Turing and Turing–Hopf Bifurcations for a Reaction Diffusion Equation with Nonlocal Advection

Abstract

In this paper, we study the stability and the bifurcation properties of the positive interior equilibrium for a reaction–diffusion equation with nonlocal advection. Under rather general assumption on the nonlocal kernel, we first study the local well posedness of the problem in suitable fractional spaces and we obtain stability results for the homogeneous steady state. As a special case, we obtain that “standard” kernels such as Gaussian, Cauchy, Laplace and triangle, will lead to stability. Next we specify the model with a given step function kernel and investigate two types of bifurcations, namely Turing bifurcation and Turing–Hopf bifurcation. In fact, we prove that a single scalar equation may display these two types of bifurcations with the dominant wave number as large as we want. Moreover, similar instabilities can also be observed by using a bimodal kernel. The resulting complex spatiotemporal dynamics are illustrated by numerical simulations.

Appendix

1.1 Reduced System

In this subsection, we give a brief calculation of the center manifold reduction as a supplement of Theorem 3.8. Recall our equation reads as follows:

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}\begin{pmatrix}w(t)\\ \gamma (t)\end{pmatrix}=L\begin{pmatrix}w(t)\\ \gamma (t)\end{pmatrix}+R\begin{pmatrix}w(t)\\ \gamma (t)\end{pmatrix}, \end{aligned}$$

wherein we have set

$$\begin{aligned} L=\begin{pmatrix} \left( A+\partial _w {\tilde{F}}(0,\gamma _0)\right) &{} 0\\ 0 &{} 0\end{pmatrix}\in {\mathcal {L}}\left( H^2\times \mathbb {R},H^0\times \mathbb {R}\right) , \end{aligned}$$

and

$$\begin{aligned} R\begin{pmatrix}w\\ \gamma \end{pmatrix}=\begin{pmatrix}\tilde{F}(w,\gamma )-\partial _w{\tilde{F}}(0,\gamma _0)w\\ 0\end{pmatrix}. \end{aligned}$$

Recall also that ${\tilde{F}}$ and B are defined by

$$\begin{aligned} \tilde{F}(w,\gamma )= & {} \frac{b-\mu }{\gamma \mu } \left( K\circ w\right) ''+B(w,w)+\left( \frac{\mu ^2}{b+\gamma \mu w}-\mu \right) w+w,\\ B(w,w)= & {} \frac{\hbox {d}}{\hbox {d}x}\left( w \frac{\hbox {d}}{\hbox {d}x}K\circ w\right) , \end{aligned}$$

and we define

$$\begin{aligned} G(w,\gamma )={\tilde{F}}(w,\gamma )-\partial _w{\tilde{F}}(0,\gamma _0)w. \end{aligned}$$

Moreover, we also define $\hat{A}= A+\partial _w {\tilde{F}}(0,\gamma _0)$ and let us observe that $\hat{A}e_n=\lambda _n(\gamma _0)e_n$ for all $n\in {\mathbb {Z}}$. Recall that the framework of 3.8 implies that we have, for some $n_0\ge 1$,

$$\begin{aligned} \lambda _{\pm n_0}(\gamma _0)\in i\mathbb {R},\quad \mathrm {Re}(\lambda _n(\gamma _0))<0,\;\forall n\ne \pm n_0. \end{aligned}$$

To perform our center manifold reduction, we will need the following computations:

• $K\circ e_n= c_n(K)e_n$ for all $n\in {\mathbb {Z}}$

• $B(e_n,e_m)=-\left( \dfrac{\pi }{L}\right) ^2c_m(K)m(m+n)e_{m+n}$ for all $(n,m)\in {\mathbb {Z}}^2$.

Define the center space ${\mathcal {E}}^c= \mathrm{span}\,\left( e_{\pm n_0}\right) \times \mathbb {R}$ and the stable space ${\mathcal {E}}^s= \mathrm{span}\,\left( e_{\pm n_0}\right) ^{\perp }\times \lbrace 0\rbrace$ where $\mathrm{span}\left( e_{\pm n_0}\right)$ denotes the vector space spanned by eigenfunctions $e_{\pm n_0}$ while $\mathrm{span}\left( e_{\pm n_0}\right) ^{\perp }$ denotes its orthogonal space for the $L^2(-L,L)-$inner product. We denote by $\tilde{\Psi }:{\mathcal {E}}^c\rightarrow {\mathcal {E}}^s$ the local center manifold and in the sequel we will make use of the following notation

$$\begin{aligned} \tilde{\Psi }(x^c,\gamma )=\left( \Psi (x^c,\gamma ),0\right) \in \mathcal E^s,\text { for }(x_c,\gamma )\in {\mathcal {E}}^c\text { close to } (0,\gamma _0), \end{aligned}$$

and $x^c=x_- e_{-n_0}+x_+e_{n_0}$. Since the center manifold is smooth (here $C^\infty )$ we rewrite it as follows:

