In the previous section, we have obtained conditions for Hopf bifurcation to occur when \(\tau=\tau_{k}^{(j)}\). In this section, we shall derive the explicit formulas for determining the direction, stability, and period of these periodic solutions bifurcating from the equilibrium \(E_{*}(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}, p^{*})\) at these critical value of τ, by using techniques from the normal form and center manifold theory [15], Throughout this section, we always assume that system (2.2) undergoes Hopf bifurcation at the equilibrium \(E_{*}(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}, p^{*})\) for \(\tau=\tau_{k}^{(j)}\), and then \(\pm{i\omega_{k}}\) are corresponding purely imaginary roots of the characteristic equation at the equilibrium \(E_{*}(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}, p^{*})\). The linear part of system (1.4) at \(E_{*}(x_{1}^{*}, x_{2}^{*}, x_{3}^{*}, p^{*})\) is given by
$$ \left \{ \textstyle\begin{array}{l} \dot{y}_{1}(t)=a_{1}y_{1}(t)+a_{2}y_{4}(t), \\ \dot{y}_{2}(t)=b_{1}y_{2}(t)+b_{2}y_{4}(t), \\ \dot{y}_{3}(t)=c_{1}y_{3}(t)+c_{2}y_{4}(t), \\ \dot{y}_{4}(t)=d_{1}y_{1}(t-\tau)+d_{1}y_{2}(t-\tau)+d_{1}y_{3}(t-\tau), \end{array}\displaystyle \right .$$
(3.1)
and the non-linear part is given by
$$ f(\mu, u_{t})=(f_{1}, f_{2}, f_{3}, f_{4})^{T}, $$
(3.2)
where
$$\begin{aligned}& f_{1} = a_{3}y_{1}^{2}(t)+a_{4}y_{1}(t)y_{1}(t- \tau)+a_{5}y_{1}(t)y_{4}(t)+a_{6}y_{1}(t- \tau )y_{4}(t) \\& \hphantom{f_{1} ={}}{}+a_{7}y_{1}^{3}(t)+a_{8}y_{1}^{2}(t)y_{1}(t- \tau )+a_{9}y_{1}^{2}(t)y_{4}(t)+a_{10}y_{1}(t)y_{1}(t- \tau)y_{4}(t)+\mathrm{h.o.t.}, \\& f_{2} = b_{3}y_{2}^{2}(t)+b_{4}y_{2}(t)y_{2}(t- \tau)+b_{5}y_{2}(t)y_{4}(t)+b_{6}y_{2}(t- \tau )y_{4}(t) \\& \hphantom{f_{2} ={}}{}+b_{7}y_{2}^{3}(t)+b_{8}y_{2}^{2}(t)y_{2}(t- \tau )+b_{9}y_{2}^{2}(t)y_{4}(t)+b_{10}y_{2}(t)y_{2}(t- \tau)y_{4}(t)+\mathrm{h.o.t.}, \\& f_{3} = c_{3}y_{3}^{2}(t)+c_{4}y_{3}(t)y_{3}(t- \tau)+c_{5}y_{3}(t)y_{4}(t)+c_{6}y_{3}(t- \tau )y_{4}(t) \\& \hphantom{f_{3} ={}}{}+c_{7}y_{3}^{3}(t)+c_{8}y_{3}^{2}(t)y_{3}(t- \tau )+c_{9}y_{3}^{2}(t)y_{4}(t)+c_{10}y_{3}(t)y_{3}(t- \tau)y_{4}(t)+\mathrm{h.o.t.}, \\& f_{4} = d_{1}y_{1}(t-\tau)y_{4}(t)+d_{1}y_{2}(t- \tau)y_{4}(t)+d_{1}y_{3}(t-\tau )y_{4}(t), \end{aligned}$$
where
$$\begin{aligned}& a_{3}=k_{1}\beta_{1}p^{*}x_{1}^{*},\qquad a_{4}=-2k_{1}\beta_{1}p^{*},\qquad a_{5}=\frac{k_{1}\beta_{1}x_{1}^{*}(2p^{*}-1)}{1-p^{*}},\qquad a_{6}=\frac{k_{1}\beta_{1}x_{1}^{*}}{1-p^{*}}, \\& a_{7}=-\frac{k_{1}\beta_{1}p^{*}}{x_{1}^{*}}, \qquad a_{8}=\frac{k_{1}\beta_{1}p^{*}}{x_{1}^{*}}, \qquad a_{9}=-\frac{k_{1}\beta_{1}p^{*}}{1-p^{*}},\qquad a_{10}= \frac{k_{1}\beta_{1}(2p^{*}-1)}{1-p^{*}}, \\& b_{3}=k_{2}\beta_{2}p^{*}x_{2}^{*},\qquad b_{4}=-2k_{2}\beta_{2}p^{*},\qquad b_{5}= \frac{k_{2}\beta_{2}x_{2}^{*}(2p^{*}-1)}{1-p^{*}}, \\& b_{6}=\frac{k_{2}\beta_{2}x_{2}^{*}}{1-p^{*}},\qquad b_{7}=-\frac{k_{2}\beta_{2}p^{*}}{x_{2}^{*}},\qquad b_{8}=\frac{k_{2}\beta_{2}p^{*}}{x_{2}^{*}}, \\& b_{9}=-\frac{k_{2}\beta_{2}p^{*}}{1-p^{*}},\qquad b_{10}= \frac{k_{2}\beta_{2}(2p^{*}-1)}{1-p^{*}} ,\qquad d_{1}=k_{4}. \end{aligned}$$
Set \(\tau=\tau_{k}^{(j)}+\mu\) and denote
$$\begin{aligned} C^{k}[-\tau,0] =&\bigl\{ \varphi|\varphi:[-\tau,0]\rightarrow{R^{4}}, \mbox{each component of }\varphi \\ &\mbox{has }k\mbox{th order continuous derivative}\bigr\} . \end{aligned}$$
For convenience, denote \(C[-\tau, 0]\) by \(C^{0}[-\tau, 0]\).
For \(\varphi(\theta)=(\varphi_{1}(\theta), \varphi_{2}(\theta), \varphi_{3}(\theta), \varphi_{4}(\theta))^{T}\in{C([-\tau, 0], R^{4})}\), define a family of operators
$$ L_{\mu}\varphi=B\varphi(0)+B_{1}\varphi(-\tau) $$
(3.3)
and
$$ G(\mu,\varphi)=(k_{1}, k_{2}, k_{3}, k_{4})^{T}, $$
(3.4)
where
$$\begin{aligned}& B=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} a_{1} & 0 & 0 & a_{2} \\ 0 & b_{1} & 0 & b_{2} \\ 0 & 0 & c_{1} & c_{2} \\ 0 & 0 & 0 & 0 \end{array}\displaystyle \right ) , \qquad B_{1}= \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ d_{1} & d_{1} & d_{1} & 0 \end{array}\displaystyle \right ), \\& k_{1}=a_{3}\varphi_{1}^{2}(0)+a_{4} \varphi_{1}(0)\varphi_{1}(-\tau)+a_{5}\varphi _{1}(0)\varphi_{4}(0)+a_{6}\varphi_{1}(- \tau)\varphi_{4}(0) \\& \hphantom{k_{1}={}}{}+a_{7}\varphi_{1}^{3}(0)+a_{8} \varphi_{1}^{2}(0)\varphi_{1}(- \tau)+a_{9}\varphi _{1}^{2}(t) \varphi_{4}(0)+a_{10}\varphi_{1}(t) \varphi_{1}(-\tau)\varphi _{4}(0)+o\bigl(\|\varphi \|^{4}\bigr), \\& k_{2} = b_{3}\varphi_{2}^{2}(0)+b_{4} \varphi_{2}(0)\varphi_{2}(-\tau)+b_{5}\varphi _{2}(0)\varphi_{4}(0)+b_{6}\varphi_{2}(- \tau)y_{4}(0) \\& \hphantom{k_{2} ={}}{}+b_{7}\varphi_{2}^{3}(0)+b_{8} \varphi_{2}^{2}(0)\varphi_{2}(-\tau)+b_{9} \varphi _{2}^{2}(0)\varphi_{4}(0)+b_{10} \varphi_{2}(0)\varphi_{2}(-\tau)\varphi _{4}(0)+o \bigl(\|\varphi\|^{4}\bigr), \\& k_{3} = c_{3}\varphi_{3}^{2}(0)+c_{4} \varphi_{3}(0)\varphi_{3}(-\tau)+c_{5}\varphi _{3}(0)\varphi_{4}(0)+c_{6}\varphi_{3}(t- \tau)\varphi_{4}(0) \\& \hphantom{k_{3} ={}}{}+c_{7}\varphi_{3}^{3}(0)+c_{8} \varphi_{3}^{2}(0)\varphi_{3}\bigl(0-\tau _{k}^{(j)}\bigr)+c_{9}\varphi_{3}^{2}(0) \varphi_{4}(0)+c_{10}\varphi_{3}(0)\varphi _{3}(-\tau)\varphi_{4}(0)+o\bigl(\|\varphi\|^{4} \bigr), \\& k_{4} = d_{1}\varphi_{1}(-\tau) \varphi_{4}(0)+d_{1}\varphi_{2}(-\tau)\varphi _{4}(0)+d_{1}\varphi_{3}(-\tau) \varphi_{4}(0), \end{aligned}$$
and \(L_{\mu}\) is a one-parameter family of bounded linear operators in \(C([-\tau, 0], R^{4})\rightarrow{R^{4}}\). By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions \(\eta(\theta,\mu)\) in \([-\tau, 0]\rightarrow{R^{4^{2}}}\), such that
$$ L_{\mu}\varphi= \int_{-\tau}^{0}d\eta(\theta, \mu)\varphi(\theta). $$
(3.5)
In fact, choosing
$$ \eta(\theta,\mu)=B\delta(\theta)+B_{1}\delta(\theta+ \tau), $$
(3.6)
where \(\delta(\theta)\) is the Dirac delta function, then (3.5) is satisfied. For \((\varphi_{1}, \varphi_{2}, \varphi_{3}, \varphi_{4})\in(C^{1}[-\tau, 0], R^{4})\), define
$$ A(\mu)\varphi=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{d\varphi(\theta)}{d\theta}, &-\tau\leq\theta< 0, \\ \int_{-\tau}^{0}d\eta(s, \mu)\varphi(s), &\theta=0 \end{array}\displaystyle \right . $$
(3.7)
and
$$ R\varphi=\left \{ \textstyle\begin{array}{l@{\quad}l} 0, &-\tau\leq\theta< 0, \\ f(\mu, \varphi), &\theta=0. \end{array}\displaystyle \right . $$
(3.8)
Then (1.4) is equivalent to the abstract differential equation
$$ \dot{u_{t}}=A(\mu)u_{t}+R( \mu)u_{t}, $$
(3.9)
where \(u=(u_{1}, u_{2}, u_{3}, u_{4})^{T}\), \(u_{t}(\theta)=u(t+\theta)\), \(\theta\in[-\tau , 0]\).
