Appendix: Direction and stability of the Hopf bifurcation
In this appendix, we determine the properties of the spatially non-homogeneous periodic solutions found in Theorem 3.2. We calculate \(Re(c_1(b_j))\) for \(j\in (0, N_1]\). We set
$$\begin{aligned}&q := \cos \frac{j}{l}x \left( \begin{array}{ccc} m_j \\ n_j \end{array} \right) =\left( \begin{array}{ccc} 1 \\ \frac{d_2 j^2}{l^2 \theta }-i\frac{\omega _j}{\theta } \end{array} \right) \quad \hbox {and}\\&\quad q* := \cos \frac{j}{l}x \left( \begin{array}{ccc} m_j^* \\ n_j^*\end{array} \right) =\left( \begin{array}{ccc} \frac{1}{l \pi }+\frac{d_2 j^2}{\omega _j l^3 \pi } i \\ \frac{- \theta }{ \omega _j l \pi } i \end{array} \right) , \end{aligned}$$
where \(\omega _j=(\theta C(b_j)-\frac{d_2^2 j^4}{l^4})^{1/2}.\) By straightforward compute, we have
$$\begin{aligned}&[2i\omega _j I-L_{2j}(b_j)]^{-1}\\&\quad =(\alpha _1+\alpha _2 i)^{-1} \left( \begin{array}{ccc} 2i\omega _j+\frac{4d_2j^2}{l^2} &{}\quad -\theta \\ C(b_j) &{}\quad 2i\omega _j-\frac{(d_2-3d_1)j^2}{l^2} \end{array} \right) , \end{aligned}$$
with
$$\begin{aligned}&\alpha _1:=\frac{(12d_1d_2-3d_2^2)j^4-3\omega _0^2l^4}{l^4},\\&\alpha _2:=\frac{6\omega _j(d_1+d_2)j^2}{l^2}, \end{aligned}$$
and
$$\begin{aligned}&[2i\omega _j I-L_{0}(b_j)]^{-1}\\&\quad =(\alpha _3+\alpha _4 i)^{-1} \left( \begin{array}{ccc} 2i\omega _j &{}\quad -\theta \\ C(b_j) &{}\quad 2i\omega _j-\frac{(d_1+d_2)j^2}{l^2} \end{array} \right) , \end{aligned}$$
with
$$\begin{aligned} \alpha _3:=\frac{d_2^2j^4-3\omega _j^2l^4}{l^4},\quad \alpha _4:=-\frac{2\omega _j(d_1+d_2)j^2}{l^2}. \end{aligned}$$
Then we have
$$\begin{aligned} w_{20}&= \frac{1}{2}[2i\omega _jI-L(b_j)]^{-1}\left[ (\cos \frac{2n}{l}x+1)\left( \begin{array}{ccc} c_j\\ d_j \end{array} \right) \right] \\&= \left[ \frac{[2i\omega _jI-L_{2j}(b_j)]^{-1}}{2}\cos \frac{2j}{l}x\right. \\&\quad \left. +\,\frac{[2i\omega _jI-L_{0}(b_j)]^{-1}}{2}\right] \left( \begin{array}{ccc}c_j\\ d_j \end{array} \right) \\&= \frac{(\alpha _1\!+\!\alpha _2 i)^{-1}}{2}\left( \begin{array}{ccc} (2i\omega _j\!+\!\frac{4d_2j^2}{l^2})c_j\!-\!\theta d_j \\ C(b_j)c_j\!+\!(2i\omega _j\!-\!\frac{(d_2-3d_1)j^2}{l^2})d_j \end{array} \right) \cos \frac{2j}{l}x\\&\quad +\,\frac{(\alpha _3+\alpha _4 i)^{-1}}{2}\left( \begin{array}{ccc} 2i\omega _jc_j-\theta d_j \\ C(b_j)c_j+(2i\omega _j-\frac{(d_1+d_2)j^2}{l^2})d_j \end{array} \right) . \end{aligned}$$
