As pointed out in [18], it is interesting to determine the direction and stability of periodic solutions bifurcating from the positive equilibrium \(E^{*}\). In this section, we shall derive explicit formulae for determining the properties of the Hopf bifurcation at \(\tau_{0}\) by using the normal form theory and the center manifold theorem introduced by Hassard et al. [18].
Let \(\bar{x}_{1}(t)=x_{1}(t)-x_{1}^{\ast}\), \(\bar {x}_{2}(t)=x_{2}(t)-x_{2}^{\ast}\), \(\bar{y}(t)=y(t)-y^{\ast}\). Then system (1.1) becomes
$$ \left \{ \textstyle\begin{array}{l} \dot{\bar{x}}_{1}(t)= -(b+d_{1})\bar{x}_{1}(t)+a\bar{x}_{2}(t), \\ \dot{\bar{x}}_{2}(t)= b\bar{x}_{1}(t)+a_{21}\bar{x}_{2}(t)+a_{22}\bar{y}(t) +a_{23}\bar{x}_{2}^{2}(t)+a_{24}a_{1}\bar{x}_{2}^{3}(t) \\ \hphantom{\dot{\bar{x}}_{2}(t)={}}{} +a_{25}a_{1}\bar{x}_{2}(t)\bar{y}(t)+a_{26}a_{1}\bar{x}_{2}^{2}(t)\bar {y}(t), \\ \dot{\bar{y}}(t)= -d\bar{y}(t)+d\bar{y}(t-\tau)+a_{31}\bar {x}_{2}(t-\tau) +a_{32}\bar{x}_{2}^{2}(t-\tau)-a_{24}a_{2}\bar{x}_{2}^{3}(t-\tau) \\ \hphantom{\dot{\bar{y}}(t)={}}{} -a_{25}a_{2}\bar{x}_{2}(t-\tau)\bar{y}(t-\tau)-a_{26}a_{2}\bar {x}_{2}^{2}(t-\tau)\bar{y}(t-\tau), \end{array}\displaystyle \right . $$
(3.1)
where
$$\left \{ \textstyle\begin{array}{l} a_{21}=-c_{1}-\frac{a_{1}c_{3}}{c_{2}^{3}},\qquad a_{22}=\frac{-da_{1}}{a_{2}},\qquad a_{23}=-b_{1}+\frac{2c_{3}a_{1}x_{2}^{\ast}}{c_{2}^{4}} +\frac{a_{1}x_{2}^{\ast}y^{\ast}}{c_{2}^{3}}(c_{2}+4x_{2}^{\ast}), \\ a_{24}=\frac{2x_{2}^{\ast}y^{\ast}}{c_{2}^{3}}-(\frac{4(x_{2}^{\ast })^{2}}{c_{2}^{5}} -\frac{1}{c_{2}^{4}})c_{3}-\frac{2(x_{2}^{\ast})^{2}y^{\ast }}{c_{2}^{4}}(c_{2}+4x_{2}^{\ast}), \\ a_{25}=-\frac{m-(x_{2}^{\ast})^{2}}{c_{2}^{2}},\qquad a_{26}=\frac{x_{2}^{\ast}}{c_{2}^{2}}+\frac{2x_{2}^{\ast}(m-(x_{2}^{\ast })^{2})}{c_{2}^{3}}, \\ a_{31}=\frac{2a_{2}x_{2}^{\ast}y^{\ast}}{c_{2}^{2}}+\frac {a_{2}c_{3}}{c_{2}^{3}},\qquad a_{32}=-\frac{2a_{2}c_{3}x_{2}^{\ast}}{c_{2}^{4}}-\frac{a_{2}x_{2}^{\ast }y^{\ast}(c_{2}+4x_{2}^{\ast})}{c_{2}^{3}} \\ c_{1}=d_{2}+2b_{1}x_{2}^{\ast}+\frac{2a_{1}x_{2}^{\ast}y^{\ast }}{c_{2}^{2}},\qquad c_{2}=m+(x_{2}^{\ast})^{2}, \\ c_{3}=y^{\ast}(m^{2}-(x_{2}^{\ast})^{4}-2mx_{2}^{\ast}-2(x_{2}^{\ast})^{3}). \end{array}\displaystyle \right . $$
Let \(t=s\tau\), \(\bar{x}_{1}(s\tau)=\hat{x}_{1}(s)\), \(\bar{x}_{2}(s\tau )=\hat{x}_{2}(s)\), \(\bar{y}(s\tau)=\hat{y}(s)\), \(\tau=\tau_{0}+\mu\), \(\mu\in{\mathrm{R}}\), where \(\tau_{0}\) is defined by (2.10). We drop the hats for simplification of notation. Then system (3.1) is transformed into the following FDE in \(C=C([-1,0],{\mathrm{R}}^{3})\):
