(-{\tau }_s\right),q\left(-{\tau }_s\right)\right)+\frac{1}{2}\hat{B}\left(q\left(-{\tau }_N\right),q\left(-{\tau }_N\right)\right)\rangle \\ =\bar{l}\left({\bar{q}}_1{}^{*}{f}_0+{\bar{q}}_2{}^{*}{g}_0\right)\end{array}$

$\begin{array}{l}z\bar{z}:G_{11}=\langle q^{*}\left(0\right),B\left(q\left(0\right),\bar{q}\left(0\right)\right)+\tilde{B}\left(q\left(-{\tau }_s\right),\bar{q}\left(-{\tau }_s\right)\right)+\hat{B}\left(q\left(-{\tau }_N\right),\bar{q}\left(-{\tau }_N\right)\right)\rangle \\ =\bar{l}\left({\bar{q}}_1{}^{*}{f}_1+{\bar{q}}_2{}^{*}{g}_1\right)\end{array}$

$\begin{array}{l}{z}^3:\frac{1}{6}{G}_{30}=\langle q^{*}\left(0\right),\frac{1}{6}C\left(q\left(0\right),q\left(0\right),q\left(0\right)\right)+\frac{1}{6}\tilde{C}\left(q\left(-{\tau }_s\right),q\left(-{\tau }_s\right),q\left(-{\tau }_s\right)\right)\\ +\frac{1}{6}\hat{C}\left(q\left(-{\tau }_N\right),q\left(-{\tau }_N\right),q\left(-{\tau }_N\right)\right)+\frac{1}{2}B\left(q\left(0\right),v_{20}\right]\left(0\right)+\frac{1}{2}\tilde{B}\left(q\left(-{\tau }_s\right),v_{20}\left(-{\tau }_s\right)\right)\\ +\frac{1}{2}\hat{B}\left(q\left(-{\tau }_N\right),v_{20}\left(-{\tau }_N\right)\right)\rangle \\ =\bar{l}\left({\bar{q}}_1{}^{*}{f}_3+{\bar{q}}_2{}^{*}{g}_3\right)\end{array}$

$\begin{array}{l}{z}^2\bar{z}:\frac{1}{2}{G}_{21}=\langle q^{*}\left(0\right),\frac{1}{2}C\left(q\left(0\right),q\left(0\right),\bar{q}\left(0\right)\right)+\frac{1}{2}\tilde{C}\left(q\left(0\right),q\left(0\right),\bar{q}\left(0\right)\right)+\frac{1}{2}\hat{C}\left(q\left(0\right),q\left(0\right),\bar{q}\left(0\right)\right)\\ +\frac{1}{2}B\left(\bar{q}\left(0\right),v_{20}\left(0\right)\right)+\frac{1}{2}\tilde{B}\left(\bar{q}\left(-{\tau }_s\right),v_{20}\left(-{\tau }_s\right)\right)+\frac{1}{2}B\left(\bar{q}\left(-{\tau }_N\right),v_{20}\left(-{\tau }_N\right)\right)\\ {}_{\begin{array}{l}\\ \end{array}}{}^{}+B\left(q\left(0\right),v_{11}\left(0\right)\right)+\tilde{B}\left(q\left(-{\tau }_s\right),v_{11}\left(-{\tau }_s\right)\right)+\hat{B}\left(q\left(-{\tau }_N\right),v_{11}\left(-{\tau }_N\right)\right)\rangle \\ =\bar{l}\left({\bar{q}}_1{}^{*}{f}_4+{\bar{q}}_2{}^{*}{g}_4\right)\end{array}$

$\begin{array}{l}z{\bar{z}}^2:\frac{1}{2}{G}_{12}=\langle q^{*}\left(0\right),\frac{1}{2}C\left(q\left(0\right),\bar{q}\left(0\right),\bar{q}\left(0\right)\right)+\frac{1}{2}\tilde{C}\left(q\left(0\right),\bar{q}\left(0\right),\bar{q}\left(0\right)\right)+\frac{1}{2}\hat{C}\left(q\left(0\right),\bar{q}\left(0\right),\bar{q}\left(0\right)\right)\\ +\frac{1}{2}B\left(q\left(0\right),v_{02}\left(0\right)\right)+\frac{1}{2}\tilde{B}\left(q\left(-{\tau }_s\right),v_{02}\left(-{\tau }_s\right)\right)+\frac{1}{2}\hat{B}\left(q\left(-{\tau }_N\right),v_{02}\left(-{\tau }_N\right)\right)\\ {}_{\begin{array}{l}\\ \end{array}}^{}+B\left(\bar{q}\left(0\right),v_{11}\left(0\right)\right)+\tilde{B}\left(\bar{q}\left(-{\tau }_s\right),v_{11}\left(-{\tau }_s\right)\right)+\hat{B}\left(\bar{q}\left(-{\tau }_N\right),v_{11}\left(-{\tau }_N\right)\right)\rangle \\ =\bar{l}\left({\bar{q}}_1{}^{*}{f}_5+{\bar{q}}_2{}^{*}{g}_5\right)\end{array}$