For \(\psi\in{C([-\tau, 0], (R^{4})^{*})}\), define
$$ A^{*}\psi(s)=\left \{ \textstyle\begin{array}{l@{\quad}l} -\frac{d\psi(s)}{ds}, &s\in{(0,\tau]}, \\ \int_{-\tau}^{0}d\eta^{T}(t,0)\psi(-t),&s=0. \end{array}\displaystyle \right . $$
(3.10)
For \(\phi\in{C([-\tau, 0], R^{4})}\) and \(\psi\in{C([0, \tau], (R^{4})^{*})}\), define the bilinear form
$$ \langle\psi,\phi\rangle=\bar{\psi}(0)\phi(0)- \int_{-\tau}^{0} \int _{\xi=0}^{\theta}\psi^{T}(\xi-\theta)\, d \eta(\theta)\phi(\xi)\, d\xi, $$
(3.11)
where \(\eta(\theta)=\eta(\theta,0)\). We have the following result on the relation between the operators \(A=A(0)\) and \(A^{*}\).
Lemma 3.1
\(A=A(0)\)
and
\(A^{*}\)
are adjoint operators.
Proof
Let \(\phi\in{C^{1}([-\tau, 0], R^{4})}\) and \(\psi\in{C^{1}([0, \tau], R^{4})^{*}}\). It follows from (3.11) and the definitions of \(A=A(0)\) and \(A^{*}\) that
$$\begin{aligned} \bigl\langle \psi(s), A(0)\phi(\theta)\bigr\rangle =&\bar{\psi}(0)A(0)\phi(0)- \int_{-\tau _{k}^{(j)}}^{0} \int_{\xi=0}^{\theta} \bar{\psi}(\xi-\theta)\,d\eta( \theta)A(0)\phi(\xi)\,d\xi \\ =&\bar{\psi}(0) \int_{-\tau_{k}^{(j)}}^{0}d\eta(\theta)\phi(\theta )- \int_{-\tau_{k}^{(j)}}^{0} \int_{\xi=0}^{\theta} \bar{\psi}(\xi-\theta)\,d\eta( \theta)A(0)\phi(\xi)\,d\xi \\ =&\bar{\psi}(0) \int_{-\tau_{k}^{(j)}}^{0}d\eta(\theta)\phi(\theta )- \int_{-\tau_{k}^{(j)}}^{0}\bigl[ \bar{\psi}(\xi-\theta)\,d\eta( \theta)\phi(\xi)\bigr]_{\xi=0}^{\theta} \\ &{}+ \int_{-\tau_{k}^{(j)}}^{0} \int_{\xi=0}^{\theta} \frac{d\bar{\psi}(\xi-\theta)}{d\xi}\,d\eta(\theta)\phi( \xi)\,d\xi \\ =& \int_{-\tau_{k}^{(j)}}^{0}\bar{\psi}(-\theta)\,d\eta(\theta) \phi(0)- \int_{-\tau_{k}^{(j)}}^{0} \int_{\xi=0}^{\theta} \biggl[-\frac{d\bar{\psi}(\xi-\theta)}{d\xi} \biggr]\,d \eta(\theta)\phi(\xi )\,d\xi \\ =&A*\bar{\psi}(0)\phi(0)- \int_{-\tau_{k}^{(j)}}^{0} \int_{\xi =0}^{\theta} A^{*}\bar{\psi}(\xi-\theta)\,d\eta( \theta)\phi(\xi)\,d\xi \\ =&\bigl\langle A^{*}\psi(s), \phi(\theta)\bigr\rangle . \end{aligned}$$
This shows that \(A=A(0)\) and \(A^{*}\) are adjoint operators and the proof is complete. □
By the discussions in Section 2, we know that \(\pm{i\omega_{k}}\) are eigenvalues of \(A(0)\), and they are also eigenvalues of \(A^{*}\) corresponding to \(i\omega_{k}\) and \(-i\omega_{k}\), respectively. We have the following result.
Lemma 3.2
The vector
$$q(\theta)=(1, r_{1}, r_{2}, r_{3})^{T}e^{i\omega_{k}\theta}, \quad \theta\in[-\tau, 0], $$
is the eigenvector of
\(A(0)\)
corresponding to the eigenvalue
\(i\omega_{k}\), and
$$q^{*}(s)=D\bigl(1, r_{1}^{*}, r_{2}^{*}, r_{3}^{*} \bigr)e^{i\omega_{k}{s}}, \quad s\in[0,\tau], $$
is the eigenvector of
\(A^{*}\)
corresponding to the eigenvalue
\(-i\omega_{k}\), moreover, \(\langle q^{*}(s), q(\theta)\rangle=1\), where
$$ D=\frac{1}{C}, $$
(3.12)
where
$$C=1+\sum_{i=1}^{3}\bar{r}_{i}r_{i}^{*}+(1+ \bar{r}_{1}+\bar{r}_{2}+\bar {r}_{3})d_{1}r_{3}^{*}e^{i\omega_{k}\tau_{k}^{(j)}}. $$
Proof
Let \(q(\theta)\) be the eigenvector of \(A(0)\) corresponding to the eigenvalue \(i\omega_{k}\) and \(q^{*}(s)\) be the eigenvector of \(A^{*}\) corresponding to the eigenvalue \(-i\omega_{k}\), namely, \(A(0)q(\theta)=i\omega_{k}q(\theta)\) and \(A^{*}q(s)=-i\omega_{k}q^{*}(s)\). From the definitions of \(A(0)\) and \(A^{*}\), we have \(A(0)q(\theta)=dq(\theta)/d\theta\) and \(A^{*}q(s)=-dq^{*}(s)/ds\). Thus, \(q(\theta)=q(0)e^{i\omega_{k}\theta}\) and \(q^{*}(s)=q(0)e^{i\omega_{0}s}\). In addition,
$$\begin{aligned} \int_{-\tau_{k}^{(j)}}^{0}d\eta(\theta)q(\theta) =&\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} a_{1} & 0 & 0 & a_{2} \\ 0 & b_{1} & 0 & b_{2} \\ 0 & 0 & c_{1} & c_{2} \\ 0 & 0 & 0 & 0 \end{array}\displaystyle \right )q(0)+\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ d_{1} & d_{1} & d_{1} & 0 \end{array}\displaystyle \right )q\bigl(-\tau_{k}^{(j)}\bigr) \\ =&A(0)q(0)=i\omega_{k}q(0). \end{aligned}$$
(3.13)
That is,
$$ \left ( \textstyle\begin{array}{@{}c@{}} a_{1}+a_{2}r_{3} \\ b_{1}r_{1}+b_{2}r_{3} \\ c_{1}r_{2}+c_{2}r_{3} \\ d_{1}(1+r_{1}+r_{2})e^{-i\omega_{k}\tau_{k}^{(j)}} \end{array}\displaystyle \right ) =\left ( \textstyle\begin{array}{@{}c@{}} i\omega_{k} \\ i\omega_{k}r_{1} \\ i\omega_{k}r_{2} \\ i\omega_{k}r_{3} \end{array}\displaystyle \right ). $$
(3.14)
Therefore, we can easily obtain
$$r_{1}=\frac{b_{2}(a_{1}-i\omega_{k})}{a_{2}(b_{1}-i\omega_{k})},\qquad r_{2}=\frac{c_{2}(a_{1}-i\omega_{k})}{a_{2}(c_{1}-i\omega_{k})}, \qquad r_{3}=\frac{i\omega_{k}-a_{1}}{a_{2}}. $$
On the other hand,
$$\begin{aligned} \int_{-\tau_{k}^{(j)}}^{0}q^{*}(-t)\,d\eta(t) =&\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} a_{1} & 0 & 0 & 0\\ 0 & b_{1} & 0 & 0 \\ 0 & 0 & c_{1} & 0 \\ a_{2} & b_{2} & c_{2} & 0 \end{array}\displaystyle \right )q(0)+\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 0 & 0 & 0 & d_{1} \\ 0 & 0 & 0 & d_{1} \\ 0 & 0 & 0 & d_{1} \\ 0 & 0 & 0 & 0 \end{array}\displaystyle \right )q^{*}\bigl(-\tau_{k}^{(j)}\bigr) \\ =&A^{*}q^{*}(0)=-i\omega_{0}q^{*}(0). \end{aligned}$$
(3.15)
Namely,