$$\left \{ \textstyle\begin{array}{l} \dot{x_{1}}(t)= (\tau_{0}+\mu)[-(b+d_{1})x_{1}(t)+ax_{2}(t)], \\ \dot{x_{2}}(t)= (\tau_{0}+\mu )[bx_{1}(t)+a_{21}x_{2}(t)+a_{22}y(t)+a_{23}x_{2}^{2}(t)+a_{24}a_{1}x_{2}^{3}(t) \\ \hphantom{\dot{x_{2}}(t)={}}{}+a_{25}a_{1}x_{2}(t)y(t)+a_{26}a_{1}x_{2}^{2}(t)y(t)], \\ \dot{y}(t)= (\tau_{0}+\mu )[-dy(t)+dy(t-1)+a_{31}x_{2}(t-1)+a_{32}x_{2}^{2}(t-1) \\ \hphantom{\dot{y}(t)={}}{}-a_{24}a_{2}x_{2}^{3}(t-1)-a_{25}a_{2}x_{2}(t-1)y(t-1)-a_{26}a_{2}x_{2}^{2}(t-1)y(t-1)]. \end{array}\displaystyle \right . $$
We rewrite this system in the matrix form
$$ \dot{x}(t)=L_{\mu}x_{t}+f(\mu,x_{t}). $$
(3.2)
where \(x(t)=(x_{1}(t),x_{2}(t),y(t))^{\mathrm{T}}\in{\mathrm{R}}^{3}\), and \(L_{\mu }:C\rightarrow{\mathrm{R}}^{3}\), \(f:{\mathrm{R}}\times C\rightarrow{\mathrm{R}}^{3}\) are given, respectively, as follows: for \(\phi(t)=(\phi_{1}(t),\phi_{2}(t),\phi_{3}(t))^{\mathrm{T}}\in C([-\tau ,0],{\mathrm{R}}^{3})\), we define
$$ L_{\mu}\phi=D_{1}\phi(0)+D_{2}\phi(-1) $$
(3.3)
and
$$ f(\mu,\phi)=(\tau_{0}+\mu)M, $$
(3.4)
where
$$ \begin{aligned} &D_{1}=(\tau_{0}+\mu)\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} -(b+d_{1}) & a & 0\\ b & a_{21} & a_{22}\\ 0 & 0 & -d \end{array}\displaystyle \right ), \qquad D_{2}=(\tau_{0}+ \mu)\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 0 & 0 & 0\\ 0 & 0 & 0\\ 0 & a_{31}& d \end{array}\displaystyle \right ), \\ &M=\left ( \textstyle\begin{array}{@{}c@{}} 0\\ a_{23}\phi_{2}^{2}(0)+a_{24}a_{1}\phi_{2}^{3}(0)+a_{25}a_{1}\phi _{2}(0)\phi_{3}(0)+a_{26}a_{1}\phi_{2}^{2}(0)\phi_{3}(0)\\ a_{32}\phi_{2}^{2}(-1)-a_{24}a_{2}\phi_{2}^{3}(-1)-a_{25}a_{2}\phi _{2}(-1)\phi_{3}(-1)-a_{26}a_{2}\phi_{2}^{2}(-1)\phi_{3}(-1) \end{array}\displaystyle \right ). \end{aligned} $$
(3.5)
By the Riesz representation theorem there exists a matrix function \(\eta (\cdot,\mu):[-1,0] \rightarrow{\mathrm{R}}^{3}\) such that
$$ L_{\mu}(\phi)= \int_{-1}^{0}\mathrm{d}\eta(\theta,\mu)\phi(\theta) \quad \mbox{for } \phi\in C. $$
(3.6)
In fact, we can choose
$$ \eta(\theta,\mu)=D_{1}\delta(\theta)+D_{2}\delta(\theta+1), $$
(3.7)
where δ is the Dirac delta function. For \(\phi\in C^{1}([-1,0],{\mathrm{R}}^{3})\), define
$$ A(\mu)\phi(\theta)= \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{\mathrm{d}\phi(\theta)}{\mathrm{d}\theta}, &-1\leq\theta< 0, \\ \int_{-1}^{0}\mathrm{d}\eta(s,\mu)\phi(s), &\theta=0 \end{array}\displaystyle \right . $$
and
$$ R(\mu)\phi(\theta)= \left \{ \textstyle\begin{array}{l@{\quad}l} 0, &-1\leq\theta< 0, \\ f(\mu,\theta), &\theta=0. \end{array}\displaystyle \right . $$
Hence, system (3.2) is equivalent to the operator equation
$$\dot{x}_{t}=A(\mu)x_{t}+R(\mu)x_{t}, $$
where \(x_{t}(t)=x(t+\theta)\) for \(\theta\in[-1,0]\).
