In the previous section, we have shown that the controlled system (1.3) undergoes Hopf bifurcation for different combinations of \(\tau_{1}\) and \(\tau_{2}\). In this section, we will investigate the direction of Hopf bifurcation and the stability of bifurcating periodic solutions of the controlled system (1.3). Throughout this section, we assume that system (1.3) undergoes a Hopf bifurcation for \(\tau _{2}^{*} \in (0,\tau_{2_{0}})\) and \(\tau_{1} = \tau_{1_{0}}\). The theoretical approach we apply is based on the normal form theory and center manifold theory [37].
Without loss of generality, we assume that \(\tau_{2}^{*} < \tau_{1_{0}}\), where \(\tau_{2}^{*} \in(0,\tau_{2_{0}})\). For convenience, let \(\bar{u}_{i}(t) = u_{i}(\tau t)\) (\(i = 1,2\)) and \(\tau_{1} = \tau_{1_{0}} + \mu\), where \(\tau_{1_{0}}\) is defined by Eq. (2.14) and \(\mu \in R\), then system (1.3) can be written as a functional differential equation (FDE) in \(C = C([ - 1,0],R^{2})\):
$$ u'(t) = L_{\mu} (u_{t}) + F( \mu,u_{t}), $$
(3.1)
where \(u(t) = (x(t),y(t))^{T} \in C\), \(u_{t}(\theta) = u(t + \theta ) = (x(t + \theta),y(t + \theta))^{T} \in C\), and \(L_{\mu}: C \to R\), \(F:R \times C \to R\) are given by
$$ L_{\mu} (\varphi) = (\tau_{1} + \mu)B\left ( \textstyle\begin{array}{c} \varphi_{1}(0) \\ \varphi_{2}(0) \end{array}\displaystyle \right ) + (\tau_{1} + \mu)C\left ( \textstyle\begin{array}{c} \varphi_{1}( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}) \\ \varphi_{2}( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}) \end{array}\displaystyle \right ) + (\tau_{1} + \mu)D\left ( \textstyle\begin{array}{c} \varphi_{1}( - 1) \\ \varphi_{2}( - 1) \end{array}\displaystyle \right ) $$
(3.2)
and
$$ F(\mu,\varphi) = (\tau_{1} + \mu) (f_{1},f_{2})^{T}, $$
(3.3)
with
$$B = \left ( \textstyle\begin{array}{c@{\quad}c} m_{1} & 0 \\ 0 & n_{1} \end{array}\displaystyle \right ),\qquad C = \left ( \textstyle\begin{array}{c@{\quad}c} 0 & m_{3} \\ n_{2} & 0 \end{array}\displaystyle \right ),\qquad D = \left ( \textstyle\begin{array}{c@{\quad}c} m_{2} & 0 \\ 0 & n_{3} \end{array}\displaystyle \right ), $$
and
$$\begin{gathered} f_{1} = m_{4} \varphi_{1}(0)\varphi_{1}( - 1) + m_{5} \varphi_{1}(0)\varphi_{2}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr), \\ f_{2} = n_{4}\varphi_{1}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr)\varphi_{2}(0) + n_{5} \varphi_{2}(0)\varphi_{2}( - 1). \end{gathered} $$
By the Riesz representation theorem, there exists a \(2\times2\) matrix function \(\eta(\theta,\mu)\), \(\theta\in[ - 1,0]\), whose elements are of bounded variation, such that
$$ L_{\mu} \varphi= \int_{ - 1}^{0} d\eta(\theta,\mu) \varphi(\theta ) \quad\mbox{for } \varphi\in C. $$
(3.4)
In fact, we can choose
$$ \eta(\theta,\mu) = \left \{ \textstyle\begin{array}{l@{\quad}l} (\tau_{1_{0}} + \mu)(B + C + D),& \theta= 0, \\ (\tau_{1_{0}} + \mu)(C + D), &\theta\in[ - \frac{\tau_{2}^{*}}{\tau_{1_{0}}},0), \\ (\tau_{1_{0}} + \mu)D, &\theta\in( - 1, - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}), \\ 0, &\theta= - 1. \end{array}\displaystyle \right . $$
(3.5)
For \(\varphi\in C([ - 1,0],R^{2})\), define
$$ A(\mu)\varphi= \left \{ \textstyle\begin{array}{l@{\quad}l} \frac{d\varphi(\theta )}{d\theta},& - 1 \le\theta< 0, \\ \int_{ - 1}^{0} d\eta(s,\mu)\varphi(s),& \theta= 0, \end{array}\displaystyle \right . $$
(3.6)
and
$$ R_{\mu} (\varphi) = \left \{ \textstyle\begin{array}{l@{\quad}l} 0, &- 1 \le\theta< 0, \\ F(\mu,\varphi), &\theta= 0. \end{array}\displaystyle \right . $$
(3.7)
Then Eq. (3.1) can be transformed into the following operator equation:
$$ u'_{t} = A(\mu)u_{t} + R( \mu)u_{t}, $$
(3.8)
where \(u_{t} = u(t + \theta) = (u_{1}(t + \theta),u_{2}(t + \theta))\), \(\theta\in[ - 1,0]\).