$$ \left ( \textstyle\begin{array}{@{}c@{}} a_{1}+d_{1}r_{3}^{*}e^{-i\omega_{k}\tau_{k}^{(j)}} \\ b_{1}r_{1}^{*}+d_{1}r_{3}^{*}e^{-i\omega_{k}\tau_{k}^{(j)}} \\ c_{1}r_{2}^{*}+d_{1}r_{3}^{*}e^{-i\omega_{k}\tau_{k}^{(j)}} \\ a_{2}+b_{2}r_{1}^{*}+c_{2}r_{3}^{*} \end{array}\displaystyle \right ) =\left ( \textstyle\begin{array}{@{}c@{}} -i\omega_{k} \\ -i\omega_{k}r_{1}^{*} \\ -i\omega_{k}r_{2}^{*} \\ -i\omega_{k}r_{3}^{*} \end{array}\displaystyle \right ). $$
(3.16)
Therefore, we can easily obtain
$$r_{1}^{*}=\frac{a_{1}+i\omega_{k}}{b_{1}+i\omega_{k}},\qquad r_{2}^{*}= \frac{a_{1}+i\omega_{k}}{c_{1}+i\omega_{k}},\qquad r_{3}^{*}=-\frac{a_{1}+i\omega_{k}}{d_{1}e^{-i\omega_{k}\tau_{k}^{(j)}}}. $$
In the sequel, we shall verify that \(\langle q^{*}(s), q(\theta)\rangle=1\). In fact, from (3.11), we have
$$\begin{aligned} \bigl\langle q^{*}(s), q(\theta)\bigr\rangle =&\bar{D}\bigl(1, \bar{r}_{1}^{*}, \bar{r}_{2}^{*}, \bar{r}_{3}^{*}\bigr) (1, r_{1}, r_{2}, r_{3})^{T} \\ &{}- \int_{-\tau_{k}^{(j)}}^{0} \int_{\xi=0}^{\theta}\bar{D}\bigl(1, \bar{r}_{1}^{*}, \bar{r}_{2}^{*}, \bar{r}_{3}^{*}\bigr) e^{-i\omega_{k}(\xi-\theta)} \,d\eta( \theta) (1, r_{1}, r_{2}, r_{3})^{T}e^{i\omega_{k}\xi} \,d\xi \\ =&\bar{D} \Biggl[1+\sum_{i=1}^{3}r_{i} \bar{r}_{i}^{*}- \int_{-\tau_{k}^{(j)}}^{0}\bigl(1, \bar{r}_{1}^{*}, \bar{r}_{2}^{*}, \bar{r}_{3}^{*}\bigr){\theta}e^{i\omega_{k}\theta}\,d \eta(\theta) (1, r_{1}, r_{2}, r_{3})^{T} \Biggr] \\ =&\bar{D} \Biggl\{ 1+\sum_{i=1}^{3}r_{i} \bar{r}_{i}^{*}+\bigl(1, \bar{r}_{1}^{*}, \bar{r}_{2}^{*}, \bar{r}_{3}^{*}\bigr) B_{1}e^{-i\omega_{k}\tau_{k}^{(j)}}(1, r_{1}, r_{2}, r_{3})^{T} \Biggr\} \\ =&\bar{D} \Biggl[1+\sum_{i=1}^{3}r_{i} \bar{r}_{i}^{*}+(1+r_{1}+r_{2}+r_{3})d_{1} \bar {r}_{3}^{*}e^{-i\omega_{k}\tau_{k}^{(j)}} \Biggr]=1. \end{aligned}$$
□
Next, we use the same notations as those in Hassard et al. [15], and we first compute the coordinates to describe the center manifold \(C_{0}\) at \(\mu=0\). Let \(y_{t}\) be the solution of equation (1.4) when \(\mu=0\).
Define
$$ z(t)=\bigl\langle q^{*}, y_{t}\bigr\rangle , \qquad W(t, \theta)=y_{t}(\theta)-2\operatorname{Re}\bigl\{ z(t)q(\theta)\bigr\} $$
(3.17)
on the center manifold \(C_{0}\), and we have
$$ W(t, \theta)=W\bigl(z(t), \bar{z}(t), \theta\bigr), $$
(3.18)
where
$$ W\bigl(z(t), \bar{z}(t), \theta\bigr)=W(z, \bar{z})=W_{20} \frac{z^{2}}{2}+W_{11}z\bar{z}+W_{02} \frac{\bar{z}^{2}}{2}+ \cdots $$
(3.19)
and z and z̄ are local coordinates for center manifold \(C_{0}\) in the direction of \(q^{*}\) and \(\bar{q}^{*}\). Noting that W is also real if \(y_{t}\) is real, we consider only real solutions. For solutions \(y_{t}\in{C_{0}}\) of (1.4),
$$\begin{aligned} \dot{z}(t) =&\bigl\langle q^{*}(s), \dot{x}_{t}\bigr\rangle =\bigl\langle q^{*}(s), A(0)u_{t}+R(0)u_{t}\bigr\rangle \\ =&\bigl\langle q^{*}(s), A(0)y_{t}\bigr\rangle +\bigl\langle q^{*}(s), R(0)y_{t}\bigr\rangle \\ =&\bigl\langle A^{*}q^{*}(s), y_{t}\bigr\rangle +\bar{q}^{*}(0)R(0)y_{t} \\ &- \int_{-\tau_{k}^{(j)}}^{0} \int_{\xi=0}^{\theta} \bar{q}^{*}(\xi-\theta)\,d\eta( \theta)A(0)R(0)y_{t}(\xi)\,d\xi \\ =&\bigl\langle i\omega_{k}q^{*}(s), y_{t}\bigr\rangle + \bar{q}^{*}(0)f\bigl(0, y_{t}(\theta)\bigr) \\ \stackrel{\mathrm{def}}{=}&i\omega_{k}z(t)+\bar{q}^{*}(0)f_{0} \bigl(z(t), \bar{z}(t)\bigr). \end{aligned}$$
(3.20)
That is,
$$ \dot{z}(t)=i\omega_{k}{z}+g(z, \bar{z}), $$
(3.21)
where
$$ g(z, \bar{z})=g_{20}\frac{z^{2}}{2}+g_{11}z \bar{z}+g_{02} \frac{\bar{z}^{2}}{2}+g_{21}\frac{z^{2}\bar{z}}{2}+ \cdots. $$
(3.22)
Hence, we have
$$\begin{aligned} g(z, \bar{z}) =&\bar{q}^{*}(0)f_{0}(z, \bar{z})=f(0, y_{t}) \\ =&\bar{D}\bigl(1, \bar{r}_{1}^{*}, \bar{r}_{2}^{*}, \bar{r}_{3}^{*}\bigr) \bigl(f_{1}(0, y_{t}), f_{2}(0, y_{t}), f_{3}(0, y_{t}), f_{4}(0, y_{t})\bigr)^{T}, \end{aligned}$$
(3.23)
where
$$\begin{aligned}& f_{1}(0,y_{t}) = a_{3}y_{1t}^{2}(0)+a_{4}y_{1t}(0)y_{1t} \bigl(-\tau _{k}^{(j)}\bigr)+a_{5}y_{1t}(0)y_{4t}(0)+a_{6}y_{1t} \bigl(-\tau _{k}^{(j)}\bigr)y_{4t}(0) \\& \hphantom{f_{1}(0,y_{t}) ={}}{}+a_{7}y_{1t}^{3}(0)+a_{8}y_{1t}^{2}(0)y_{1t} \bigl(-\tau _{k}^{(j)}\bigr)+a_{9}y_{1t}^{2}(0)y_{4t}(0) \\& \hphantom{f_{1}(0,y_{t}) ={}}{}+a_{10}y_{1t}(0)y_{1t} \bigl(-\tau _{k}^{(j)}\bigr)y_{4t}(0)+ \mathrm{h.o.t.}, \\& f_{2}(0,y_{t}) = b_{3}y_{2t}^{2}(0)+b_{4}y_{2t}(0)y_{2t} \bigl(-\tau _{k}^{(j)}\bigr)+b_{5}y_{2t}(0)y_{4t}(0)+b_{6}y_{2t} \bigl(-\tau _{k}^{(j)}\bigr)y_{4t}(0) \\& \hphantom{f_{2}(0,y_{t}) ={}}{}+b_{7}y_{2t}^{3}(0)+b_{8}y_{2t}^{2}(0)y_{2t} \bigl(-\tau _{k}^{(j)}\bigr)+b_{9}y_{2t}^{2}(0)y_{4t}(0) \\& \hphantom{f_{2}(0,y_{t}) ={}}{}+b_{10}y_{2t}(0)y_{2t} \bigl(-\tau _{k}^{(j)}\bigr)y_{4t}(0)+ \mathrm{h.o.t.}, \\& f_{3}(0,y_{t}) = c_{3}y_{3t}^{2}(0)+c_{4}y_{3t}(0)y_{3t} \bigl(-\tau _{k}^{(j)}\bigr)+c_{5}y_{3t}(0)y_{4t}(0)+c_{6}y_{3t} \bigl(-\tau _{k}^{(j)}\bigr)y_{4t}(0) \\& \hphantom{f_{3}(0,y_{t}) ={}}{}+c_{7}y_{3t}^{3}(0)+c_{8}y_{3t}^{2}(0)y_{3t} \bigl(-\tau _{k}^{(j)}\bigr)+c_{9}y_{3t}^{2}(0)y_{4t}(0) \\& \hphantom{f_{3}(0,y_{t}) ={}}{}+c_{10}y_{3t}(0)y_{3t} \bigl(-\tau _{k}^{(j)}\bigr)y_{4t}(0)+ \mathrm{h.o.t.}, \\& f_{4}(0,y_{t}) = d_{1}y_{1t}\bigl(- \tau_{k}^{(j)}\bigr)y_{4t}(0)+d_{1}y_{2t} \bigl(-\tau _{k}^{(j)}\bigr)y_{4t}(0)+d_{1}y_{3t} \bigl(-\tau_{k}^{(j)}\bigr)y_{4t}(0). \end{aligned}$$