Likewise we have
$$\begin{aligned} w_{11}&= -\frac{1}{2}[L(b_j)]^{-1}\left[ (\cos \frac{2n}{l}x+1)\left( \begin{array}{ccc} e_j\\ f_j \end{array} \right) \right] \\&= \frac{\alpha _5^{-1}}{2}\left( \begin{array}{ccc} \frac{4d_2j^2}{l^2}e_j-\theta f_j \\ C(b_j)e_j-\frac{(d_2-3d_1)j^2}{l^2})f_j \end{array} \right) \cos \frac{2j}{l}x\\&\quad -\,\frac{1}{2\theta C(b_j)}\left( \begin{array}{ccc} \theta f_j \\ -C(b_j)e_j+\frac{(d_1+d_2)j^2}{l^2}f_j \end{array} \right) \end{aligned}$$
with \(\alpha _5:=[(12d_1d_2-3d_2^2)j^4+\omega _0^2l^4]/l^4\). From computation, it follows that
$$\begin{aligned}&c_j\!=\!\frac{-2 l^2 p \theta \left( 2 s^2-5 s \theta +4 \theta ^2\right) -4 s (s-\theta )^2 \sqrt{\frac{\theta }{s-\theta }} b_j \left( j^2 d_2-i l^2 \omega _j\right) }{l^2 s^2 \theta },\\&d_j=\frac{2 (s-\theta )}{l^2 s^2 \theta } \left( l^2 p (s-4 \theta ) \theta +\frac{2 s \theta b_j \left( j^2 d_2-i l^2 \omega _j\right) }{\sqrt{\frac{\theta }{s-\theta }}}\right) ,\\&e_j=-\frac{2}{l^2 s^2 \theta } \left( l^2 p \theta \left( 2 s^2-5 s \theta +4 \theta ^2\right) \right. \\&\quad \quad \left. +\,2 j^2 s (s-\theta )^2 \sqrt{\frac{\theta }{s-\theta }} b_j d_2\right) ,\\&f_j=\frac{2 (s-\theta )}{l^2 s^2 \theta } \left( l^2 p (s-4 \theta ) \theta +\frac{2 j^2 s \theta b_j d_2}{\sqrt{\frac{\theta }{s-\theta }}}\right) ,\\&g_j=\frac{2 (s-\theta )^2 b_j}{l^2 s^3 \theta } \left( 12 l^2 p (s-2 \theta ) \theta \sqrt{\frac{\theta }{s-\theta }}\right. \\&\quad \quad \left. -\,s (s-4 \theta ) b_j \left( 3 j^2 d_2-i l^2 \omega _j\right) \right) ,\\&h_j=\frac{2 (s-\theta )^2 b_j}{l^2 s^3 \theta } \left( 12 l^2 p \theta \sqrt{\frac{\theta }{s-\theta }} (-s+2 \theta )\right. \\&\quad \quad \left. +\,s (s-4 \theta ) b_j \left( 3 j^2 d_2-i l^2 \omega _j\right) \right) . \end{aligned}$$
Then we have
$$\begin{aligned}&Q_{w_{20}\overline{q}} \\&=\left( \begin{array}{ccc} f_\mathrm{uu}\xi _1+f_\mathrm{uv}(\xi _1\overline{n_j}+\xi _2)+f_{\mathrm{vv}}\xi _2\overline{n_j} \\ g_\mathrm{uu}\xi _1+g_\mathrm{uv}(\xi _1\overline{n_j}+\xi _2)+g_{\mathrm{vv}}\xi _2\overline{n_j} \end{array} \right) \cos \frac{2j}{l}x \cos \frac{j}{l}x\\&\quad \quad +\left( \begin{array}{l} f_\mathrm{uu}\eta _1+f_\mathrm{uv}(\eta _1\overline{n_j}+\eta _2)+f_{\mathrm{vv}}\eta _2\overline{n_j} \\ g_\mathrm{uu}\eta _1 +g_\mathrm{uv}(\eta _1\overline{n_j}+\eta _2)+g_{\mathrm{vv}}\eta _2\overline{n_j} \end{array} \right) \cos \frac{j}{l}x, \end{aligned}$$
