## 3 Main Results
### Direction and Stability of Bifurcating Periodic Solutions
**Lemma 4**. : _The evolution of \(z\) concerning time t satisfies the following form:_  
\[\dot{z}(t)=j\omega_{0}z+g_{20}z^{2}+g_{11}z\bar{z}+g_{02}\bar{z}^{2}+g_{21}z^{ 2}z+\cdots \tag{23}\]  
_in which_  
\[g_{20}=\bar{D}f_{20},g_{11}=\bar{D}f_{11},g_{02}=\bar{D}f_{02}\,and\ \ g_{21}=\bar{D}f_{21} \tag{24}\]  
_within_  
\[ \begin{array}{l}f_{20}=2h_{2}e^{-2j\omega_{0}\tau_{0}}\\ f_{11}=2h_{2}\\ f_{02}=2h_{2}e^{2j\omega_{0}\tau_{0}}\\ f_{21}=2[2h_{2}e^{-j\omega_{0}\tau_{0}}y_{11}^{(2)}(-\tau)+h_{2}e^{j\omega_{0 }\tau_{0}}y_{20}^{(2)}(-\tau)+3h_{3}e^{-2j\omega_{0}\tau_{0}}]\end{array} \]  
Proof of Lemma 4.: Near the origin O, take q and \(\bar{q}\) as local coordinate direction, and \(z(t)\) and \(\bar{z}(t)\) is the manifold manifold in the \(C\). Based on Equations (21) and (23), we can derive the following equation:  
\[\dot{z}(t) = <q^{*},\dot{u}_{t}>=<q^{*},Lu_{t}+Fu_{t}>\] \[= j\omega_{0}z(t)+\tilde{q}^{\top}(0)Fu_{t}(0)\] \[= j\omega_{0}z+\bar{D}\left[1\;\tilde{q}_{2}^{*}\right]\left[ \begin{array}{c}f_{w}\\ 0\end{array} \right]\] \[= j\omega_{0}z+\bar{D}\left[1\;\tilde{q}_{2}^{*}\right]\left[F_{2 0}z^{2}+F_{11}z\bar{z}+F_{02}z^{2}+F_{21}z^{2}z+\cdots\right]\]  
in which  
Therefore, we summarized the calculations on \(y_{20}(-\tau)\) and \(y_{11}(-\tau)\) in the following lemma. **Lemma 5** ([19]). : _The coefficients \(y_{20}^{(2)}(-\tau),y_{11}^{(2)}(-\tau)\) of the center manifold \(y(t)\) are such that_  
\[y_{20}^{(2)}(-\tau)=\frac{2j}{\omega}\bar{D}l_{2}e^{-4j\omega\tau}+\frac{2j}{3 \omega}Dl_{2}+C_{202}e^{-2j\omega\tau} \tag{25}\]  
\[y_{11}^{(2)}(-\tau)=-\frac{2j}{\omega}\bar{D}h_{2}e^{-j\omega\tau}+\frac{2j}{ \omega}Dh_{2}e^{j\omega\tau}+C_{112}, \tag{26}\]  
_in which_  
\[C_{20}=\left[ \begin{array}{cc}2j\omega_{0}+\tau_{0}\mu&\tau_{0} Ke^{-2j\omega_{0}\tau_{0}}\\ -\tau_{0}&2j\omega_{0}+\tau_{0}d\end{array} \right]^{-1}\left[ \begin{array}{c }f_{20}\\ 0\end{array} \right]=\left[ \begin{array}{c}C_{201}\\ C_{202}\end{array} \right]\]  
According to Equations (25) and (26), up to this point, we can obtain all expressions of \(g_{ij}\). Therefore, we have the following theorem about the nonlinear dynamics of the system. For applying the Hassard formula,  
\[\left\{ \begin{aligned} &\mu_{2}=-\frac{\text{Re}[C_{1}(0)]}{a^{ \prime}(0)]}\\ &\tau_{2}=-\frac{\text{Im}[C_{1}(0)]+\mu_{2}\omega^{\prime}(0)}{ \omega_{0}}\\ &\beta_{2}=2\text{Re}[C_{1}(0)]\end{aligned} \right. \tag{27}\]  
with  
\[C_{1}(0)= \frac{j}{2\omega_{0}}[4h_{2}^{2}\bar{D}^{2}e^{-2j\omega_{0}\tau_ {0}}-4h_{2}|\bar{D}|^{2}-\frac{2}{3}h_{2}|\bar{D}e^{2j\omega_{0}\tau_{0}}|^{2}]\] \[+2h_{2}\bar{D}e^{-j\omega_{0}\tau_{0}}y_{11}^{(2)}(-\tau)+h_{2}e^ {j\omega_{0}\tau_{0}}y_{20}^{(2)}(-\tau)+3h_{3}e^{-j\omega_{0}\tau_{0}}\]  
It is noticed that all the bifurcation parameters can be determined in a systematic way, with the value of  
\[-\frac{\partial P/\partial\tau}{\partial P/\partial\lambda}\Big{|}_{(\tau_{0},j\omega_{0})}\;=a^{\prime}(0)+\omega^{\prime}(0)j\]  
**Theorem 2**. : _In Formulas (27), \(\mu_{2}\) determines the direction of the bifurcation; if \(\mu\,>0\) (\(<\)0), the periodic solution is called supercritical (subcritical). \(\tau_{2}\) determines the period of the periodic solution of the bifurcation; if \(\tau_{2}>0\) (\(<\)0), then the period increases (decreases). \(\beta_{2}\) determines the stability of the periodic solution. If (\(\beta_{2}<0\)) (\(>\)0), the periodic solution is asymptotically stable (unstable)._