$\begin{array}{l}{\bar{z}}^3:\frac{1}{6}{G}_{03}=\langle q^{*}\left(0\right),\frac{1}{6}C\left(\bar{q}\left(0\right),\bar{q}\left(0\right),\bar{q}\left(0\right)\right)+\frac{1}{6}\tilde{C}\left(\bar{q}\left(0\right),\bar{q}\left(0\right),\bar{q}\left(0\right)\right)+\frac{1}{6}\hat{C}\left(\bar{q}\left(0\right),\bar{q}\left(0\right),\bar{q}\left(0\right)\right)\rangle \\ +\frac{1}{2}B\left(q\left(0\right),v_{20}\left(0\right)\right)+\frac{1}{2}B\left(q\left(-{\tau }_s\right),v_{20}\left(-{\tau }_s\right)\right)+\frac{1}{2}B\left(q\left(-{\tau }_N\right),v_{20}\left(-{\tau }_N\right)\right)\\ =\bar{l}\left({\bar{q}}_1{}^{*}{f}_6+{\bar{q}}_2{}^{*}{g}_6\right)\end{array}$

其中，

$\begin{array}{c}{f}_0=\left[2{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{\prime }\left(Q^{*}\right)q_1^2+{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)Q^{*}{q}_1^2\right]{\text{e}}^{-2i{\omega }_0{\tau }_s}+\left[-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)Q^{*}{q}_1^2-{\beta }^{\prime }\left(Q^{*}\right)q_1^2\right]\\ -\frac{1}{2}{k^{″}}_N\left(N^{*}\right)Q^{*}{q}_2^2-{k^{\prime }}_N\left(N^{*}\right)q_1{q}_2\end{array}$

$\begin{array}{c}{f}_1=2\left[{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{\prime }\left(Q^{*}\right)q_1{\bar{q}}_1+{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)Q^{*}{q}_1{\bar{q}}_1\right]+2\left[-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)Q^{*}{q}_1{\bar{q}}_1-{\beta }^{\prime }\left(Q^{*}\right)q_1{\bar{q}}_1\right]\\ -{k^{″}}_N\left(N^{*}\right)Q^{*}{q}_2{\bar{q}}_2-{k^{\prime }}_N\left(N^{*}\right)\left({\bar{q}}_1{q}_2+q_1{\bar{q}}_2\right)\end{array}$

$\begin{array}{c}{f}_2=\left[2{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{\prime }\left(Q^{*}\right){\bar{q}}_1^2+{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)Q^{*}{\bar{q}}_1^2\right]{\text{e}}^{2i{\omega }_0{\tau }_s}+\left[-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)Q^{*}{\bar{q}}_1^2-{\beta }^{\prime }\left(Q^{*}\right){\bar{q}}_1^2\right]\\ -\frac{1}{2}{k^{″}}_N\left(N^{*}\right)Q^{*}{\bar{q}}_2^2-{k^{\prime }}_N\left(N^{*}\right){\bar{q}}_1{\bar{q}}_2\end{array}$

$\begin{array}{c}{f}_3=\left[2{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{\prime }\left(Q^{*}\right)q_1+{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)Q^{*}{q}_1\right]{v_{20}^{\left(1\right)}|}_{\theta =-{\tau }_s}{\text{e}}^{-i{\omega }_0{\tau }_s}\\ +\left[{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)q_1^3+\frac{1}{3}{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{‴}\left(Q^{*}\right)Q^{*}{q}_1^3\right]{\text{e}}^{-3i{\omega }_0{\tau }_s}+\left[-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)Q^{*}{q}_1-{\beta }^{\prime }\left(Q^{*}\right)q_1\right]{v_{20}^{\left(1\right)}|}_{\theta =0}\\ +\left[-\frac{1}{6}{\beta }^{‴}\left(Q^{*}\right)Q^{*}{q}_1^3-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)q_1^3\right]-\frac{1}{2}{k^{″}}_N\left(N^{*}\right)Q^{*}{v_{20}^{\left(2\right)}|}_{\theta =0}{q}_2\\ -\frac{1}{6}{k^{‴}}_N\left(N^{*}\right)Q^{*}{q}_2^3-{k^{\prime }}_N\left(N^{*}\right)\left(\frac{{v_{20}^{\left(1\right)}|}_{\theta =0}}{2}{q}_2+\frac{{v_{20}^{\left(1\right)}|}_{\theta =0}}{2}{q}_1\right)-\frac{1}{2}{k^{″}}_N\left(N^{*}\right)q_1{q}_2^2\end{array}$