For \(\phi\in C([ - 1,0],(R^{2})^{ *} )\), where \((R^{2})^{ *} \) is the 2-dimensional space of row vectors, we further define the adjoint operator \(A^{ *} \) of \(A(0)\):
$$A^{ *} \phi(s) = \left \{ \textstyle\begin{array}{l@{\quad}l} - \frac{d\phi(s)}{ds}, &s \in (0,1], \\ \int_{ - 1}^{0} d\eta^{T}(t,0)\phi( - t), &s = 0. \end{array}\displaystyle \right . $$
For \(\varphi\in C([ - 1,0],R^{2})\) and \(\phi\in C([ - 1,0],(R^{2})^{ *} )\), define the bilinear form
$$\bigl\langle \phi(s),\varphi(s) \bigr\rangle = \bar{\phi} (0)\varphi(0) - \int_{ - 1}^{0} \int_{\xi= 0}^{\theta} \phi(\xi- \theta)\,d\eta (\theta ) \varphi(\xi)\,d\xi, $$
where \(\eta(\theta) = \eta(\theta,0)\), \(A = A(0)\) and \(A^{ *} \) are adjoint operators. From Section 2, we know that \(\pm i\omega^{ *} \tau_{1_{0}}\) are eigenvalues of \(A(0)\), and they are also the eigenvalues of \(A^{ *} \) corresponding to \(i\omega^{ *} \tau_{1_{0}}\) and \(- i\omega^{ *} \tau_{1_{0}}\). Further, we suppose \(q(\theta) = (1,\alpha )^{T}e^{i\omega^{ *} \tau_{1_{0}}\theta} \) is the eigenvector of \(A(0)\) corresponding to \(i\omega^{ *} \tau_{1_{0}}\) and \(q^{ *} (s) = M(1,\alpha^{ *} )e^{i\omega^{ *} \tau_{1_{0}}s}\) is the eigenvector of \(A^{ *} \) corresponding to \(- i\omega^{ *} \tau_{1_{0}}\), where \(M = 1/D\).
By the direct calculation, we obtain
$$\begin{gathered} \alpha= \frac{i\omega^{ *} - m_{1} - m_{2}e^{ - i\omega^{ *} \tau_{1_{0}}}}{m_{3}e^{ - i\omega^{ *} \tau_{{2}}^{*}}},\qquad\alpha ^{ *} = - \frac{i\omega^{ *} + m_{1} + m_{2}e^{ - i\omega^{ *} \tau_{1_{0}}}}{n_{2}e^{ - i\omega^{ *} \tau_{{2}}^{*}}}, \\ D = 1 + \bar{\alpha} \alpha^{ *} + m_{2} \tau_{1_{0}}e^{i\omega^{ *} \tau_{1_{0}}} + n_{2}\alpha^{ *} \tau_{{2}}^{*}e^{i\omega^{ *} \tau_{2}^{ *}} + m_{3}\bar{\alpha} \tau_{{2}}^{*}e^{i\omega^{ *} \tau_{2}^{ *}} + n_{3}\bar{\alpha} \alpha^{ *} \tau_{1_{0}}e^{i\omega^{ *} \tau_{1_{0}}}. \end{gathered} $$
Then we have \(\langle q^{ *} (s),q(\theta) \rangle= 1\), \(\langle q^{ *} (s),\bar{q}(\theta) \rangle= 0\).