For \(\psi\in C([-1,0],({\mathrm{R}}^{3})^{\ast})\), define
$$ A^{\ast}(\mu)\psi(s)= \left \{ \textstyle\begin{array}{l@{\quad}l} -\frac{\mathrm{d}\psi(s)}{\mathrm{d} s}, & 0< s\leq1, \\ \int_{-1}^{0}\mathrm{d}\eta(s,\mu)\psi(-s), & s=0, \end{array}\displaystyle \right . $$
and the bilinear inner product
$$ \langle\phi,\psi\rangle=\bar{\psi}^{\mathrm{T}}(0)\phi(0)- \int_{-1}^{0} \int_{\xi=0}^{\theta}\bar{\psi}^{\mathrm{T}}(\xi-\theta)\, \mathrm{d}\eta(\theta)\phi (\xi)\, \mathrm{d}\xi, $$
(3.8)
where \(\eta(\theta)=\eta(\theta,0)\). Then \(A(0)\) and \(A^{\ast}\) are adjoint operators.
By the discussion in Section 2 we know that \(\pm i\omega_{0}\tau_{0}\) are eigenvalues of \(A(0)\). Hence, they are also eigenvalues of \(A^{\ast}\).
Assume that \(q(\theta)=(1, q_{1}, q_{2})^{\mathrm{T}}e^{i\omega_{0}\tau _{0}\theta}\) is the eigenvector of \(A(0)\) corresponding to \(i\omega_{0}\tau_{0}\). Then \(A(0)q(\theta)=i\omega_{0}\tau_{0}q(\theta )\). From the definition of \(A(0)\) and from (3.3), (3.6), and (3.7), for \(q(-1)=q(0)e^{-i\omega_{0}\tau_{0}}\), we have
$$\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} -(b+d_{1}) & a & 0\\ b & a_{21} & a_{22}\\ 0 & 0 & -d \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} 1\\ q_{1}(0)\\ q_{2}(0) \end{array}\displaystyle \right ) =i\omega_{0}\tau_{0}\left ( \textstyle\begin{array}{@{}c@{}} 1\\ q_{1}(0)\\ q_{2}(0) \end{array}\displaystyle \right ). $$
Then we obtain
$$q_{1}=\frac{b+d_{1}+i\omega_{0}}{a},\qquad q_{2}=\frac{-a_{31}e^{-i\omega_{0}\tau_{0}}(b+d_{1}+i\omega _{0})}{a[de^{-i\omega_{0}\tau_{0}}-(d+i\omega_{0})]}. $$
Similarly, we can calculate the eigenvector \(q^{*}(s)=D(1,q_{1}^{*}, q_{2}^{*})^{\mathrm{T}}e^{i\omega_{0}\tau_{0}}\) of A corresponding to \(-i\omega_{0}\tau_{0}\), where
$$q^{\ast}_{1}=\frac{b+d_{1}-i\omega_{0}}{b},\qquad q^{\ast}_{2}= \frac{-a_{22}(b+d_{1}-i\omega_{0})}{b[de^{i\omega_{0}\tau _{0}}-(d-i\omega_{0})]}. $$
We normalize q and \(q^{*}\) by the condition \(\langle q^{*}(s), q(\theta)\rangle=1\). Clearly, \(\langle q^{*}(s), {q}(\theta)\rangle=0\). In order to ensure that \(\langle q^{*}(s), q(\theta)\rangle=1\), we need to determine the value of D. By (3.8) we can choose