$\begin{array}{c}{f}_4=\left[2{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{\prime }\left(Q^{*}\right)+{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)Q^{*}\right]\left({2v_{11}^{\left(1\right)}|}_{\theta =-{\tau }_s}{q}_1{\text{e}}^{-i{\omega }_0{\tau }_s}+{v_{20}^{\left(1\right)}|}_{\theta =-{\tau }_s}{\bar{q}}_1{\text{e}}^{i{\omega }_0{\tau }_s}\right)\\ +3\left[{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)+\frac{1}{3}{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{‴}\left(Q^{*}\right)Q^{*}\right]q_1^2{\bar{q}}_1{\text{e}}^{-i{\omega }_0{\tau }_s}\\ +\left[-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)Q^{*}-{\beta }^{\prime }\left(Q^{*}\right)\right]\left(2{q_1{v}_{11}^{\left(1\right)}|}_{\theta =0}+{{\bar{q}}_1{v}_{20}^{\left(1\right)}|}_{\theta =0}\right)\\ +3\left[-\frac{1}{6}{\beta }^{‴}\left(Q^{*}\right)Q^{*}{q}_1^2{\bar{q}}_1-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)q_1^2{\bar{q}}_1\right]-\frac{1}{2}{k^{″}}_N\left(N^{*}\right)Q^{*}\left({2v_{11}^{\left(2\right)}|}_{\theta =0}{q}_2+{v_{20}^{\left(2\right)}|}_{\theta =0}{\bar{q}}_2\right)\\ -\frac{1}{2}{k^{‴}}_N\left(N^{*}\right)Q^{*}{q}_2^2{\bar{q}}_2-{k^{\prime }}_N\left(N^{*}\right)\left(\frac{{v_{20}^{\left(1\right)}|}_{\theta =0}}{2}{\bar{q}}_2+\frac{{v_{20}^{\left(2\right)}|}_{\theta =0}}{2}{\bar{q}}_1+{v_{11}^{\left(1\right)}|}_{\theta =0}{q}_2+{v_{11}^{\left(2\right)}|}_{\theta =0}{q}_1\right)\\ -\frac{1}{2}{k^{″}}_N\left(N^{*}\right)\left(2q_1{q}_2{\bar{q}}_2+{\bar{q}}_1{q}_2^2\right)\end{array}$

$\begin{array}{c}{f}_5=\left[2{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{\prime }\left(Q^{*}\right)+{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)Q^{*}\right]\left({2v_{11}^{\left(1\right)}|}_{\theta =-{\tau }_s}{\bar{q}}_1{\text{e}}^{i{\omega }_0{\tau }_s}+{v_{02}^{\left(1\right)}|}_{\theta =-{\tau }_s}{q}_1{\text{e}}^{-i{\omega }_0{\tau }_s}\right)\\ +3\left[{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)+\frac{1}{3}{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{‴}\left(Q^{*}\right)Q^{*}\right]q_1{\bar{q}}_1^2{\text{e}}^{i{\omega }_0{\tau }_s}\\ +\left[-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)Q^{*}-{\beta }^{\prime }\left(Q^{*}\right)\right]\left(2{\bar{q}}_1{{v_{11}^{\left(1\right)}|}_{\theta =0}+q_1{v}_{02}^{\left(1\right)}|}_{\theta =0}\right)\\ +3\left[-\frac{1}{6}{\beta }^{‴}\left(Q^{*}\right)Q^{*}-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)\right]q_1{\bar{q}}_1^2-\frac{1}{2}{k^{″}}_N\left(N^{*}\right)Q^{*}\left({2v_{11}^{\left(2\right)}|}_{\theta =0}{\bar{q}}_2+{v_{02}^{\left(2\right)}|}_{\theta =0}{q}_2\right)\\ -\frac{1}{2}{k^{‴}}_N\left(N^{*}\right)Q^{*}{q}_2{\bar{q}}_2^2-{k^{\prime }}_N\left(N^{*}\right)\left(\frac{{v_{02}^{\left(1\right)}|}_{\theta =0}}{2}{q}_2+\frac{{v_{02}^{\left(2\right)}|}_{\theta =0}}{2}{q}_1+{v_{11}^{\left(1\right)}|}_{\theta =0}{\bar{q}}_2+{v_{11}^{\left(2\right)}|}_{\theta =0}{\bar{q}}_1\right)\\ -\frac{1}{2}{k^{″}}_N\left(N^{*}\right)\left(2{\bar{q}}_1{q}_2{\bar{q}}_2+q_1{\bar{q}}_2^2\right)\end{array}$