Noticing that
$$y_{t}(\theta)=\bigl(y_{1t}(\theta), y_{2t}( \theta), y_{3t}(\theta ),y_{4t}(\theta)\bigr)^{T} =W(t,\theta)+zq(\theta)+\bar{z}\bar{q} $$
and
$$q(\theta)=(1, r_{1}, r_{2}, r_{3})^{T}e^{i\omega_{k}\theta}, $$
we have
$$\begin{aligned}& y_{1t}(0) = z+\bar{z}+W_{20}^{(1)}(0) \frac{z^{2}}{2}+W_{11}^{(1)}(0)z\bar {z}+W_{02}^{(1)}(0) \frac{\bar{z}^{2}}{2}+\cdots, \\& y_{2t}(0) = r_{1}z+\bar{r}_{1} \bar{z}+W_{20}^{(2)}(0)\frac{z^{2}}{2} +W_{11}^{(2)}(0)z \bar{z}+W_{02}^{(2)}(0)\frac{\bar{z}^{2}}{2}+\cdots, \\& y_{3t}(0) = r_{2}z+\bar{r}_{2} \bar{z}+W_{20}^{(3)}(0)\frac {z^{2}}{2}+W_{11}^{(3)}(0)z \bar{z} +W_{02}^{(3)}(0)\frac{\bar{z}^{2}}{2}+\cdots, \\& y_{4t}(0) = r_{3}z+\bar{r}_{3} \bar{z}+W_{20}^{(4)}(0)\frac {z^{2}}{2}+W_{11}^{(4)}(0)z \bar{z} +W_{02}^{(4)}(0)\frac{\bar{z}^{2}}{2}+\cdots, \\& y_{1t}\bigl(-\tau_{k}^{(j)}\bigr) = e^{-i\omega_{k}\tau_{k}^{(j)}}z+e^{i\omega_{k}\tau _{k}^{(j)}}\bar{z} +W_{20}^{(1)}\bigl(- \tau_{k}^{(j)}\bigr)\frac{z^{2}}{2} \\& \hphantom{y_{1t}\bigl(-\tau_{k}^{(j)}\bigr) ={}}{}+W_{11}^{(1)} \bigl(-\tau _{k}^{(j)}\bigr)z\bar{z}+W_{02}^{(1)} \bigl(-\tau_{k}^{(j)}\bigr)\frac{\bar{z}^{2}}{2}+\cdots, \\& y_{2t}\bigl(-\tau_{k}^{(j)}\bigr) = r_{1}e^{-i\omega_{k}\tau_{k}^{(j)}}z+\bar {r}_{1}e^{i\omega_{k}\tau_{k}^{(j)}}\bar{z} +W_{20}^{(2)}\bigl(-\tau_{k}^{(j)}\bigr) \frac{z^{2}}{2} \\& \hphantom{y_{2t}\bigl(-\tau_{k}^{(j)}\bigr) ={}}{}+W_{11}^{(2)}\bigl(-\tau _{k}^{(j)} \bigr)z\bar{z}+W_{02}^{(2)}\bigl(-\tau_{k}^{(j)} \bigr)\frac{\bar{z}^{2}}{2}+\cdots , \\& y_{3t}\bigl(-\tau_{k}^{(j)}\bigr) = r_{2}e^{-i\omega_{k}\tau_{k}^{(j)}}z+\bar {r}_{2}e^{i\omega_{k}\tau_{k}^{(j)}}\bar{z} +W_{20}^{(3)}\bigl(-\tau_{k}^{(j)}\bigr) \frac{z^{2}}{2} \\& \hphantom{y_{3t}\bigl(-\tau_{k}^{(j)}\bigr) ={}}{}+W_{11}^{(3)}\bigl(-\tau _{k}^{(j)} \bigr)z\bar{z}+W_{02}^{(3)}\bigl(-\tau_{k}^{(j)} \bigr)\frac{\bar{z}^{2}}{2}+\cdots , \\& y_{4t}\bigl(-\tau_{k}^{(j)}\bigr) = r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}z+\bar {r}_{3}e^{i\omega_{k}\tau_{k}^{(j)}}\bar{z} +W_{20}^{(4)}\bigl(-\tau_{k}^{(j)}\bigr) \frac{z^{2}}{2} \\& \hphantom{y_{4t}\bigl(-\tau_{k}^{(j)}\bigr) ={}}{}+W_{11}^{(4)}\bigl(-\tau _{k}^{(j)} \bigr)z\bar{z}+W_{02}^{(4)}\bigl(-\tau_{k}^{(j)} \bigr)\frac{\bar{z}^{2}}{2}+\cdots. \end{aligned}$$
From (3.22) and (3.23), we have
$$\begin{aligned} g(z, \bar{z}) =&\bar{q}^{*}(0)f_{0}(z,\bar{z}) \\ =&\bar{D} \bigl[f_{1}(0, y_{t})+\bar{r}_{1}^{*}f_{2}(0, x_{t})+\bar{r}_{2}^{*}f_{3}(0, y_{t})+ \bar{r}_{3}^{*}f_{4}(0, y_{t}) \bigr] \\ =&\bar{D} \bigl[a_{3}+a_{4}e^{-i\omega_{k}\tau _{k}^{(j)}}+a_{5}r_{3}+a_{6}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \\ &{}+\bar{r}_{1}^{*} \bigl(b_{3}r_{1}^{2}+b_{4}r_{1}^{2}e^{-i\omega_{k}\tau _{k}^{(j)}}+b_{5}r_{1}r_{3}+b_{6}r_{1}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr) \\ &{}+\bar{r}_{2}^{*} \bigl(c_{3}r_{2}^{2}+c_{4}r_{2}^{2}e^{-i\omega_{k}\tau _{k}^{(j)}}+c_{5}r_{2}r_{3}+c_{6}r_{2}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr) \\ &{} +\bar{r}_{3}^{*} \bigl(d_{1}r_{3}e^{-i\omega_{k}\tau _{k}^{(j)}}+d_{1}r_{1}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}+d_{1}r_{2}r_{3}e^{-i\omega_{k}\tau _{k}^{(j)}} \bigr) \bigr]z^{2} \\ &{}+\bar{D} \bigl[2a_{3}+2a_{4}\operatorname{Re}\bigl\{ r_{3}e^{i\omega_{k}\tau_{k}^{(j)}}\bigr\} +2a_{5}\operatorname{Re} \{r_{3}\}+2a_{6}\operatorname{Re}\bigl\{ r_{3}e^{i\omega_{k}\tau_{k}^{(j)}} \bigr\} \\ &{}+r_{1}^{*} \bigl(2b_{3}|r_{1}|^{2}+2b_{4} \operatorname{Re}\bigl\{ r_{1}\bar{r}_{1}e^{-i\omega_{k}\tau _{k}^{(j)}} \bigr\} +2b_{5}\operatorname{Re}\{r_{1}\bar{r}_{3} \}+2b_{6}\operatorname{Re}\bigl\{ r_{1}\bar {r}_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr\} \bigr) \\ &{}+r_{2}^{*} \bigl(2c_{3}|r_{2}|^{2}+2c_{4} \operatorname{Re}\bigl\{ |r_{2}|^{2}e^{i\omega_{k}\tau _{k}^{(j)}}\bigr\} +2c_{5}\operatorname{Re}\{r_{2}\bar{r}_{3} \}+2c_{6}\operatorname{Re}\bigl\{ r_{2}\bar {r}_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr\} \bigr) \\ &{} +r_{3}^{*} \bigl(2d_{1}\operatorname{Re}\bigl\{ r_{3}e^{i\omega_{k}\tau_{k}^{(j)}}\bigr\} +2d_{1}\operatorname{Re}\bigl\{ r_{1}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr\} +2d_{1}\operatorname{Re}\bigl\{ r_{2}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr\} \bigr) \bigr]z\bar{z} \\ &{}+\bar{D} \bigl[a_{3}+a_{4}e^{i\omega_{k}\tau_{k}^{(j)}}+a_{5} \bar{r}_{3}+a_{6}\bar {r}_{3}e^{i\omega_{k}\tau_{k}^{(j)}} \\ &{}+r_{1}^{*} \bigl(b_{3}\bar{r}_{1}^{2}+b_{4} \bar{r}_{1}^{2}e^{i\omega_{k}\tau _{k}^{(j)}}+b_{5} \bar{r}_{1}\bar{r}_{3}+b_{6}\bar{r}_{1} \bar{r}_{3}e^{i\omega_{k}\tau _{k}^{(j)}} \bigr) \\ &{}+r_{2}^{*} \bigl(c_{3}\bar{r}_{2}^{2}+c_{4} \bar{r}_{2}^{2}e^{i\omega_{k}\tau _{k}^{(j)}}+c_{5} \bar{r}_{2}r_{3}+c_{6}\bar{r}_{2} \bar{r}_{3}e^{i\omega_{k}\tau _{k}^{(j)}} \bigr) \\ &{} +r_{3} \bigl(d_{1}\bar{r}_{3}e^{i\omega_{k}\tau_{k}^{(j)}}+d_{1} \bar {r}_{1}\bar{r}_{3}e^{i\omega_{k}\tau_{k}^{(j)}}+d_{1} \bar{r}_{2}\bar{r}_{3}e^{i\omega _{k}\tau_{k}^{(j)}} \bigr) \bigr] \bar{z}^{2} \\ &{}+\bar{D} \biggl\{ r_{1}^{*} \biggl[b_{3} \bigl(2r_{1}W_{11}^{(2)}(0)+W_{201}^{(2)}(0) \bar{r}_{1} \bigr) \\ &{}+b_{4} \biggl(r_{1}W_{11}^{(2)} \bigl(-\tau_{k}^{(j)}\bigr)+\frac{1}{2}\bar {r}_{1}W_{20}^{(2)}\bigl(-\tau_{k}^{(j)} \bigr) \\ &{} +\frac{1}{2}\bar{r}_{1}W_{20}^{(2)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}} +r_{1}W_{11}^{(2)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}} \biggr)+b_{5} \biggl(r_{1}W_{11}^{(4)}(0)+ \frac{1}{2}\bar{r}_{1}W_{20}^{(4)}(0) \\ &{} +\frac{1}{2}\bar{r}_{3}W_{20}^{(2)}(0)+r_{3}W_{11}^{(2)}(0) \biggr) +b_{6} \biggl(r_{1}W_{11}^{(4)}(0)+ \frac{1}{2}\bar {r}_{1}W_{20}^{(4)}(0)e^{i\omega_{k}\tau_{k}^{(j)}} \\ &{} +\frac{1}{2}\bar{r}_{3}W_{20}^{(2)}(- \tau)e^{i\omega_{k}\tau _{k}^{(j)}}+r_{3}W_{11}^{(2)}\bigl(- \tau_{k}^{(j)}\bigr) \biggr) \\ &{}+3b_{7}r_{1}^{2} \bar{r}_{1}+b_{8} \bigl(r_{1}^{2} \bar{r}_{1}e^{i\omega_{k}\tau _{k}^{(j)}}+2r_{1}^{2} \bar{r}_{1}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr) \\ &{} +b_{9}\bigl(r_{1}^{2}\bar{r}_{3}+2|r_{1}|^{2}r_{3} \bigr)+b_{10} \bigl(r_{1}^{2}e^{-i\omega _{k}\tau_{k}^{(j)}}+|r_{1}|^{2}r_{3}e^{i\omega_{k}\tau_{k}^{(j)}} +|r_{1}|^{2}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr) \biggr] \\ &{}+r_{2}^{*} \biggl[3c_{3}r_{2}^{2} \bar{r}_{2}+c_{4} \biggl(r_{2}W_{11}^{(3)} \bigl(-\tau _{k}^{(j)}\bigr)+\frac{1}{2} \bar{r}_{2}W_{20}^{(3)}\bigl(-\tau_{k}^{(j)} \bigr) \\ &{}+\frac{1}{2}\bar{r}_{2}W_{20}^{(3)} \bigl(-\tau_{k}^{(j)}\bigr)e^{i\omega_{k}\tau _{k}^{(j)}}+r_{2}W_{11}^{(3)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}} \biggr) \\ &{} +c_{5} \biggl(r_{2}W_{11}^{(4)}(0)+ \frac{1}{2}\bar{r}_{2}W_{20}^{(4)}(0) + \frac{1}{2}\bar{r}_{3}W_{20}^{(3)}(0)+r_{3}W_{11}^{(3)}(0) \biggr) \\ &{}+c_{6} \biggl(r_{2}W_{11}^{(4)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}}+ \frac {1}{2}\bar{r}_{2}W_{20}^{(4)}(0)e^{i\omega_{k}\tau_{k}^{(j)}} \\ &{}+\frac{1}{2}\bar{r}_{3}W_{20}^{(3)}\bigl(- \tau_{k}^{(j)}\bigr)+r_{3}W_{11}^{(3)} \bigl(-\tau _{k}^{(j)}\bigr) \biggr) \\ &{}+3c_{7}r_{2}^{2}\bar{r}_{2}+c_{8} \bigl(r_{2}^{2}\bar{r}_{2}e^{i\omega_{k}\tau _{k}^{(j)}}+2r_{2}^{2} \bar{r}_{2}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr) \\ &{}+c_{9} \bigl(r_{2}^{2}\bar{r}_{3}+2|r_{2}|^{2}r_{3} \bigr)+c_{10} \bigl(r_{2}^{2}\bar {r}_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \\ &{} +|r_{2}|^{2}r_{3}e^{-i\omega_{k}\tau _{k}^{(j)}}+|r_{2}|^{2}r_{3}e^{i\omega_{k}\tau_{k}^{(j)}} \bigr) \biggr] +\bar{r}_{3}^{*}d_{1}\biggl[W_{11}^{(4)}(0)+ \frac{1}{2}W_{20}^{(4)}(0)+\frac {1}{2} \bar{r}_{3}W_{20}^{(1)}(0) \\ &{}+r_{3}W_{11}^{(1)}(0)+r_{1}W_{11}^{(4)}(0)e^{-i\omega_{k}\tau _{k}^{(j)}}+ \frac{1}{2}\bar{r}_{1}W_{20}^{(4)}(0)e^{i\omega_{k}\tau_{k}^{(j)}} \\ &{}+\frac{1}{2}\bar{r}_{3}W_{20}^{(4)}\bigl(- \tau_{k}^{(j)}\bigr)+r_{3}W_{11}^{(2)} \bigl(-\tau _{k}^{(j)}\bigr)+r_{2}W_{11}^{(4)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}} \\ &{}+ \frac{1}{2}\bar {r}_{2}W_{20}^{(4)}(0)e^{i\omega_{k}\tau_{k}^{(j)}} +\frac{1}{2}\bar{r}_{3}W_{20}^{(3)}\bigl(- \tau_{k}^{(j)}\bigr)e^{i\omega_{k}\tau _{k}^{(j)}}+r_{3}W_{11}^{(3)} \bigl(-\tau_{k}^{(j)}\bigr)\biggr]z^{2}\bar{z}+\cdots \biggr\} \end{aligned}$$
and we obtain
$$\begin{aligned}& g_{20} = 2\bar{D}\bigl[a_{3}+a_{4}e^{-i\omega_{k}\tau _{k}^{(j)}}+a_{5}r_{3}+a_{6}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \\ & \hphantom{g_{20} ={}}{}+\bar{r}_{1}^{*}\bigl(b_{3}r_{1}^{2}+b_{4}r_{1}^{2}e^{-i\omega_{k}\tau _{k}^{(j)}}+b_{5}r_{1}r_{3}+b_{6}r_{1}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr) \\& \hphantom{g_{20} ={}}{}+\bar{r}_{2}^{*}\bigl(c_{3}r_{2}^{2}+c_{4}r_{2}^{2}e^{-i\omega_{k}\tau _{k}^{(j)}}+c_{5}r_{2}r_{3}+c_{6}r_{2}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr) \\& \hphantom{g_{20} ={}}{}+\bar{r}_{3}^{*}\bigl(d_{1}r_{3}e^{-i\omega_{k}\tau _{k}^{(j)}}+d_{1}r_{1}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}+d_{1}r_{2}r_{3}e^{-i\omega_{k}\tau _{k}^{(j)}} \bigr)\bigr], \\& g_{11} = \bar{D}\bigl[2a_{3}+2a_{4} \operatorname{Re}\bigl\{ r_{3}e^{i\omega_{k}\tau _{k}^{(j)}}\bigr\} +2a_{5} \operatorname{Re}\{r_{3}\}+2a_{6}\operatorname{Re}\bigl\{ r_{3}e^{i\omega_{k}\tau _{k}^{(j)}}\bigr\} \\& \hphantom{g_{11} ={}}{}+r_{1}^{*}\bigl(2b_{3}|r_{1}|^{2}+2b_{4} \operatorname{Re}\bigl\{ r_{1}\bar{r}_{1}e^{-i\omega_{k}\tau _{k}^{(j)}} \bigr\} +2b_{5}\operatorname{Re}\{r_{1}\bar{r}_{3} \}+2b_{6}\operatorname{Re}\bigl\{ r_{1}\bar {r}_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr\} \bigr) \\& \hphantom{g_{11} ={}}{}+r_{2}^{*}\bigl(2c_{3}|r_{2}|^{2}+2c_{4} \operatorname{Re}\bigl\{ |r_{2}|^{2}e^{i\omega_{0}\tau _{k}^{(j)}}\bigr\} +2c_{5}\operatorname{Re}\{r_{2}\bar{r}_{3} \}+2c_{6}\operatorname{Re}\bigl\{ r_{2}\bar {r}_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr\} \bigr) \\& \hphantom{g_{11} ={}}{}+r_{3}^{*}\bigl(2d_{1}\operatorname{Re}\bigl\{ r_{3}e^{i\omega_{k}\tau_{k}^{(j)}}\bigr\} +2d_{1}\operatorname{Re}\bigl\{ r_{1}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr\} +2d_{1}\operatorname{Re}\bigl\{ r_{2}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr\} \bigr)\bigr], \\& g_{02} = 2\bar{D}\bigl[a_{3}+a_{4}e^{i\omega_{k}\tau_{k}^{(j)}}+a_{5} \bar {r}_{3}+a_{6}\bar{r}_{3}e^{i\omega_{k}\tau_{k}^{(j)}} \\& \hphantom{g_{02} ={}}{}+r_{1}^{*}\bigl(b_{3}\bar{r}_{1}^{2}+b_{4} \bar{r}_{1}^{2}e^{i\omega_{k}\tau _{k}^{(j)}}+b_{5} \bar{r}_{1}\bar{r}_{3}+b_{6}\bar{r}_{1} \bar{r}_{3}e^{i\omega_{k}\tau _{k}^{(j)}}\bigr) \\& \hphantom{g_{02} ={}}{}+r_{2}^{*}\bigl(c_{3}\bar{r}_{2}^{2}+c_{4} \bar{r}_{2}^{2}e^{i\omega_{k}\tau _{k}^{(j)}}+c_{5} \bar{r}_{2}r_{3}+c_{6}\bar{r}_{2} \bar{r}_{3}e^{i\omega_{k}\tau _{k}^{(j)}}\bigr) \\& \hphantom{g_{02} ={}}{}+r_{3}\bigl(d_{1}\bar{r}_{3}e^{i\omega_{k}\tau_{k}^{(j)}}+d_{1} \bar {r}_{1}\bar{r}_{3}e^{i\omega_{k}\tau_{k}^{(j)}}+d_{1} \bar{r}_{2}\bar{r}_{3}e^{i\omega _{k}\tau_{k}^{(j)}}\bigr)\bigr], \\& g_{21} = 2\bar{D} \biggl\{ r_{1}^{*}\biggl[b_{3} \bigl(2r_{1}W_{11}^{(2)}(0)+W_{201}^{(2)}(0) \bar{r}_{1}\bigr) +b_{4}\biggl(r_{1}W_{11}^{(2)} \bigl(-\tau_{k}^{(j)}\bigr)+\frac{1}{2}\bar {r}_{1}W_{20}^{(2)}\bigl(-\tau_{k}^{(j)} \bigr) \\& \hphantom{g_{21} ={}}{}+\frac{1}{2}\bar{r}_{1}W_{20}^{(2)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}} +r_{1}W_{11}^{(2)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}} \biggr)+b_{5} \biggl(r_{1}W_{11}^{(4)}(0)+ \frac{1}{2}\bar{r}_{1}W_{20}^{(4)}(0) \\& \hphantom{g_{21} ={}}{}+\frac{1}{2}\bar{r}_{3}W_{20}^{(2)}(0)+r_{3}W_{11}^{(2)}(0) \biggr) +b_{6}\biggl(r_{1}W_{11}^{(4)}(0)+ \frac{1}{2}\bar {r}_{1}W_{20}^{(4)}(0)e^{i\omega_{k}\tau_{k}^{(j)}} \\& \hphantom{g_{21} ={}}{}+\frac{1}{2}\bar{r}_{3}W_{20}^{(2)} \bigl(-\tau_{k}^{(j)}\bigr)e^{i\omega _{k}\tau_{k}^{(j)}}+r_{3}W_{11}^{(2)} \bigl(-\tau_{k}^{(j)}\bigr)\biggr) \\& \hphantom{g_{21} ={}}{}+3b_{7}r_{1}^{2} \bar{r}_{1}+b_{8}\bigl(r_{1}^{2} \bar{r}_{1}e^{i\omega_{k}\tau _{k}^{(j)}}+2r_{1}^{2} \bar{r}_{1}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr) \\& \hphantom{g_{21} ={}}{}+b_{9}\bigl(r_{1}^{2} \bar{r}_{3}+2|r_{1}|^{2}r_{3} \bigr)+b_{10}\bigl(r_{1}^{2}e^{-i\omega _{k}\tau_{k}^{(j)}}+|r_{1}|^{2}r_{3}e^{i\omega_{k}\tau_{k}^{(j)}} +|r_{1}|^{2}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr)\biggr] \\& \hphantom{g_{21} ={}}{}+r_{2}^{*}\biggl[3c_{3}r_{2}^{2} \bar{r}_{2}+c_{4}\biggl(r_{2}W_{11}^{(3)} \bigl(-\tau _{k}^{(j)}\bigr)+\frac{1}{2} \bar{r}_{2}W_{20}^{(3)}\bigl(-\tau_{k}^{(j)} \bigr) +\frac{1}{2}\bar{r}_{2}W_{20}^{(3)}(- \tau)e^{i\omega_{k}\tau_{k}^{(j)}} \\& \hphantom{g_{21} ={}}{}+r_{2}W_{11}^{(3)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}} \biggr)+c_{5} \biggl(r_{2}W_{11}^{(4)}(0)+ \frac{1}{2}\bar{r}_{2}W_{20}^{(4)}(0) + \frac{1}{2}\bar{r}_{3}W_{20}^{(3)}(0)+r_{3}W_{11}^{(3)}(0) \biggr) \\& \hphantom{g_{21} ={}}{}+c_{6}\biggl(r_{2}W_{11}^{(4)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}}+ \frac {1}{2}\bar{r}_{2}W_{20}^{(4)}(0)e^{i\omega_{k}\tau_{k}^{(j)}} +\frac{1}{2}\bar{r}_{3}W_{20}^{(3)}\bigl(- \tau_{k}^{(j)}\bigr)+r_{3}W_{11}^{(3)} \bigl(-\tau _{k}^{(j)}\bigr)\biggr) \\& \hphantom{g_{21} ={}}{}+3c_{7}r_{2}^{2} \bar{r}_{2}+c_{8}\bigl(r_{2}^{2} \bar{r}_{2}e^{i\omega_{k}\tau _{k}^{(j)}}+2r_{2}^{2} \bar{r}_{2}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr) +c_{9}\bigl(r_{2}^{2} \bar{r}_{3}+2|r_{2}|^{2}r_{3} \bigr)+c_{10}\bigl(r_{2}^{2}\bar {r}_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \\& \hphantom{g_{21} ={}}{}+|r_{2}|^{2}r_{3}e^{-i\omega_{k}\tau _{k}^{(j)}}+|r_{2}|^{2}r_{3}e^{i\omega_{k}\tau_{k}^{(j)}} \bigr)\biggr] +\bar{r}_{3}^{*}d_{1}\biggl[W_{11}^{(4)}(0)+ \frac{1}{2}W_{20}^{(4)}(0)+\frac {1}{2} \bar{r}_{3}W_{20}^{(1)}(0) \\& \hphantom{g_{21} ={}}{}+r_{3}W_{11}^{(1)}(0)+r_{1}W_{11}^{(4)}(0)e^{-i\omega_{k}\tau _{k}^{(j)}}+ \frac{1}{2}\bar{r}_{1}W_{20}^{(4)}(0)e^{i\omega_{k}\tau_{k}^{(j)}} \\& \hphantom{g_{21} ={}}{}+\frac{1}{2}\bar{r}_{3}W_{20}^{(4)}\bigl(- \tau_{k}^{(j)}\bigr)+r_{3}W_{11}^{(2)} \bigl(-\tau _{k}^{(j)}\bigr)+r_{2}W_{11}^{(4)}(0)e^{-i\omega_{k}\tau_{k}^{(j)}} \\& \hphantom{g_{21} ={}}{}+ \frac{1}{2}\bar {r}_{2}W_{20}^{(4)}(0)e^{i\omega_{k}\tau_{k}^{(j)}} +\frac{1}{2}\bar{r}_{3}W_{20}^{(3)}\bigl(- \tau_{k}^{(j)}\bigr)e^{i\omega_{k}\tau _{k}^{(j)}}+r_{3}W_{11}^{(3)} \bigl(-\tau_{k}^{(j)}\bigr)\biggr]\biggr\} . \end{aligned}$$
For
$$\begin{aligned}& W_{20}^{(2)}(0), \qquad W_{11}^{(2)}(0), \qquad W_{11}^{(2)}\bigl(-\tau_{k}^{(j)} \bigr),\qquad W_{20}^{(2)}\bigl(-\tau_{k}^{(j)} \bigr),\qquad W_{11}^{(4)}(0),\qquad W_{20}^{(4)}(0), \\& W_{11}^{(4)}(0),\qquad W_{20}^{(4)}(0), \qquad W_{11}^{(2)}\bigl(-\tau_{k}^{(j)} \bigr),\qquad W_{11}^{(3)}\bigl(-\tau_{k}^{(j)}\bigr), \qquad W_{11}^{(2)}(0),\qquad W_{20}^{(3)}(0), \\& W_{11}^{(3)}(0),\qquad W_{11}^{(4)}(0), \qquad W_{20}^{(3)}\bigl(-\tau_{k}^{(j)} \bigr),\qquad W_{11}^{(3)}\bigl(-\tau_{k}^{(j)} \bigr),\qquad W_{20}^{(3)}\bigl(-\tau_{k}^{(j)} \bigr) \end{aligned}$$
unknown in \(g_{21}\), we still need to compute them.