and
$$\begin{aligned}&Q_{w_{11}q} \\&\quad =\left( \begin{array}{l} f_\mathrm{uu}\tau _1+f_\mathrm{uv}(\tau _1\overline{n_j}+\tau _2)+f_{\mathrm{vv}}\tau _2\overline{n_j} \\ g_\mathrm{uu}\tau _1+g_\mathrm{uv}(\tau _1\overline{n_j}+\tau _2)+g_{\mathrm{vv}}\tau _2\overline{n_j} \end{array} \right) \cos \frac{2j}{l}x \cos \frac{j}{l}x\\&\quad \quad +\left( \begin{array}{l} f_\mathrm{uu}\chi _1+f_\mathrm{uv}(\chi _1\overline{n_j}+\chi _2)+f_{\mathrm{vv}}\chi _2\overline{n_j} \\ g_\mathrm{uu}\chi _1+g_\mathrm{uv}(\chi _1\overline{n_j}+\chi _2)+g_{\mathrm{vv}}\chi _2\overline{n_j} \end{array} \right) \cos \frac{j}{l}x, \end{aligned}$$
where
$$\begin{aligned}&\left\{ \begin{array}{ll} f_\mathrm{uu}=-\frac{2 p \left( 2 s^2-5 s \theta +4 \theta ^2\right) }{s^2},&{}\quad f_\mathrm{uv} =-\frac{2 b (s-\theta )^2 \sqrt{\frac{\theta }{s-\theta }}}{s}, \quad f_{\mathrm{vv}}=0,\\ g_\mathrm{uu}=\frac{2 p \left( s^2-5 s \theta +4 \theta ^2\right) }{s^2},&{}\quad g_\mathrm{uv}=\frac{2 b (s-\theta )^2 \sqrt{\frac{\theta }{s-\theta }}}{s}, \quad g_{\mathrm{vv}}=0,\\ \end{array}\right. \end{aligned}$$
(5.1)
$$\begin{aligned}&\left\{ \begin{array}{ll} \xi _1=\xi _{11}+i* \xi _{12}, &{} \xi _2=\xi _{21}+i*\xi _{22},\\ \eta _1=\eta _{11}+i*\eta _{12}, &{} \eta _2=\eta _{21}+i*\eta _{22},\\ \tau _1=\frac{\alpha _5^{-1}}{2}\left( \frac{4d_2j^2}{l^2}e_j-\theta f_j\right) , &{} \tau _2 =\frac{\alpha _5^{-1}}{2}\left( C(b_j)e_j-\frac{(d_2-3d_1)j^2}{l^2}f_j\right) ,\\ \chi _1=-\frac{1}{2C(b_j)}f_j, &{} \chi _2=\frac{1}{2\theta C(b_j)}\left( C(b_j)e_j-\frac{(d_1+d_2)j^2}{l^2}f_j \right) ,\\ \end{array} \right. \end{aligned}$$
(5.2)
and
$$\begin{aligned} \xi _{11}&= \frac{2 j^2 d_2 f_{\mathrm{uv}} \left( 2 j^2 d_2 \alpha _1-l^2 \alpha _2 \omega _j\right) }{l^4 \theta \left( \alpha _1^2+\alpha _2^2\right) }\\&\quad +\,\frac{\left( f_{\mathrm{uu}}-g_{\mathrm{uv}}\right) \alpha _2 \omega _j}{\alpha _1^2+\alpha _2^2} +\frac{\alpha _1 \left( \theta ^2 g_{\mathrm{uu}}+4 f_{\mathrm{uv}} \omega _j^2\right) }{2 \theta \left( \alpha _1^2+\alpha _2^2\right) }\\&\quad +\,\frac{j^2 d_2 \left( 2 f_{\mathrm{uu}}+g_{\mathrm{uv}}\right) \alpha _1}{l^2 \left( \alpha _1^2+\alpha _2^2\right) },\\ \xi _{12}&= -\frac{2 j^2 d_2 f_{\mathrm{uv}} \left( 2 j^2 d_2 \alpha _2+l^2 \alpha _1 \omega _j\right) }{l^4 \theta \left( \alpha _1^2+\alpha _2^2\right) }+\frac{\left( f_{\mathrm{uu}}-g_{\mathrm{uv}}\right) \alpha _1 \omega _j}{\alpha _1^2+\alpha _2^2}\\&\quad -\,\frac{\alpha _2 \left( \theta ^2 g_{\mathrm{uu}}+4 f_{\mathrm{uv}} \omega _j^2\right) }{2 \theta \left( \alpha _1^2+\alpha _2^2\right) }-\frac{j^2 d_2 \left( 2 f_{\mathrm{uu}}+g_{\mathrm{uv}}\right) \alpha _2}{l^2 \left( \alpha _1^2+\alpha _2^2\right) },\\ \xi _{21}&= \frac{j^2 \left( 3 d_1-d_2\right) g_{\mathrm{uu}} \alpha _1}{2 l^2 \left( \alpha _1^2+\alpha _2^2\right) } +\frac{j^4 \left( 3 d_1-d_2\right) d_2 g_{\mathrm{uv}} \alpha _1}{l^4 \theta \left( \alpha _1^2+\alpha _2^2\right) }\\&\quad +\,\frac{\omega _j \left( -C\left( b_j\right) f_{\mathrm{uv}} \alpha _2+2 g_{\mathrm{uv}} \alpha _1 \omega _j\right) }{\theta \left( \alpha _1^2+\alpha _2^2\right) }\\&\frac{C\left( b_j\right) f_{\mathrm{uu}} \alpha _1+2 g_{\mathrm{uu}} \alpha _2 \omega _j}{2 \left( \alpha _1^2+\alpha _2^2\right) }+\frac{3 j^2 \left( -d_1+d_2\right) g_{\mathrm{uv}} \alpha _2 \omega _j}{l^2 \theta \left( \alpha _1^2+\alpha _2^2\right) }\\&\quad +\,\frac{C\left( b_j\right) j^2 \alpha _1 d_2 f_{\mathrm{uv}}}{l^2 \left( \alpha _1{}^2+\alpha _2{}^2\right) \theta },\\ \xi _{22}&= \frac{j^2 \left( -3 d_1+d_2\right) g_{\mathrm{uu}} \alpha _2}{2 l^2 \left( \alpha _1^2+\alpha _2^2\right) } +\frac{j^4 d_2 \left( -3 d_1+d_2\right) g_{\mathrm{uv}} \alpha _2}{l^4 \theta \left( \alpha _1^2+\alpha _2^2\right) }\\&\quad -\,\frac{\omega _j \left( C \left( b_j\right) f_{\mathrm{uv}} \alpha _1+2 g_{\mathrm{uv}} \alpha _2 \omega _j\right) }{\theta \left( \alpha _1^2+\alpha _2^2\right) }\\&\quad +\,\frac{3 j^2 \left( -d_1\!+\!d_2\right) g_{\mathrm{uv}} \alpha _1 \omega _j}{l^2 \theta \left( \alpha _1^2 \!+\!\alpha _2^2\right) }\!+\!\frac{-C\left( b_j\right) f_{\mathrm{uu}} \alpha _2\!+\!2 g_{\mathrm{uu}} \alpha _1 \omega _j}{2 \left( \alpha _1^2\!+\!