## 3 Main Results
### Direction and Stability of Bifurcating Periodic Solutions  
In this section, we will consider the bifurcation solution of the system (1). In addition, we use the time delay \(\tau\) as a bifurcation parameter. The system Equation (1) can be rewritten by Taylor expansion as follows:  
\[ \begin{array}{l}\dot{m}(t)=-Kp(t-\tau)-\mu m(t)+h_{2}p^{2}(t-\tau)+h_{3}p^{ 3}(t-\tau)\\ \dot{p}(t)=m(t)-dp(t)\end{array}  \tag{11}\]  
We could write system (11) as the matrix form  
\[\dot{u}(t)=Au(t)+Bu(t-\tau)+F \tag{12}\]  
within,  
\[F=\left[ \begin{array}{c}f_{1}\\ f_{2}\end{array} \right]=\left[ \begin{array}{c}h_{2}p^{2}(t-\tau)+h_{3}p^{3}( t-\tau)\\ 0\end{array} \right]\]  
within \(h_{2}=\frac{\delta m(\delta n-\pi+\delta+1)}{2(\delta+1)^{3}p_{2}^{2}}\), \(h_{3}=\frac{\delta n[6\delta n(1+\delta)(n-1)-(n-1)(n-2)(1+\delta)^{2}-6(\beta n )^{2}]}{6(\delta+1)^{4}p_{2}^{2}}\). In order to apply [PERSON] bifurcation theorem, it is necessary to turn Equation (12) to abstract differential equations, and the notation \(u_{t}(\theta)\) is introduced:\[u_{t}(\theta)=u(t+\theta)=\left[ \begin{array}{c}m(t+\theta)\\ p(t+\theta)\end{array} \right]\text{, }-\tau\leq\theta\leq 0 \tag{13}\]  
Equation (13) can induce the differential operator:  
\[Lu_{t}(\theta)=\left\{ \begin{array}{c}\frac{du_{t}(\theta)}{d\theta},\qquad- \tau\leq\theta<0\\ \\ \int_{-\tau}^{0}d\eta(\theta)u_{t}(\theta),\ \theta=0\end{array} \right. \tag{14}\]  
where \(L\) is a continuous linear function mapping \(C\) into \(R^{2}\), and \(\eta\) is an \(2\times 2\) matrix function of bounded variation  
\[d\eta(\theta)=[A\delta(\theta)+B\delta(\theta+\tau)]d\theta\]  
and, within \(\delta(\cdot)\), is the Dirac delta function. It should be noted that, in the form of the following inner product [19], we can derive the conjugate linear operator of this linear operator  
\[<\mu^{*},v>=\tilde{\mu}^{*T}(0)v(0)+\int_{-\tau}^{0}\tilde{\mu}^{*T}(\xi+\tau )Bv(\xi) \tag{15}\]  
and the conjugate linear operator is shown below  
\[L^{*}u_{t}^{*}(\theta)=\left\{ \begin{array}{c}-\frac{du_{t}^{*}}{d\theta}, \qquad-\tau\leq\theta<0\\ \\ \int_{-\tau}^{0}d\eta^{\mathrm{T}}(\theta)u_{t}^{*}(-\theta),\ \theta=0\end{array} \right. \tag{16}\]  
with  
\[d\eta^{\mathrm{T}}(\theta)=[A^{\mathrm{T}}\delta(\theta)+B^{\mathrm{T}}\delta (\theta+\tau)]d\theta\]  
Therefore, Equation (14) serves as the boundary value condition of the differential operator, and the following abstract differential equation can be derived  
\[\dot{u}(t)=Lu_{t}+Fu_{t} \tag{17}\]  
within, the nonlinear part in Equation (17) goes  
\[F(u_{t})=\left\{ \begin{array}{c}\left[\begin{array}{c}0\\ 0\end{array}\right]\text{, }-\tau\leq\theta<0\\ \\ f_{w}\\ 0\end{array} \right.\qquad\theta=0 \tag{18}\]  
with \(f_{w}=h_{2}p^{2}(t-\tau)+h_{3}p^{3}(t-\tau)\) in Equation (18). According to the theory of eigendecomposition, what we need to calculate is the eigenvector \(q(\theta)\) corresponding to \(j\omega_{0}\) of the linear operator \(L\). We give the following lemma. Lemma 2: \(q(\theta)\) _is encoded in Equation (14) as the eigenvector of \(L\), associated with \(j\omega_{0}\) and satisfying_  
\[q(\theta)=\left[ \begin{array}{c}q_{1}\\ 1\end{array} \right]e^{j\omega_{0}\theta} \tag{19}\]  
_within \(q_{1}=j\omega_{0}+d\)_  
Proof of Lemma 2: From Equation (14), we can see that, when the range of \(\theta\) is \([-\tau,0)\), the action of the linear operator \(L\) can be colloquially understood as taking the derivative.