Next, we use the same notations as those in Hassard [37] and firstly compute the coordinates to describe the center manifold \(C_{0}\) at \(\mu= 0\). Let \(u_{t}\) be the solution of Eq. (3.1) when \(\mu= 0\).
Define
$$\begin{aligned}& z(t) = \bigl\langle q^{ *},u_{t} \bigr\rangle , \\& W(t,\theta) = u_{t}(\theta) - 2\operatorname{Re} \bigl\{ z(t)q(\theta) \bigr\} , \end{aligned}$$
(3.9)
on the center manifold \(C_{0}\), and we have
$$ W(t,\theta) = W\bigl(z(t),\bar{z}(t),\theta\bigr), $$
(3.10)
where
$$ W\bigl(z(t),\bar{z}(t),\theta\bigr) = W(z,\bar{z}) = W_{20} \frac{z^{2}}{2} + W_{11}z\bar{z} + W_{02}\frac{\bar{z}^{2}}{2} + \cdots, $$
(3.11)
and z and z̄ are the local coordinates for the center manifold \(C_{0}\) in the direction of \(q^{ *} \) and \(\bar{q}^{*}\). Noting that W is also real if \(u_{t}\) is real, we consider real solutions. For solutions \(u_{t} \in C_{0}\) of Eq. (3.1),
$$\dot{z}(t) = i\omega^{ *} \tau_{1_{0}}z + \bar{q}^{*}( \theta )F\bigl(0,W(z,\bar{z},\theta)\bigr) + 2\operatorname{Re} \bigl\{ zq(\theta) \bigr\} . $$
We define this equation as
$$\dot{z}(t) = i\omega^{ *} \tau_{1_{0}}z + \bar{q}^{*}(0)F_{0}. $$
That is,
$$\dot{z}(t) = i\omega^{ *} \tau_{1_{0}}z + g(z,\bar{z}), $$
where
$$ \begin{aligned}[b] g(z,\bar{z}) &= \bar{q}^{ *} (0)F_{0}(z,\bar{z}) = F(0,u_{t}) \\ &= g_{20}\frac{z^{2}}{2} + g_{11}z\bar{z} + g_{02}\frac{\bar{z}}{2}^{2} + g_{21} \frac{z^{2}\bar{z}}{2} + \cdots. \end{aligned} $$
(3.12)
Noticing \(u_{t}(\theta) = (x_{t}(\theta),y_{t}(\theta)) = W(t,\theta) + zq(\theta) + \bar{z}\bar{q}(\theta)\) and \(q(\theta) = (1,\alpha )^{T}e^{i\omega^{ *} \tau_{1_{0}}\theta} \) by Eq. (3.9), we have
$$\begin{gathered} x_{t}(0) = z + \bar{z} + \frac{1}{2}W_{20}^{(1)}(0)z^{2} + W_{11}^{(1)}(0)z\bar{z} + \frac{1}{2}W_{02}^{(1)}(0) \bar{z}^{2} + \cdots, \\ y_{t}(0) = \alpha z + \bar{\alpha} \bar{z} + \frac{1}{2}W_{20}^{(2)}(0)z^{2} + W_{11}^{(2)}(0)z\bar{z} + \frac{1}{2}W_{02}^{(2)}(0) \bar{z}^{2} + \cdots, \\ x_{t}( - 1) = ze^{ - i\omega^{ *} \tau_{1_{0}}} + \bar{z}e^{i\omega^{ *} \tau_{1_{0}}} + \frac{1}{2}W_{20}^{(1)}( - 1)z^{2} + W_{11}^{(1)}( - 1)z\bar{z} + \frac{1}{2}W_{02}^{(1)}( - 1)\bar{z}^{2} + \cdots, \\ y_{t}( - 1) = \alpha ze^{ - i\omega^{ *} \tau_{1_{0}}} + \bar{\alpha} \bar{z}e^{i\omega^{ *} \tau_{1_{0}}} + \frac{1}{2}W_{20}^{(2)}( - 1)z^{2} + W_{11}^{(2)}( - 1)z\bar{z} + \frac{1}{2}W_{02}^{(2)}( - 1)\bar{z}^{2} + \cdots, \\ \begin{aligned}x_{t}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr) ={}& ze^{ - i\omega^{ *} \tau_{2}^{ *}} + \bar{z}e^{i\omega^{ *} \tau_{2}^{*}} + \frac{1}{2}W_{20}^{(1)} \biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr)z^{2} + W_{11}^{(1)} \biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr)z\bar{z}\\ & + \frac{1}{2}W_{02}^{(1)} \biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr)\bar {z}^{2} + \cdots,\end{aligned} \\ \begin{aligned}y_{t}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr) ={}& \alpha ze^{ - i\omega^{ *} \tau_{2}^{*}} + \bar{\alpha} \bar{z}e^{i\omega^{ *} \tau_{2}^{*}} + \frac{1}{2}W_{20}^{(2)} \biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr)z^{2} + W_{11}^{(2)} \biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr)z\bar{z} \\&+ \frac{1}{2}W_{02}^{(2)} \biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr)\bar {z}^{2} + \cdots. \end{aligned}\end{gathered} $$
Then from Eq. (3.12) we have
$$\begin{aligned} g(z,\bar{z}) ={}& \bar{M}\tau_{1_{0}}\bigl[ \bigl(m_{4}e^{ - i\omega^{ *} \tau_{1_{0}}} + m_{5}\alpha e^{ - i\omega^{ *} \tau_{2}^{*}} \bigr) + \bar{\alpha}^{*}\bigl(n_{4}\alpha e^{ - i\omega^{ *} \tau_{2}^{*}} + n_{5}\alpha^{2}e^{ - i\omega^{ *} \tau_{1_{0}}}\bigr)\bigr]z^{2} \\ &+ 2\bar{M}\tau_{1_{0}}\bigl[\bigl(m_{4}\operatorname{Re} \bigl\{ e^{ - i\omega^{ *} \tau _{1_{0}}}\bigr\} + m_{5}\operatorname{Re} \bigl\{ \alpha e^{ - i\omega^{ *} \tau_{2}^{*}}\bigr\} \bigr)\\ & + \bar{\alpha}^{*}\bigl(n_{4} \operatorname{Re} \bigl\{ \alpha e^{i\omega^{ *} \tau_{2}^{*}}\bigr\} + n_{5} \operatorname{Re} \bigl\{ | \alpha |^{2}e^{i\omega^{ *} \tau_{1_{0}}}\bigr\} \bigr) \bigr]z\bar{z} \\ &+ \bar{M}\tau_{1_{0}}\bigl[\bigl(m_{4}e^{i\omega^{ *} \tau_{1_{0}}} + m_{5}\bar{\alpha} e^{i\omega^{ *} \tau_{2}^{*}}\bigr) + \bar{\alpha}^{*} \bigl(n_{4}\bar{\alpha} e^{i\omega\tau_{2}^{*}} + n_{5}\bar{ \alpha}^{2}e^{i\omega^{ *} \tau_{1_{0}}}\bigr)\bigr]\bar{z}^{2} \\ &+ \bar{M}\tau_{1_{0}}\biggl[m_{4}\biggl(W_{11}^{(1)}( - 1) + \frac {1}{2}W_{20}^{(1)}( - 1) + \frac{1}{2}W_{20}^{(1)}(0)e^{i\omega^{ *} \tau_{1_{0}}} + W_{11}^{(1)}(0)e^{ - i\omega^{ *} \tau_{1_{0}}}\biggr) \\ &+ m_{5}\biggl(W_{11}^{(2)}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr) + \frac{1}{2}W_{20}^{(2)} \biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr) + \frac{1}{2}W_{20}^{(1)}(0) \bar{\alpha} e^{i\omega^{ *} \tau_{2}^{*}} + W_{11}^{(1)}(0)\alpha e^{ - i\omega^{ *} \tau_{2}^{*}}\biggr) \\ &+ \bar{\alpha}^{*}\biggl[n_{4}\biggl(W_{11}^{(2)}(0)e^{ - i\omega^{ *} \tau_{{2}}^{*}} + \frac{1}{2}W_{20}^{(2)}(0)e^{i\omega^{ *} \tau_{2}^{*}} \\ &+ \frac{1}{2}\bar{\alpha} W_{20}^{(1)}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr) + \alpha W_{11}^{(1)}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr)\biggr) \\ &+ n_{5}\biggl(\alpha W_{11}^{(2)}( - 1) + \frac{1}{2}\bar{\alpha} W_{20}^{(2)}( - 1)\\& + \frac{1}{2}\bar{\alpha} W_{20}^{(2)}(0)e^{i\omega^{ *} \tau _{1_{0}}} + \alpha W_{11}^{(2)}(0)e^{ - i\omega^{ *} \tau_{1_{0}}}\biggr)\biggr] \biggr]z^{2}\bar{z} + \cdots. \end{aligned} $$
Comparing the coefficients with Eq. (3.12), we obtain
$$\begin{gathered} g_{20} = 2\bar{M}\tau_{1_{0}}\bigl[ \bigl(m_{4}e^{ - i\omega^{ *} \tau_{1_{0}}} + m_{5}\alpha e^{ - i\omega^{ *} \tau_{2}^{*}} \bigr) + \bar{\alpha}^{*}\bigl(n_{4}\alpha e^{ - i\omega^{ *} \tau_{2}^{*}} + n_{5}\alpha^{2}e^{ - i\omega^{ *} \tau_{1_{0}}}\bigr)\bigr], \\ \begin{aligned}g_{11} ={}& 2\bar{M}\tau_{1_{0}}\bigl[ \bigl(m_{4}\operatorname{Re} \bigl\{ e^{ - i\omega ^{ *} \tau_{1_{0}}}\bigr\} + m_{5}\operatorname{Re} \bigl\{ \alpha e^{ - i\omega^{ *} \tau_{2}^{*}}\bigr\} \bigr)\\ & + \bar{\alpha}^{*}\bigl(n_{4}\operatorname{Re} \bigl\{ \alpha e^{i\omega^{*}\tau_{2}^{*}}\bigr\} + n_{5}\operatorname{Re} \bigl\{ \vert \alpha \vert ^{2}e^{i\omega^{ *} \tau_{1_{0}}}\bigr\} \bigr)\bigr],\end{aligned}\\ g_{02} = 2\bar{M}\tau_{1_{0}}\bigl[\bigl(m_{4}e^{i\omega^{ *} \tau_{1_{0}}} + m_{5}\bar{\alpha} e^{i\omega^{ *} \tau_{2}^{*}}\bigr) + \bar{\alpha}^{*} \bigl(n_{4}\bar{\alpha} e^{i\omega^{*}\tau_{2}^{*}} + n_{5}\bar{ \alpha}^{2}e^{i\omega^{ *} \tau_{1_{0}}}\bigr)\bigr], \\ \begin{aligned} g_{21} ={}& 2\bar{M} \tau_{1_{0}}\biggl[m_{4}\biggl(W_{11}^{(1)}( - 1) + \frac{1}{2}W_{20}^{(1)}( - 1) + \frac{1}{2}W_{20}^{(1)}(0)e^{i\omega^{ *} \tau_{1_{0}}} + W_{11}^{(1)}(0)e^{ - i\omega^{ *} \tau_{1_{0}}}\biggr) \\ &+ m_{5}\biggl(W_{11}^{(2)}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr) + \frac{1}{2}W_{20}^{(2)} \biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr) + \frac{1}{2}W_{20}^{(1)}(0) \bar{\alpha} e^{i\omega^{ *} \tau_{2}^{*}} + W_{11}^{(1)}(0)\alpha e^{ - i\omega^{ *} \tau_{2}^{*}}\biggr) \\ &+ \bar{\alpha}^{*}\biggl[n_{4}\biggl(W_{11}^{(2)}(0)e^{ - i\omega^{ *} \tau _{2}^{*}} + \frac{1}{2}W_{20}^{(2)}(0)e^{i\omega^{ *} \tau_{2}^{*}} + \frac{1}{2}\bar{\alpha} W_{20}^{(1)}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr) + \alpha W_{11}^{(1)}\biggl( - \frac{\tau_{2}^{*}}{\tau_{1_{0}}}\biggr)\biggr) \\ &+ n_{5}\biggl(\alpha W_{11}^{(2)}( - 1) + \frac{1}{2}\bar{\alpha} W_{20}^{(2)}( - 1) + \frac{1}{2}\bar{\alpha} W_{20}^{(2)}(0)e^{i\omega^{ *} \tau _{1_{0}}} + \alpha W_{11}^{(2)}(0)e^{ - i\omega^{ *} \tau_{1_{0}}}\biggr)\biggr]\biggr]. \end{aligned} \end{gathered} $$