\alpha _2^2\right) }\\&\quad -\,\frac{C\left( b_j\right) j^2 \alpha _2 d_2 f_{\mathrm{uv}}}{l^2 \left( \alpha _1{}^2+\alpha _2{}^2\right) \theta },\\ \eta _{11}&= \frac{\left( f_{\mathrm{uu}}+g_{\mathrm{uv}}\right) \alpha _4 \omega _j}{\alpha _3^2+\alpha _4^2} -\frac{j^2 d_2 \left( \theta g_{\mathrm{uv}} \alpha _3-2 f_{\mathrm{uv}} \alpha _4 \omega _j\right) }{l^2 \theta \left( \alpha _3^2+\alpha _4^2\right) }\\&\quad -\,\frac{\alpha _3 \left( \theta ^2 g_{\mathrm{uu}} -4 f_{\mathrm{uv}} \omega _j^2\right) }{2 \theta \left( \alpha _3^2+\alpha _4^2\right) },\\ \eta _{12}&= \frac{\left( f_{\mathrm{uu}}+g_{\mathrm{uv}}\right) \alpha _3 \omega _j}{\alpha _3^2+\alpha _4^2} +\frac{\alpha _4 \left( \theta ^2 g_{\mathrm{uu}}-4 f_{\mathrm{uv}} \omega _j^2\right) }{2 \theta \left( \alpha _3^2 +\alpha _4^2\right) }\\&\quad +\,\frac{j^2 d_2 \left( \theta g_{\mathrm{uv}} \alpha _4 +2 f_{\mathrm{uv}} \alpha _3 \omega _j\right) }{l^2 \theta \left( \alpha _3^2+\alpha _4^2\right) }, \end{aligned}$$
$$\begin{aligned} \eta _{21}&= \frac{\omega _j \left( -C\left( b_j\right) f_{\mathrm{uv}} \alpha _4 +2 g_{\mathrm{uv}} \alpha _3 \omega _j\right) }{\theta \left( \alpha _3^2+\alpha _4^2\right) } \\&\quad +\,\frac{C\left( b_j\right) f_{\mathrm{uu}} \alpha _3+2 g_{\mathrm{uu}} \alpha _4 \omega _j}{2 \left( \alpha _3^2 +\alpha _4^2\right) }+\frac{C\left( b_j\right) j^2 \alpha _3 d_2 f_{\mathrm{uv}}}{l^2 \left( \alpha _3{}^2+\alpha _4{}^2\right) \theta }\\&\quad -\,\frac{j^2 \left( d_1+d_2\right) g_{\mathrm{uu}} \alpha _3}{2 l^2 \left( \alpha _3^2+\alpha _4^2\right) } -\frac{j^4 d_2 \left( d_1+d_2\right) g_{\mathrm{uv}} \alpha _3}{l^4 \theta \left( \alpha _3^2+\alpha _4^2\right) } \\&\quad +\,\frac{j^2 \left( d_1+3 d_2\right) g_{\mathrm{uv}} \alpha _4 \omega _j}{l^2 \theta \left( \alpha _3^2+\alpha _4^2\right) },\\ \eta _{22}&= -\frac{\omega _j \left( C\left( b_j\right) f_{\mathrm{uv}} \alpha _3+2 g_{\mathrm{uv}} \alpha _4 \omega _j\right) }{\theta \left( \alpha _3^2+\alpha _4^2\right) }\\&\quad +\,\frac{-C\left( b_j\right) f_{\mathrm{uu}} \alpha _4+2 g_{\mathrm{uu}} \alpha _3 \omega _j}{2 \left( \alpha _3^2+\alpha _4^2\right) }-\frac{C\left( b_j\right) j^2 \alpha _4 d_2 f_{\mathrm{uv}}}{l^2 \left( \alpha _3{}^2+\alpha _4{}^2\right) \theta }\\&\quad +\,\frac{j^2 \left( d_1+d_2\right) g_{\mathrm{uu}} \alpha _4}{2 l^2 \left( \alpha _3^2+\alpha _4^2\right) }+\frac{j^4 d_2 \left( d_1+d_2\right) g_{\mathrm{uv}} \alpha _4}{l^4 \theta \left( \alpha _3^2+\alpha _4^2\right) }\\&\quad +\,\frac{j^2 \left( d_1+3 d_2\right) g_{\mathrm{uv}} \alpha _3 \omega _j}{l^2 \theta \left( \alpha _3^2+\alpha _4^2\right) }. \end{aligned}$$