$$\bar{D}=\bigl[1+q_{1}\bar{q}^{\ast}_{1}+q_{2} \bar{q}^{\ast }_{2}+\bigl(a_{31}q_{1} \bar{q}^{\ast}_{2} +dq_{2}\bar{q}^{\ast}_{2} \bigr)\tau_{0}e^{-i\omega_{0}\tau_{0}}\bigr]^{-1}. $$
In the following, we use the same notation as in [18], and using a computation similar to that of Wei and Ruan [19], we can obtain the coefficients that will be used for determining the important qualities:
$$\left \{ \textstyle\begin{array}{l} g_{20}=2\tau_{0}\bar{D}(k_{11}\bar{q}^{\ast}_{1}+k_{21}\bar{q}^{\ast}_{2}),\qquad g_{11}=\tau_{0}\bar{D}(k_{12}\bar{q}^{\ast}_{1}+k_{22}\bar{q}^{\ast }_{2}), \\ g_{02}=2\tau_{0}\bar{D}(k_{13}\bar{q}^{\ast}_{1}+k_{23}\bar{q}^{\ast}_{2}),\qquad g_{21}=2\tau_{0}\bar{D}(k_{14}\bar{q}^{\ast}_{1}+k_{24}\bar{q}^{\ast }_{2}), \end{array}\displaystyle \right . $$
where
$$\begin{aligned}& k_{11}= a_{23}q_{1}^{2}+a_{25}a_{1}q_{1}q_{2}, \\& k_{12}= 2a_{23}q_{1}\bar{q}_{1}+a_{25}a_{1}(q_{1} \bar{q}_{2}+q_{2}\bar {q}_{1}), \\& k_{13}= a_{23}\bar{q}_{1}^{2}+a_{25}a_{1} \bar{q}_{1}\bar{q_{2}}, \\& k_{14}= a_{23}\bigl[\bar {q}_{1}W_{20}^{(2)}(0)+2q_{1}W_{11}^{(2)}(0) \bigr]+3a_{24}a_{1}q_{1}^{2} \bar{q}_{1} +a_{26}a_{1}\bigl(q_{1}^{2} \bar{q}_{2}+2q_{1}q_{2}\bar{q}_{1}\bigr) \\& \hphantom{k_{14}={}}{} +a_{25}a_{1}\biggl[\frac{1}{2} \bar{q}_{2}W_{20}^{(2)}(0)+q_{2}W_{11}^{(2)}(0) +\frac{1}{2}\bar{q}_{1}W_{20}^{(3)}(0)+q_{1}W_{11}^{(3)}(0) \biggr], \\& k_{21}= \bigl(a_{32}q_{1}^{2}-a_{25}a_{2}q_{1}q_{2} \bigr)e^{-2i\omega_{0}\tau _{0}}, \\& k_{22}= \bigl[2a_{32}q_{1}\bar{q}_{1}-a_{25}a_{2}(q_{1} \bar {q}_{2}+q_{2}\bar{q_{1}})\bigr]e^{-2i\omega_{0}\tau_{0}}, \\& k_{23}= \bigl(a_{32}\bar{q}_{1}^{2}-a_{25}a_{2} \bar{q}_{1}\bar {q}_{2}\bigr)e^{-2i\omega_{0}\tau_{0}}, \\& k_{24}= a_{32}\bigl[\bar {q}_{1}W_{20}^{(2)}(-1)+2q_{1}W_{11}^{(2)}(-1) \bigr]e^{-i\omega_{0}\tau_{0}} -3a_{24}a_{2}q_{1}^{2} \bar{q}_{1}e^{-3i\omega_{0}\tau_{0}} \\& \hphantom{k_{24}={}}{} -a_{26}a_{2}\bigl(q_{1}^{2} \bar{q}_{2}+2q_{1}q_{2}\bar{q}_{1} \bigr)e^{-3i\omega _{0}\tau_{0}} \\& \hphantom{k_{24}={}}{} -a_{25}a_{2}\biggl[\frac{1}{2}W_{20}^{(2)}(-1) \bar{q}_{2}+q_{2}W_{11}^{(2)}(-1) + \frac{1}{2}\bar {q}_{1}W_{20}^{(3)}(-1)+q_{1}W_{11}^{(3)}(-1) \biggr]e^{-i\omega_{0}\tau_{0}} \end{aligned}$$
and