Since there are \(W_{20}(\theta)\) and \(W_{11}(\theta)\) in \(g_{21}\), in the sequel, we shall compute these quantities. From Eqs. (3.8) and (3.9), we have
$$ \begin{aligned}[b] W' &= \left \{ \textstyle\begin{array}{l@{\quad}l} AW - 2\operatorname{Re} \{ \bar{q}^{ *} (0)F_{0}q(\theta)\}, &- 1 \le\theta< 0, \\ AW - 2\operatorname{Re} \{ \bar{q}^{ *} (0)F_{0}q(\theta)\} + F_{0},& \theta= 0, \end{array}\displaystyle \right . \\ &= AW + H(z,\bar{z},\theta), \end{aligned} $$
(3.13)
where
$$ H(z,\bar{z},\theta) = H_{20}(\theta)\frac{z^{2}}{2} + H_{11}(\theta )z\bar{z} + H_{02}(\theta)\frac{\bar{z}}{2}^{2} + H_{21}(\theta )\frac{z^{2}\bar{z}}{2} + \cdots. $$
(3.14)
Comparing the coefficients, we obtain
$$\begin{aligned}& \bigl(AW - 2i\tau_{1_{0}}\omega^{ *} \bigr)W_{20} = - H_{20}(\theta ), \end{aligned}$$
(3.15)
$$\begin{aligned}& AW_{11}(\theta) = - H_{11}(\theta). \end{aligned}$$
(3.16)
From Eq. (3.13), we know that for \(\theta\in[ - 1,0)\),
$$ H(z,\bar{z},\theta) = - \bar{q}^{ *} (0)f_{0}q(\theta) - q^{ *} (0)\bar{f}_{0}\bar{q}(\theta) = - gq(\theta) - \bar{g} \bar{q}(\theta). $$
(3.17)
Comparing the coefficients with Eq. (3.14) gives
$$ H_{20}(\theta) = - g_{20}q(\theta) - \bar{g}_{02} \bar{q}(\theta ) $$
(3.18)
and
$$ H_{11}(\theta) = - g_{11}q(\theta) - \bar{g}_{11} \bar{q}(\theta). $$
(3.19)
From Eqs. (3.15), (3.18) and the definition of A, it follows that
$$ W'_{20}(\theta) = 2i\omega^{ *} \tau_{1_{0}}W_{20}(\theta) + g_{20}q(\theta) + \bar{g}_{02}\bar{q}(\theta). $$
(3.20)
Notice that \(q(\theta) = (1,\alpha)^{T}e^{i\omega^{ *} \tau _{1_{0}}\theta} \), hence
$$ W_{20}(\theta) = \frac{ig_{20}}{\omega^{ *} \tau_{1_{0}}}q(0)e^{i\omega^{ *} \tau_{1_{0}}\theta} + \frac{i\bar{g}_{02}}{3\omega^{ *} \tau_{1_{0}}}\bar{q}(0)e^{ - i\omega^{ *} \tau_{1_{0}}\theta} + E_{1}e^{2i\omega^{ *} \tau_{1_{0}}\theta}, $$
(3.21)
where \(E_{1} = (E_{1}^{(1)},E_{1}^{(2)})^{T} \in R^{2}\) is a constant vector. Similarly, from Eqs. (3.16) and (3.19), we obtain
$$\begin{aligned}& W'_{11}(\theta) = g_{11}q(\theta) + \bar{g}_{11}\bar{q}(\theta ), \end{aligned}$$
(3.22)
$$\begin{aligned}& W_{11}(\theta) = - \frac{ig_{11}}{\omega^{ *} \tau_{1_{0}}}q(0)e^{i\omega^{ *} \tau_{1_{0}}\theta} + \frac{i\bar{g}_{11}}{\omega^{ *} \tau_{1_{0}}}\bar{q}(0)e^{ - i\omega^{ *} \tau_{1_{0}}\theta} + E_{2}, \end{aligned}$$
(3.23)
where \(E_{2} = (E_{2}^{(1)},E_{2}^{(2)})^{T} \in R^{2}\) is also a constant vector.