From (3.9), (3.23), we have
$$\begin{aligned} W' =&\left \{ \textstyle\begin{array}{l@{\quad}l} AW-2\operatorname{Re}{\{\bar{q}^{*}(0)f_{0}q(\theta)\}}, &-\tau_{k}^{(j)}\leq \theta< 0, \\ AW-2\operatorname{Re}{\{\bar{q}^{*}(0)f_{0}q(\theta)\}}+f_{0}, &\theta=0 \end{array}\displaystyle \right . \\ \stackrel{\mathrm{def}}{=}&AW+H(z, \bar{z}, \theta), \end{aligned}$$
(3.24)
where
$$ H(z,\bar{z},\theta)=H_{20}(\theta)\frac{z^{2}}{2}+H_{11}( \theta)z\bar {z}+H_{02}(\theta) \frac{\bar{z}^{2}}{2}+\cdots. $$
(3.25)
Comparing the coefficients, we obtain
$$\begin{aligned}& (A-2i\omega_{0})W_{20}(\theta)=-H_{20}( \theta), \end{aligned}$$
(3.26)
$$\begin{aligned}& AW_{11}(\theta)=-H_{11}(\theta), \\& \ldots. \end{aligned}$$
(3.27)
And we know that for \(\theta\in[-\tau_{k}^{(j)}, 0)\),
$$ H(z,\bar{z}, \theta)=-\bar{q}^{*}(0)f_{0}q(\theta)-q^{*}(0) \bar{f}_{0}\bar{q}(\theta) =-g(z, \bar{z})q(\theta)-\bar{g}(z, \bar{z}) \bar{q}(\theta). $$
(3.28)
Comparing the coefficients of (3.25) with (3.28) gives
$$\begin{aligned}& H_{20}(\theta)=-g_{20}q(\theta)- \bar{g}_{02}\bar{q}(\theta), \end{aligned}$$
(3.29)
$$\begin{aligned}& H_{11}(\theta)=-g_{11}q(\theta)-\bar{g}_{11} \bar{q}(\theta). \end{aligned}$$
(3.30)
From (3.26), (3.29), and the definition of A, we get
$$ \dot{W}_{20}(\theta)=2i\omega_{k}W_{20}( \theta)+g_{20}q(\theta) +\bar{g}_{02}\bar{q}(\theta). $$
(3.31)
Noting that \(q(\theta)=q(0)e^{i\omega_{0}\theta}\), we have
$$ W_{20}(\theta)=\frac{ig_{20}}{\omega_{k}}q(0) e^{i\omega_{k}\theta}+ \frac{i\bar{g}_{02}}{3\omega_{k}} \bar{q}(0)e^{-i\omega_{k}\theta} +E_{1}e^{2i\omega_{k}\theta}, $$
(3.32)
where \(E_{1}\) is a constant vector. Similarly, from (3.27), (3.30), and the definition of A, we have
$$\begin{aligned}& \dot{W}_{11}(\theta)=g_{11}q(\theta) +\bar{g}_{11} \bar{q}(\theta), \end{aligned}$$
(3.33)
$$\begin{aligned}& W_{11}(\theta)=-\frac{ig_{11}}{\omega_{k}}q(0) e^{i\omega_{k}\theta}+ \frac{i\bar{g}_{11}}{\omega_{k}} \bar{q}(0)e^{-i\omega_{k}\theta}+E_{2}, \end{aligned}$$
(3.34)
where \(E_{2}\) is a constant vector.
In the following, we shall seek appropriate \(E_{1}\), \(E_{2}\) in (3.32), (3.34), respectively. It follows from the definition of A and (3.29), (3.30) that
$$ \int_{-\tau_{k}^{(j)}}^{0}d\eta(\theta)W_{20}(\theta)= 2i\omega_{k}W_{20}(0)-H_{20}(0) $$
(3.35)
and
$$ \int_{-\tau_{k}^{(j)}}^{0}d\eta(\theta)W_{11}( \theta)=-H_{11}(0), $$
(3.36)
where \(\eta(\theta)=\eta(0, \theta)\).
From (3.30), we have
$$ H_{20}(0)=-g_{20}q(0)-\bar{g}_{02} \bar{q}(0)+(H_{1}, H_{2}, H_{3}, H_{4})^{T}, $$
(3.37)
where
$$\begin{aligned}& H_{1}=2\bigl(a_{3}+a_{4}e^{-i\omega_{k}\tau_{k}^{(j)}}+a_{5}r_{3}+a_{6}r_{3}e^{-i\omega_{k}\tau _{k}^{(j)}} \bigr), \\& H_{2}=2 \bigl(b_{3}r_{1}^{2}+b_{4}r_{1}^{2}e^{-i\omega_{k}\tau _{k}^{(j)}}+b_{5}r_{1}r_{3}+b_{6}r_{1}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr), \\& H_{3}=2 \bigl(c_{3}r_{2}^{2}+c_{4}r_{2}^{2}e^{-i\omega_{k}\tau _{k}^{(j)}}+c_{5}r_{2}r_{3}+c_{6}r_{2}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr), \\& H_{4}=2 \bigl(d_{1}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}+d_{1}r_{1}r_{3}e^{-i\omega_{k}\tau _{k}^{(j)}}+d_{1}r_{2}r_{3}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr). \end{aligned}$$
From (3.31), we have
$$ H_{11}(0)=-g_{11}q(0)-\bar{g}_{11}(0) \bar{q}(0)+(P_{1}, P_{2}, P_{3}, P_{4})^{T}, $$
(3.38)
where
$$\begin{aligned}& P_{1}=2a_{3}+2a_{4}\operatorname{Re}\bigl\{ r_{3}e^{i\omega_{k}\tau_{k}^{(j)}}\bigr\} +2a_{5}\operatorname{Re}\{ r_{3}\}+2a_{6}\operatorname{Re}\bigl\{ r_{3}e^{i\omega_{k}\tau_{k}^{(j)}} \bigr\} , \\& P_{2}=2b_{3}|r_{1}|^{2}+2b_{4} \operatorname{Re}\bigl\{ r_{1}\bar{r}_{1}e^{-i\omega_{k}\tau _{k}^{(j)}} \bigr\} +2b_{5}\operatorname{Re}\{r_{1}\bar{r}_{3} \}+2b_{6}\operatorname{Re}\bigl\{ r_{1}\bar {r}_{3}e^{-i\omega_{k}\tau_{k}^{(j)}}\bigr\} , \\& P_{3}=2c_{3}|r_{2}|^{2}+2c_{4} \operatorname{Re}\bigl\{ |r_{2}|^{2}e^{i\omega_{k}\tau_{k}^{(j)}}\bigr\} +2c_{5}\operatorname{Re}\{r_{2}\bar{r}_{3} \}+2c_{6}\operatorname{Re}\bigl\{ r_{2}\bar{r}_{3}e^{-i\omega _{k}\tau_{k}^{(j)}} \bigr\} , \\& P_{4}=2d_{1}\operatorname{Re}\bigl\{ r_{3}e^{i\omega_{k}\tau_{k}^{(j)}} \bigr\} +2d_{1}\operatorname{Re}\bigl\{ r_{1}e^{-i\omega_{k}\tau_{k}^{(j)}} \bigr\} +2d_{1}\operatorname{Re}\bigl\{ r_{2}e^{-i\omega_{k}\tau _{k}^{(j)}} \bigr\} . \end{aligned}$$
From (3.26), (3.27), and the definition of A, we have
$$ \left \{ \textstyle\begin{array}{l} BW_{20}(0)+B_{1}W_{20}(-\tau_{k}^{(j)})=2i\omega_{k}W_{20}(0)-H_{20}(0), \\ BW_{11}(0)+B_{1}W_{11}(-\tau_{k}^{(j)})=-H_{11}(0). \end{array}\displaystyle \right . $$
(3.39)
Noting that
$$\begin{aligned}& \biggl(i\omega_{k}I- \int_{-\tau_{k}^{(j)}}^{0}e^{i\omega_{k}\theta} \,d\eta(\theta) \biggr)q(0)=0, \end{aligned}$$
(3.40)
$$\begin{aligned}& \biggl(-i\omega_{k}I- \int_{-\tau_{k}^{(j)}}^{0}e^{-i\omega_{k}\theta} \,d\eta(\theta) \biggr) \bar{q}(0)=0, \end{aligned}$$
(3.41)
and substituting (3.36) and (3.41) into (3.39), we have
$$ \biggl(2i\omega_{k}I- \int_{-\tau_{k}^{(j)}}^{0}e^{2i\omega_{k}\theta} \,d\eta(\theta) \biggr)E_{1}=(H_{1}, H_{2}, H_{3}, H_{4})^{T}. $$
(3.42)
That is,