$$\left \{ \textstyle\begin{array}{l} W_{20}(\theta)=\frac{ig_{20}}{\omega_{0}\tau_{0}}q(0)e^{i\omega_{0}\tau _{0}\theta} +\frac{i\bar{g}_{02}}{3\omega_{0}\tau_{0}}\bar{q}(0)e^{-i\omega_{0}\tau _{0}\theta} +E_{1}e^{2i\omega_{0}\tau_{0}\theta}, \\ W_{11}(\theta)=-\frac{ig_{11}}{\omega_{0}\tau_{0}}q(0)e^{i\omega _{0}\tau_{0}\theta} +\frac{i\bar{g}_{11}}{\omega_{0}\tau_{0}}\bar{q}(0)e^{-i\omega_{0}\tau _{0}\theta}+E_{2}. \end{array}\displaystyle \right . $$
Moreover \(E_{1}\) and \(E_{2}\) satisfy the following equations:
$$\begin{aligned}& \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 2i\omega_{0}+(b+d_{1}) & -a & 0\\ -b & 2i\omega_{0}-a_{21} & -a_{22}\\ 0 & -a_{31}e^{-2i\omega_{0}\tau_{0}} & 2i\omega_{0}-d-de^{-2i\omega _{0}\tau_{0}} \end{array}\displaystyle \right )E_{1}=2\left ( \textstyle\begin{array}{@{}c@{}} 0\\ k_{11}\\ k_{21} \end{array}\displaystyle \right ), \\& \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} b+d_{1} & -a & 0\\ -b & -a_{21} & -a_{22}\\ 0 & -a_{31} & 0 \end{array}\displaystyle \right )E_{2}=\left ( \textstyle\begin{array}{@{}c@{}} 0\\ k_{12}\\ k_{22} \end{array}\displaystyle \right ). \end{aligned}$$
Furthermore, \(g_{ij}\) is expressed by the parameters and delay in (1.1). Thus, we can compute the following values:
$$ \left \{ \textstyle\begin{array}{l} C_{1}(0)=\frac{i}{2\omega_{0}\tau_{0}}(g_{20}g_{11}-2|g_{11}|^{2}-\frac {|g_{02}|^{2}}{3})+\frac{g_{21}}{2}, \\ \mu_{2}=-\frac{\operatorname{Re}\{C_{1}(0)\}}{\operatorname{Re}\{\frac{\mathrm{d}\lambda (\tau_{0})}{\mathrm{d}\tau}\}}, \\ \beta_{2}=2\operatorname{Re}\{C_{1}(0)\}, \\ T_{2}=\frac{\operatorname{Im}\{C_{1}(0)\}+\mu_{2}\operatorname{Im}\{\frac{\mathrm{d}\lambda (\tau_{0})}{\mathrm{d}\tau}\}}{\omega_{0}\tau_{0}},\quad k=0,1,2,\ldots. \end{array}\displaystyle \right . $$
(3.9)
By the result of Hassard et al. [18] we have the following theorem.
Theorem 3.1
In view of (3.9), the following results hold:
-
(i)
the sign of
\(\mu_{2}\)
determines the directions of the Hopf bifurcation: if
\(\mu_{2}>0\) (\(\mu_{2}<0\)), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for
\(\tau>\tau^{*}\) (\(\tau<\tau_{0}\));
-
(ii)
the sign of
\(\beta_{2}\)
determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if
\(\beta _{2}<0\) (\(\beta_{2}>0\));
-
(iii)
the sign of
\(T_{2}\)
determines the period of the bifurcating periodic solutions: the period is increasing (decreasing) if
\(\beta_{2}>0\) (\(\beta_{2}<0\)).