In what follows, we shall seek appropriate \(E_{1}\) and \(E_{2}\) in Eqs. (3.21) and (3.23), respectively. It follows from the definition of A and Eqs. (3.18), (3.19) that
$$ \int_{ - 1}^{0} d\eta(\theta) W_{20}(\theta) = 2i\omega^{ *} \tau_{1_{0}}W_{20}(0) - H_{20}(0) $$
(3.24)
and
$$ \int_{ - 1}^{0} d\eta(\theta) W_{11}(\theta) = - H_{11}(0), $$
(3.25)
where \(\eta(\theta) = \eta(0,\theta)\). From Eqs. (3.15) and (3.16) we have
$$\begin{aligned}& H_{20}(0) = - g_{20}(0)q(0) - \bar{g}_{02}(0) \bar{q}(0) + 2\tau_{1_{0}}(H_{1},H_{2})^{T}, \end{aligned}$$
(3.26)
$$\begin{aligned}& H_{11}(0) = - g_{11}(0)q(0) - \bar{g}_{11}(0)\bar{q}(0) + 2\tau_{1_{0}}(p_{1},p_{2})^{T}, \end{aligned}$$
(3.27)
where
$$\begin{gathered} H_{1} = m_{4}e^{ - i\omega^{ *} \tau_{1_{0}}} + m_{5}\alpha e^{ - i\omega^{ *} \tau_{2}^{*}}, \\ H_{2} = n_{4}\alpha e^{ - i\omega^{ *} \tau_{2}^{*}} + n_{5}\alpha ^{2}e^{ - i\omega^{ *} \tau_{1_{0}}}, \\ p_{1} = m_{4}\operatorname{Re} \bigl\{ e^{ - i\omega^{ *} \tau_{1_{0}}}\bigr\} + m_{5}\operatorname{Re} \bigl\{ \alpha e^{ - i\omega^{ *} \tau_{2}^{*}}\bigr\} , \\ p_{2} = n_{4}\operatorname{Re} \bigl\{ \alpha e^{i\omega^{ *} \tau _{2}^{*}}\bigr\} + n_{5}\operatorname{Re} \bigl\{ \vert \alpha \vert ^{2}e^{i\omega^{ *} \tau_{1_{0}}}\bigr\} . \end{gathered} $$
Noting that
$$\begin{gathered} \biggl(i\omega^{ *} \tau_{1_{0}}I - \int_{ - 1}^{0} e^{i\omega^{ *} \tau_{1_{0}}\theta}\, d\eta(\theta) \biggr)q(0) = 0, \\ \biggl( - i\omega^{ *} \tau_{1_{0}}I - \int_{ - 1}^{0} e^{ - i\omega^{ *} \tau_{1_{0}}\theta} \,d\eta(\theta) \biggr) \overline{q}(0) = 0, \end{gathered} $$
and substituting Eqs. (3.21) and (3.26) into Eq. (3.24), we have
$$\biggl(2i\omega^{ *} \tau_{1_{0}}I - \int_{ - 1}^{0} e^{2i\omega^{ *} \tau_{1_{0}}\theta} \,d\eta(\theta) \biggr)E_{1} = 2\tau_{1_{0}}(H_{1},H_{2})^{T}. $$
That is,
$$\left ( \textstyle\begin{array}{c@{\quad}c} 2i\omega^{ *} - m_{1} - m_{2}e^{ - 2i\omega^{ *} \tau_{1_{0}}} & - m_{3}e^{ - 2i\omega^{ *} \tau_{2}^{*}} \\ - n_{2}e^{ - 2i\omega^{ *} \tau_{2}^{*}} & 2i\omega^{ *} - n_{1} - n_{3}e^{ - 2i\omega^{ *} \tau_{1_{0}}} \end{array}\displaystyle \right )E_{1} = 2(H_{1},H_{2})^{T}. $$
It follows that
$$ E_{1}^{(1)} = \frac{\Delta_{11}}{\Delta_{1}},\qquad E_{1}^{(2)} = \frac{\Delta_{12}}{\Delta_{1}}, $$
(3.28)
with
$$\Delta_{1} = \det \left ( \textstyle\begin{array}{c@{\quad}c} v_{1} & v_{2} \\ v_{3} & v_{4} \end{array}\displaystyle \right ),\qquad\Delta_{11} = 2\det \left ( \textstyle\begin{array}{c@{\quad}c} H_{1} & v_{2} \\ H_{2} & v_{4} \end{array}\displaystyle \right ),\qquad\Delta_{12} = 2\det \left ( \textstyle\begin{array}{c@{\quad}c} v_{1} & H_{1} \\ v_{3} & H_{2} \end{array}\displaystyle \right ), $$
