## 3 Main Results
### Direction and Stability of Bifurcating Periodic Solutions
In addition, in the linear part of the system analyzed earlier, the differential operator corresponds to the eigenvalue of \(\pm j\omega_{0}\), we denote the associating eigenvector \(L\) with \(j\omega_{0}\), and we can obtain the following formula:  
\[q(\theta)=q(0)e^{j\omega_{0}\theta}\]  
and, when \(\theta=0\), the action of the linear operator L is equivalent to the differentiation, from which we can derive the following characteristic equation:  
\[Lq(\theta)=j\omega_{0}q(\theta)=j\omega_{0}q(0)e^{j\omega_{0}\theta}\]  
The above formula is expanded as  
\[Aq(0)+Bq(0)e^{-j\omega_{0}\overline{\omega}_{0}}=j\omega_{0}q(0)\]  
thus  
\[-\mu q_{1}-Kq_{2}e^{-j\omega_{0}\overline{\omega}_{0}}=j\omega_{0}q_{1}\]  
\[q_{1}-dq_{2}=j\omega_{0}q_{2}\]  
Set \(q_{2}=1\), then \(q_{1}=j\omega_{0}+d\),  
\[q(\theta)=q(0)e^{j\omega_{0}\theta}=\left[ \begin{array}{c}q_{1}\\ 1\end{array} \right]e^{j\omega_{0}\theta},\;q_{1}=j\omega_{0}+d\]  
Thus, we obtain the result in (19). In addition, corresponding to the above discussions, there is the following statement for conjugate eigenvectors. **Lemma 3**. : \(q^{*}(\theta)\) _is encoded as an adjoint eigenvector of \(L^{*}\) in Equation (16) associating with \(-j\omega_{0}\) and such that_  
\[q^{*}(\theta)=\bar{D}\left[ \begin{array}{c}1\\ q_{2}^{*}\end{array} \right]e^{j\omega_{0}\theta} \tag{20}\]  
_within \(q_{2}^{*}=-j\omega_{0}+\mu\), \(D=\frac{1}{<q_{1}q_{2}^{*}>}=\frac{1}{\mu+d-2j\omega_{0}-\tau Ke^{j\omega_{0} \overline{\omega}_{0}}}\)_  
Proof of Lemma 3.: From Equation (16), we can observe a statement similar to the above, that is, when the range of \(\theta\) is \([-\tau,0)\), the action of the conjugate linear operator \(L^{*}\) can be colloquially understood as taking the derivative, and \(-j\omega_{0}\) is an eigenvalue of \(L^{*}\), we can obtain the following equation:  
\[q_{1}^{*}(\theta)=q_{1}^{*}(0)e^{-j\omega_{0}\theta}\]  
and, when \(\theta=0\), the action of the conjugate linear operator \(L^{*}\) is equivalent to the differentiation, from which we can derive the following characteristic equation:  
\[L^{*}q_{1}^{*}(\theta)=-j\omega_{0}q_{1}^{*}(\theta)=-j\omega_{0}q_{1}^{*}(0)e ^{j\omega_{0}\theta}\]  
Expanding the above formula, we can obtain the adjoint matrix equation  
\[A^{\mathrm{T}}q_{1}^{*}(0)+B^{\mathrm{T}}q_{1}^{*}(0)e^{j\omega_{0}\overline{ \omega}_{0}}=-j\omega_{0}q_{1}^{*}(0)\]  
thus  
\[-\mu q_{01}^{*}+q_{2}^{*}=-j\omega_{0}q_{01}^{*}\] \[-dq_{2}^{*}-Kq_{01}^{*}e^{j\omega_{0}\overline{\omega}_{0}}=-j \omega_{0}q_{2}^{*}\]Setting \(q_{01}^{*}=1\), we have \(q_{2}^{*}=-j\omega_{0}+\mu\),  
\[q_{1}^{*}(\theta)=q_{1}^{*}(0)e^{-j\omega_{0}\theta}=\left[ \begin{array}{c}1\\ q_{2}^{*}\end{array} \right]e^{-j\omega_{0}\theta}\]  
In this case, the \(q_{1}^{*}(\theta)\) is not the normal orthogonal basis; in order to construct manifold near the origin \(C(\mu)\) on the manifold \(C(0)\), \(\mu=0\) coordinate system, we need to normalize it by introducing the inner product form references Equation (15), and the normalized eigenvector \(q^{*}(\theta)=Dq_{1}^{*}(\theta)\) should be such that \(<q^{*},q>=1\), and we have  
\[<q,q_{1}^{*}>=\widetilde{q}_{1}^{\top}(0)q(0)+e^{j\omega_{0}\tau_{0}} \tilde{q}_{1}^{\top}(0)Bq(0)=\frac{1}{D}\]  
thus  
\[D=\frac{1}{<q,q_{1}^{*}>}=\frac{1}{\mu+d-2j\omega_{0}-\tau Ke^{j\omega_{0}\tau_ {0}}}\]  
Thus, we obtain the result in (20). In the above procedure, we give the adjoint form and the adjoint equation of the solution operator to the similarity transformation, and \(q(\theta)\) and \(q^{*}(\theta)\) can be obtained. Then, we can use the conjugate eigenvector \(q^{*}(\theta)\) to project the central epidemic. After determining the accompanying vectors, the solution space can be divided into central and stable manifolds Like in literature [7], we define  
\[z=<q^{*},u_{t}> \tag{21}\]  
and  
\[y(t,\theta)=u_{t}(\theta)-z(t)q(\theta)-z(t)\bar{q}(\theta) \tag{22}\]  
The solution of the system is composed of popular linear combinations, \(z\) is the projected solution at time t, and \(y\) is a high-order term.