$$ \det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 2i\omega_{k}-a_{1} & 0 & 0 & -a_{2} \\ 0 & 2i\omega_{k}-b_{1} & 0 & -b_{2} \\ 0 & 0 & 2i\omega_{k}-c_{1} & -c_{2} \\ -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} &-d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & 2i\omega_{k} \end{array}\displaystyle \right )\left ( \textstyle\begin{array}{@{}c@{}} E_{1}^{(1)} \\ E_{1}^{(2)} \\ E_{1}^{(3)} \\ E_{1}^{(4)} \end{array}\displaystyle \right )=\left ( \textstyle\begin{array}{@{}c@{}} H_{1} \\ H_{2} \\ H_{3} \\ H_{4} \end{array}\displaystyle \right ). $$
(3.43)
Hence,
$$ E_{1}^{(1)}=\frac{\Delta_{11}}{\Delta_{1}},\qquad E_{1}^{(2)}=\frac{\Delta _{12}}{\Delta_{1}},\qquad E_{1}^{(3)}= \frac{\Delta_{13}}{\Delta_{1}},\qquad E_{1}^{(4)}=\frac{\Delta _{14}}{\Delta_{1}}, $$
(3.44)
where
$$\begin{aligned}& \Delta_{1}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 2i\omega_{k}-a_{1} & 0 & 0 & -a_{2} \\ 0 & 2i\omega_{k}-b_{1} & 0 & -b_{2} \\ 0 & 0 & 2i\omega_{k}-c_{1} & -c_{2} \\ -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} &-d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & 2i\omega_{k} \end{array}\displaystyle \right ), \\& \Delta_{11}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} H_{1} & 0 & 0 & -a_{2} \\ H_{2} & 2i\omega_{k}-b_{1} & 0 & -b_{2} \\ H_{3}1 & 0 & 2i\omega_{k}-c_{1} & -c_{2} \\ H_{4} & -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} &-d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & 2i\omega_{k} \end{array}\displaystyle \right ), \\& \Delta_{12}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 2i\omega_{k}-a_{1} & H_{1} & 0 & -a_{2} \\ 0 & H_{2} & 0 & -b_{2} \\ 0 & H_{3} & 2i\omega_{k}-c_{1} & -c_{2} \\ -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & H_{4} &-d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & 2i\omega_{k} \end{array}\displaystyle \right ), \\& \Delta_{13}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 2i\omega_{k}-a_{1} & 0 & H_{1} & -a_{2} \\ 0 & 2i\omega_{k}-b_{1} & H_{2} & -b_{2} \\ 0 & 0 & H_{3} & -c_{2} \\ -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} &H_{4} & 2i\omega_{k} \end{array}\displaystyle \right ), \\& \Delta_{14}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} 2i\omega_{k}-a_{1} & 0 & 0 & H_{1} \\ 0 & 2i\omega_{k}-b_{1} & 0 & H_{2} \\ 0 & 0 & 2i\omega_{k}-c_{1} & H_{3} \\ -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & -d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} &-d_{1}e^{-2i\omega_{k}\tau_{k}^{(j)}} & H_{4} \end{array}\displaystyle \right ). \end{aligned}$$
Similarly, substituting (3.37) and (3.42) into (3.40), we have
$$ \biggl( \int_{-\tau_{k}^{(j)}}^{0} d\eta(\theta) \biggr)E_{2}=(P_{1}, P_{2}, P_{3}, P_{4})^{T}. $$
(3.45)
That is,
$$ \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} a_{1} & 0 & 0 & a_{2} \\ 0 & b_{1} & 0 & b_{2} \\ 0 & 0 & c_{1} & c_{2} \\ d_{1} & d_{1} & d_{1} & 0 \end{array}\displaystyle \right )\left ( \textstyle\begin{array}{@{}c@{}} E_{2}^{(1)} \\ E_{2}^{(2)} \\ E_{2}^{(3)} \\ E_{2}^{(4)} \end{array}\displaystyle \right ) =\left ( \textstyle\begin{array}{@{}c@{}} P_{1} \\ P_{2} \\ P_{3}\\ P_{4} \end{array}\displaystyle \right ). $$
(3.46)
Hence,
$$ E_{2}^{(1)}=\frac{\Delta_{21}}{\Delta_{2}},\qquad E_{2}^{(2)}=\frac{\Delta _{22}}{\Delta_{2}},\qquad E_{2}^{(3)}= \frac{\Delta_{23}}{\Delta_{2}},\qquad E_{2}^{(4)}=\frac{\Delta _{24}}{\Delta_{2}}, $$
(3.47)
where
$$\begin{aligned}& \Delta_{2}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} a_{1} & 0 & 0 & a_{2} \\ 0 & b_{1} & 0 & b_{2} \\ 0 & 0 & c_{1} & c_{2} \\ d_{1} & d_{1} & d_{1} & 0 \end{array}\displaystyle \right ), \\& \Delta_{21}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} -P_{1} & 0 & 0 & a_{2} \\ -P_{2} & b_{1} & 0 & b_{2} \\ -P_{3} & 0 & c_{1} & c_{2} \\ -P_{4} & d_{1} & d_{1} & 0 \end{array}\displaystyle \right ), \\& \Delta_{22}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} a_{1} & -P_{1} & 0 & a_{2} \\ 0 & -P_{2} & 0 & b_{2} \\ 0 & -P_{3} & c_{1} & c_{2} \\ d_{1} & -P_{4} & d_{1} & 0 \end{array}\displaystyle \right ), \\& \Delta_{23}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} a_{1} & 0 & -P_{1} & a_{2} \\ 0 & b_{1} & -P_{2} & b_{2} \\ 0 & 0 & -P_{3} & c_{2} \\ d_{1} & d_{1} & -P_{4} & 0 \end{array}\displaystyle \right ), \\& \Delta_{24}=\det \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} a_{1} & 0 & 0 & -P_{1} \\ 0 & b_{1} & 0 & -P_{2} \\ 0 & 0 & c_{1} & -P_{3} \\ d_{1} & d_{1} & d_{1} & -P_{4} \end{array}\displaystyle \right ). \end{aligned}$$
From (3.32), (3.34), (3.44), (3.47), we can calculate \(g_{21}\) and derive the following values:
$$\begin{aligned}& c_{1}(0)=\frac{i}{2\omega_{k}} \biggl(g_{20}g_{11}-2|g_{11}|^{2}- \frac {|g_{02}|^{2}}{3} \biggr)+\frac{g_{21}}{2}, \\& \mu_{2}=-\frac{\operatorname{Re}\{c_{1}(0)\}}{\operatorname{Re}\{\lambda'(\tau _{k}^{(j)})\} }, \\& \beta_{2}=2\operatorname{Re} {\bigl(c_{1}(0)\bigr)}, \\& T_{2}=-\frac{\operatorname{Im}{\{c_{1}(0)\}}+\mu_{2}\operatorname{Im}\{\lambda'(\tau _{k}^{(j)})\}}{\omega_{k}}. \end{aligned}$$
These formulas give a description of the Hopf bifurcation periodic solutions of (1.4) at \(\tau=\tau_{k}^{(j)}\) on the center manifold. From the discussion above, we have the following result.
Theorem 3.3
For system (1.4), if (H1)-(H4) hold, the periodic solution is supercritical (subcritical) if
\(\mu_{2}>0\) (\(\mu_{2}<0\)); the bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable) if
\(\beta_{2}<0\) (\(\beta_{2}>0\)); the periods of the bifurcating periodic solutions increase (decrease) if
\(T_{2}>0\) (\(T_{2}<0\)).