$$\begin{aligned} \Psi (x^c,\gamma )= & {} \sum _{n\ne \pm n_0}\Psi _n(x^c,\gamma )e_n=\sum _{n\ne \pm n_0}P_n(x^c,\gamma ) e_n\\&+\,O\left( \left( (\Vert x_c\Vert +|\gamma -\gamma _0|\right) ^3\right) \text { in }H^2, \end{aligned}$$

where, for each $n\in {\mathbb {Z}}\setminus \{\pm n_0\}$, $P_n(x^c,\gamma )$ is homogeneous polynomial of degree 2 for the variables $x_-$, $x_+$ and $(\gamma -\gamma _0)$. For notational simplicity, we also denote by $P_{\pm n_0}(x^c,\gamma )$ the first-order polynomials

$$\begin{aligned} P_{-n_0}(x^c,\gamma )=x_-\text { and }P_{n_0}(x^c,\gamma )=x_+. \end{aligned}$$

Note that since the center manifold is real valued, one has

$$\begin{aligned} x_+=\overline{x}_-\text { and }\Psi _{-n}(x^c,\gamma )=\overline{\Psi _n(x^c,\gamma )},\;\forall n\ne \pm n_0. \end{aligned}$$

To compute the—center manifold—reduced system, let us introduce the center and stable projectors $\Pi ^c$ and $\Pi ^s$ as follows:

$$\begin{aligned} \Pi ^c\varphi =\sum _{n=\pm n_0} c_n(\varphi )e_n\text { and }\Pi ^s\varphi =\sum _{n\ne \pm n_0} c_n(\varphi )e_n, \end{aligned}$$

as well as the center and stable part of the linear operator ${\hat{A}}$, respectively, denoted by ${\hat{A}}^c$ and ${\hat{A}}^s$ and defined by

$$\begin{aligned} {\hat{A}}^c\varphi =\sum _{n=\pm n_0} c_n(\varphi )\lambda _n(\gamma _0)e_n\text { and }{\hat{A}}^s \varphi =\sum _{n\ne \pm n_0}\lambda _n(\gamma _0)c_n(\varphi )e_n. \end{aligned}$$

Next, the reduced system reads as

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\hbox {d} x^c(t)}{\hbox {d}t}={\hat{A}}^c x^c(t)+\Pi ^c G\left( x^c(t)+\Psi \left( x^c(t),\gamma (t)\right) ,\gamma (t)\right) ,\\ \dfrac{\hbox {d}\gamma (t)}{\hbox {d}t}=0, \end{array}\right. } \end{aligned}$$

(5.1)

and the center manifold satisfies the following equation in the neighborhood of $(x^c,\gamma )=(0,,\gamma _0)$:

$$\begin{aligned}&\partial _{x^c}\Psi (x^c,\gamma ) \left[ A^c x^c+ \Pi ^c G(x^c+\Psi (x^c,\gamma ),\gamma )\right] \nonumber \\&\quad =A^s \Psi (x^c,\gamma )+\Pi ^s G(x^c+\Psi (x^c,\gamma ),\gamma ). \end{aligned}$$

(5.2)

Our goal is to obtain a Taylor expansion up to order 3 of the above reduced system. To that aim we shall first compute a Taylor expansion of $\Pi ^c G(x^c+\Psi (x^c,\gamma ),\gamma )$ and $\Pi ^s G(x^c+\Psi (x^c,\gamma ),\gamma )$, respectively, up to order 3 and 2. To do so, first note that for $\Vert w\Vert$ small enough and $\gamma$ close to $\gamma _0$ one has the series expansion

$$\begin{aligned} \tilde{F}(w,\gamma )=\frac{b-\mu }{\gamma \mu } \left( K\circ w\right) ''+B(w,w)+w(1-\mu )+\frac{\mu ^2}{b}\sum _{p=0}^{\infty }\frac{\gamma ^p\mu ^p w^{p+1}}{b^p}, \end{aligned}$$

and

$$\begin{aligned} \partial _w{\tilde{F}}(0,\gamma _0)w=\frac{b-\mu }{\gamma _0\mu } \left( K\circ w\right) ''+w(1-\mu )+\frac{\mu ^2}{b}w. \end{aligned}$$

As a consequence one has, for all w and $|\gamma -\gamma _0|$ small enough,

$$\begin{aligned} G(w,\gamma )=\frac{b-\mu }{\mu }\frac{\gamma _0-\gamma }{\gamma _0\gamma } \left( K\circ w\right) ''+B(w,w)+\frac{\mu ^2}{b}\sum _{p=1}^{\infty }\frac{\gamma ^p\mu ^p w^{p+1}}{b^p}. \end{aligned}$$

Hence, this leads us to the following order 3 Taylor expansion

$$\begin{aligned} \begin{aligned} G(w,\gamma )=&\frac{b-\mu }{\gamma _0\mu }\frac{(2\gamma _0-\gamma )(\gamma _0-\gamma )}{\gamma _0^2} \left( K\circ w\right) ''+B(w,w)+\frac{\mu ^3\gamma _0}{b^2}w^2\\&+\,\frac{\mu ^3(\gamma -\gamma _0)}{b^2}w^2+ \frac{\mu ^4\gamma _0^2}{b^3}w^3 +O((|w|+|\gamma -\gamma _0|)^4). \end{aligned} \end{aligned}$$