Notice that for any \(j\in \text{ N }\),
$$\begin{aligned}&\int _{0}^{l\pi } \cos ^2\frac{jx}{l} dx=\frac{1}{2}l\pi ,\\&\int _{0}^{l\pi } \cos \frac{2jx}{l} \cos ^2\frac{jx}{l} dx =\frac{1}{4}l\pi ,\\&\int _{0}^{l\pi } \cos ^4\frac{jx}{l} dx=\frac{3}{8}l\pi , \end{aligned}$$
we have
$$\begin{aligned} <q^*,Q_{w_{20\overline{q}}}>&= \frac{l\pi }{4}\overline{m_j^*}((f_\mathrm{uu}\xi _1+f_\mathrm{uv}(\xi _1\overline{n_j}+\xi _2))\\&\quad +\,\overline{n_j^*}(g_\mathrm{uu}\xi _1+g_\mathrm{uv}(\xi _1\overline{n_j}+\xi _2)))\\&\quad +\,\frac{l\pi }{2}\overline{m_j^*}((f_\mathrm{uu}\eta _1+f_\mathrm{uv}(\eta _1\overline{n_j}+\eta _2))\\&\quad +\,\overline{n_j^*}(g_\mathrm{uu}\eta _1+g_\mathrm{uv}(\eta _1\overline{n_j}+\eta _2))),\\ <q^*,Q_{w_{11q}}>&= \frac{l\pi }{4}\overline{m_j^*}((f_\mathrm{uu}\tau _1+f_\mathrm{uv}(\tau _1n_j+\tau _2))\\&\quad +\,\overline{n_j^*}(g_\mathrm{uu}\tau _1+g_\mathrm{uv}(\tau _1n_j+\tau _2)))\\&\quad +\,\frac{l\pi }{2}\overline{m_j^*}((f_\mathrm{uu}\chi _1+f_\mathrm{uv}(\chi _1n_j+\chi _2))\\&\quad +\,\overline{n_j^*}(g_\mathrm{uu}\chi _1+g_\mathrm{uv}(\chi _1n_j+\chi _2))),\\ <q^*,C_{w_{qq\overline{q}}}>&= \frac{3l\pi }{8}(\overline{m_j^*}g_j+\overline{n_j^*}h_j). \end{aligned}$$
When \(j\in \text{ N }\), it follows that \(<q^*,Q_{qq}>=<q^*,Q_{q\overline{q}}>=0\). Thus we have
$$\begin{aligned} Re(c_1(b_j))&= \frac{3}{16} \left( f_{\mathrm{uuu}}+g_{\mathrm{uuv}}+\frac{2 j^2 d_2 f_{\mathrm{uuv}}}{l^2 \theta }\right) \\&\quad +\,\frac{\left( 2 \eta _{12}+\xi _{12}\right) }{8 l^2 \pi \omega _j}\left( j^2 \pi d_2 f_{\mathrm{uu}} -l \theta \left( f_{\mathrm{uv}}+l \pi g_{\mathrm{uu}}\right) \right) \\&\quad +\,\frac{\left( 2 \eta _{22}+\xi _{22}\right) }{8 l^2 \omega _j}\left( j^2 d_2 f_{\mathrm{uv}} -l^2 \theta g_{\mathrm{uv}}\right) \\&\quad +\,\frac{1}{8} f_{\mathrm{uv}} \left( 2 \eta _{21}+\xi _{21}+2 \tau _2+4 \chi _2\right) \\&\quad +\,\frac{1}{8} \left( 2 \eta _{11}+\xi _{11}+2 \tau _1+4 \chi _1\right) \left( f_{\mathrm{uu}} \right. \\&\quad \left. +\,\frac{\theta \left( j^2 d_2 f_{\mathrm{uv}}-l^2 \theta g_{\mathrm{uv}}\right) }{l^3 \pi \omega _j^2}\right) . \end{aligned}$$
Thus the bifurcating periodic solution is supercritical (resp. subcritical) if \(c_1(b_j)<0\) (resp. \(>0\)).