where
$$\begin{gathered} v_{1} = 2i\omega^{ *} - m_{1} - m_{2}e^{ - 2i\omega^{ *} \tau_{1_{0}}},\qquad v_{2} = - m_{3}e^{ - 2i\omega^{ *} \tau_{2}^{*}},\qquad v_{3} = - n_{2}e^{ - 2i\omega^{ *} \tau_{2}^{*}}, \\ v_{4} = 2i\omega^{ *} - n_{1} - n_{3}e^{ - 2i\omega^{ *} \tau_{1_{0}}}. \end{gathered} $$
Similarly, substituting Eqs. (3.22) and (3.27) into Eq. (3.25), we have
$$\biggl( \int_{ - 1}^{0} d\eta(\theta)\biggr) E_{2} = 2\tau_{1_{0}}(p_{1},p_{2})^{T}, $$
that is,
$$\left ( \textstyle\begin{array}{c@{\quad}c} m_{1} + m_{2} & m_{3} \\ n_{2} & n_{1} + n_{3} \end{array}\displaystyle \right )E_{2} = 2( - p_{1}, - p_{2})^{T}. $$
It follows that
$$ E_{2}^{(1)} = \frac{\Delta_{21}}{\Delta_{2}},\qquad E_{2}^{(2)} = \frac{\Delta_{22}}{\Delta_{2}}, $$
(3.29)
where
$$\begin{gathered} \Delta_{2} = \det \left ( \textstyle\begin{array}{c@{\quad}c} m_{1} + m_{2} & m_{3} \\ n_{2} & n_{1} + n_{3} \end{array}\displaystyle \right ),\qquad\Delta_{21} = 2\det \left ( \textstyle\begin{array}{c@{\quad}c} - p_{1} & m_{3} \\ - p_{2} & n_{1} + n_{3} \end{array}\displaystyle \right ), \\ \Delta_{22} = 2\det \left ( \textstyle\begin{array}{c@{\quad}c} m_{1} + m_{2} & - p_{1} \\ n_{2} & - p_{2} \end{array}\displaystyle \right ). \end{gathered} $$
From Eqs. (3.21), (3.23), (3.28), (3.29), we can determine \(g_{21}\) and derive the following values:
$$ \begin{gathered} c_{1}(0) = \frac{i}{2\omega^{ *} \tau_{1_{0}}} \biggl(g_{20}g_{11} - 2 \vert g_{11} \vert ^{2} - \frac{ \vert g_{02} \vert ^{2}}{3}\biggr) + \frac{g_{21}}{2}, \\ \mu_{2} = - \frac{\operatorname{Re} \{ c_{1}(0)\}}{\operatorname{Re} \{ \lambda'(\tau_{1_{0}})\}}, \\ \beta_{2} = 2\operatorname{Re} \bigl(c_{1}(0) \bigr), \\ T_{2} = - \frac{\operatorname{Im} \{ c_{1}(0)\} + \mu_{2}\operatorname{Im} \{ \lambda'(\tau_{1_{0}})\}}{\omega^{ *} \tau_{1_{0}}}. \end{gathered} $$
(3.30)
These formulas describe the periodic solutions of Eq. (3.1) at \(\tau= \tau_{1_{0}}\) on the center manifold. From the discussion above, we have the following result.
Theorem 3.1
The direction of Hopf bifurcation is determined by the sign of
\(\mu_{2}\): if
\(\mu_{2} > 0\) (\(\mu_{2} < 0\)), then the Hopf bifurcation is supercritical (subcritical). The stability of the bifurcating periodic solutions is determined by the sign of
\(\beta _{2}\): if
\(\beta_{2} < 0\) (\(\beta_{2} > 0\)), the bifurcating periodic solutions are stable (unstable). The period of the bifurcating periodic solutions is determined by the sign of
\(T_{2}\): if
\(T_{2} > 0\) (\(T_{2} < 0\)), the bifurcating periodic solutions increase (decrease).