$\begin{array}{c}{f}_6=\left[2{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{\prime }\left(Q^{*}\right)+{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)Q^{*}\right]{v_{02}^{\left(1\right)}|}_{\theta =-{\tau }_s}{\bar{q}}_1{\text{e}}^{i{\omega }_0{\tau }_s}\\ +\left[{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{″}\left(Q^{*}\right)+\frac{1}{3}{\text{e}}^{-{\gamma }_s{\tau }_s}{\beta }^{‴}\left(Q^{*}\right)Q^{*}\right]{\bar{q}}_1^3{\text{e}}^{3i{\omega }_0{\tau }_s}+\left[-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)Q^{*}-{\beta }^{\prime }\left(Q^{*}\right)\right]{v_{02}^{\left(1\right)}|}_{\theta =0}{\bar{q}}_1\\ +\left[-\frac{1}{6}{\beta }^{‴}\left(Q^{*}\right)Q^{*}-\frac{1}{2}{\beta }^{″}\left(Q^{*}\right)\right]{\bar{q}}_1^3-\frac{1}{2}{k^{″}}_N\left(N^{*}\right)Q^{*}{v_{02}^{\left(2\right)}|}_{\theta =0}{\bar{q}}_2\\ -\frac{1}{6}{k^{‴}}_N\left(N^{*}\right)Q^{*}{\bar{q}}_2^3-{k^{\prime }}_N\left(N^{*}\right)\left(\frac{{v_{02}^{\left(1\right)}|}_{\theta =0}}{2}{\bar{q}}_2+\frac{{v_{02}^{\left(2\right)}|}_{\theta =0}}{2}{\bar{q}}_1\right)-\frac{1}{2}{k^{″}}_N\left(N^{*}\right){\bar{q}}_1{\bar{q}}_2^2\end{array}$

$g_0={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right)q_1{q}_2{\text{e}}^{-2i{\omega }_0{\tau }_N}+\frac{1}{2}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)Q^{*}{q}_2^2{\text{e}}^{-2i{w}_0{\tau }_N}$

$g_1={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right)\left(q_1{\bar{q}}_2+{\bar{q}}_1{q}_2\right)+{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)Q^{*}{q}_2{\bar{q}}_2$

$g_2={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right){\bar{q}}_1{\bar{q}}_2{\text{e}}^{2i{\omega }_0{\tau }_N}+\frac{1}{2}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)Q^{*}{\bar{q}}_2^2{\text{e}}^{2i{w}_0{\tau }_N}$

$\begin{array}{c}{g}_3={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right)\left(\frac{{v_{20}^{\left(1\right)}|}_{\theta =-{\tau }_N}}{2}{q}_2{\text{e}}^{-i{\omega }_0{\tau }_N}+\frac{{v_{20}^{\left(2\right)}|}_{\theta =-{\tau }_N}}{2}{q}_1{\text{e}}^{-i{\omega }_0{\tau }_N}\right)+\frac{1}{2}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)q_1{q}_2^2{\text{e}}^{-3i{w}_0{\tau }_N}\\ +\frac{1}{2}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)Q^{*}{v_{20}^{\left(2\right)}|}_{\theta =-{\tau }_N}{q}_2{\text{e}}^{-i{w}_0{\tau }_N}+\frac{1}{6}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{‴}}_N\left(N^{*}\right)Q^{*}{q}_2^3{\text{e}}^{-3i{\omega }_0{\tau }_N}\end{array}$

$\begin{array}{c}{g}_4={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right)\left({v_{11}^{\left(1\right)}|}_{\theta =-{\tau }_N}{q}_2{\text{e}}^{-i{\omega }_0{\tau }_N}+{v_{11}^{\left(2\right)}|}_{\theta =-{\tau }_N}{q}_1{\text{e}}^{-i{\omega }_0{\tau }_N}\right)\\ +{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right)\left(\frac{{v_{20}^{\left(1\right)}|}_{\theta =-{\tau }_N}}{2}{\bar{q}}_2{\text{e}}^{i{\omega }_0{\tau }_N}+\frac{{v_{20}^{(2)}|}_{\theta =-{\tau }_N}}{2}{\bar{q}}_1{\text{e}}^{i{\omega }_0{\tau }_N}\right)\\ +\frac{1}{6}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)\left(2q_1{q}_2{\bar{q}}_2{\text{e}}^{-i{\omega }_0{\tau }_N}+{\bar{q}}_1{q}_2^2{\text{e}}^{-i{w}_0{\tau }_N}\right)\\ +{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)Q^{*}\left(2{v_{11}^{\left(2\right)}|}_{\theta =-{\tau }_N}{q}_2{\text{e}}^{-i{w}_0{\tau }_N}+{v_{20}^{\left(2\right)}|}_{\theta =-{\tau }_N}{\bar{q}}_2{\text{e}}^{i{\omega }_0{\tau }_N}\right)\\ +\frac{1}{6}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{‴}}_N\left(N^{*}\right)Q^{*}{q}_2^2{\bar{q}}_2{\text{e}}^{-i{\omega }_0{\tau }_N}\end{array}$

$\begin{array}{c}{g}_5={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right)\left({v_{11}^{\left(1\right)}|}_{\theta =-{\tau }_N}{\bar{q}}_2{\text{e}}^{i{\omega }_0{\tau }_N}+{v_{11}^{\left(2\right)}|}_{\theta =-{\tau }_N}{\bar{q}}_1{\text{e}}^{i{\omega }_0{\tau }_N}\right)\\ +{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right)\left(\frac{{v_{02}^{\left(1\right)}|}_{\theta =-{\tau }_N}}{2}{q}_2{\text{e}}^{-i{\omega }_0{\tau }_N}+\frac{{v_{02}^{\left(2\right)}|}_{\theta =-{\tau }_N}}{2}{q}_1{\text{e}}^{-i{\omega }_0{\tau }_N}\right)\\ +\frac{1}{6}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)\left(2{\bar{q}}_1{q}_2{\bar{q}}_2{\text{e}}^{i{\omega }_0{\tau }_N}+q_1{\bar{q}}_2^2{\text{e}}^{i{w}_0{\tau }_N}\right)\\ +{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)Q^{*}\left(2{v_{11}^{\left(2\right)}|}_{\theta =-{\tau }_N}{\bar{q}}_2{\text{e}}^{i{w}_0{\tau }_N}+{v_{02}^{\left(2\right)}|}_{\theta =-{\tau }_N}{q}_2{\text{e}}^{-i{\omega }_0{\tau }_N}\right)\\ +\frac{1}{6}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{‴}}_N\left(N^{*}\right)Q^{*}{\bar{q}}_2^2{q}_2{\text{e}}^{i{\omega }_0{\tau }_N}\end{array}$

$\begin{array}{c}{g}_6={\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{\prime }}_N\left(N^{*}\right)\left(\frac{{v_{02}^{\left(1\right)}|}_{\theta =-{\tau }_N}}{2}{\bar{q}}_2{\text{e}}^{i{\omega }_0{\tau }_N}+\frac{{v_{02}^{\left(2\right)}|}_{\theta =-{\tau }_N}}{2}{\bar{q}}_1{\text{e}}^{i{\omega }_0{\tau }_N}\right)+\frac{1}{2}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right){\bar{q}}_1{\bar{q}}_2^2{\text{e}}^{3i{w}_0{\tau }_N}\\ +\frac{1}{2}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{″}}_N\left(N^{*}\right)Q^{*}{v_{02}^{\left(2\right)}|}_{\theta =-{\tau }_N}{\bar{q}}_2{\text{e}}^{i{w}_0{\tau }_N}+\frac{1}{6}{\text{e}}^{{\tau }_{NP}{\eta }_{NP}-{\gamma }_0{\tau }_{NM}}{k^{‴}}_N\left(N^{*}\right)Q^{*}{\bar{q}}_2^3{\text{e}}^{3i{\omega }_0{\tau }_N}\end{array}$

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