Now choosing the following form for w

$$\begin{aligned} w= & {} x^c+\Psi (x^c,\gamma )=x_-e_{-n_0}+x_+ e_{n_0}+\sum _{n\ne \pm n_0}P_n(x^c,\gamma )\\&+\,O\left( \left( \Vert x^c\Vert +|\gamma -\gamma _0|\right) ^3\right) \text { in }H^2, \end{aligned}$$

yields

$$\begin{aligned} \begin{aligned} (K\circ w)''=&-\left( \left( \frac{n_0\pi }{L}\right) ^2 c_{-n_0}(K) x_- e_{-n_0} + \left( \frac{n_0\pi }{L}\right) ^2 c_{n_0}(K) x_+ e_{n_0}\right) \\&-\,\sum _{n\ne \pm n_0}\left( \frac{n\pi }{L}\right) ^2c_n(K)P_n(x^c,\gamma )e_n+O\left( \left( \Vert x^c\Vert +|\gamma -\gamma _0|\right) ^3\right) \text { in }H^0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} B(w,w)=&\sum _{m,n\in {\mathbb {Z}}^2} P_n(x^c,\gamma )P_m(x^c,\gamma )B(e_n,e_m)\\ =&-\sum _{m,n} P_n(x^c,\gamma )P_m(x^c,\gamma )\left( \dfrac{\pi }{L}\right) ^2c_m(K)m(m+n)e_{m+n}\\&+\,O\left( \left( \Vert x^c\Vert +|\gamma -\gamma _0|\right) ^4\right) \text { in }H^0, \end{aligned} \end{aligned}$$

Now, we calculate those terms of B(w, w) up to order 2, which are given by

$$\begin{aligned} \text {order 2 }{\left\{ \begin{array}{ll} -x_+^2\left( \dfrac{\pi }{L}\right) ^2c_{n_0}(K) 2n_0^2 e_{2n_0},&{} n=m=n_0; \\ -x_-^2\left( \dfrac{\pi }{L}\right) ^2c_{-n_0}(K) 2n_0^2 e_{-2n_0},&{} n=m=-n_0;\\ 0,&{}n=n_0,m=-n_0; \text { or } n=-n_0,m=n_0. \end{array}\right. } \end{aligned}$$

For further normal form computation, we list all possible situations of order 3 of $\Pi ^c B(w,w)$, that the components along the vectors $e_{n_0}$ and $e_{-n_0}$. They reads as follows:

$$\begin{aligned} \text {order 3 }{\left\{ \begin{array}{ll} 0,&{} n=n_0,\;m=0; \\ -x_+P_{-2n_0}(x^c,\gamma )\left( \dfrac{\pi }{L}\right) ^2c_{-2n_0}(K) 2n_0^2 e_{-n_0},&{} n=n_0,\;m=-2n_0;\\ 0,&{} n=-n_0,\;m=0;\\ -x_-P_{2n_0}(x^c,\gamma )\left( \dfrac{\pi }{L}\right) ^2c_{2n_0}(K) 2n_0^2 e_{n_0},&{} n=-n_0,\;m=2n_0;\\ -x_+P_{0}(x^c,\gamma )\left( \dfrac{\pi }{L}\right) ^2c_{n_0}(K) n_0^2 e_{n_0},&{} n=0,\;m=n_0;\\ x_+P_{-2n_0}(x^c,\gamma )\left( \dfrac{\pi }{L}\right) ^2c_{n_0}(K) n_0^2 e_{-n_0},&{} n=-2n_0,\;m=n_0;\\ -x_-P_{0}(x^c,\gamma )\left( \dfrac{\pi }{L}\right) ^2c_{-n_0}(K) n_0^2 e_{-n_0},&{} n=0,\;m=-n_0;\\ x_-P_{2n_0}(x^c,\gamma )\left( \dfrac{\pi }{L}\right) ^2c_{-n_0}(K) n_0^2 e_{n_0},&{} n=2n_0,\;m=-n_0; \end{array}\right. } \end{aligned}$$

Finally, we compute the term $\Pi ^c w^2$ and $\Pi ^c w^3$. To that aim, note that one has

$$\begin{aligned} w^2=&\left( x_- e_{-n_0} +x_+ e_{n_0}\; +\;\sum _{n\ne \pm n_0}P_n(x^c,\gamma )e_n\right) ^2\\ =\,&x_+^2 e_{2n_0}+2 x_+x_- e_{0}+x_-^2 e_{-2n_0}+(x_- e_{-n_0} +x_+ e_{n_0})\sum _{n\ne \pm n_0}P_n(x^c,\gamma )e_n\\&+\,O((\Vert x^c\Vert +|\gamma -\gamma _0|)^4). \end{aligned}$$

therefore

$$\begin{aligned} \Pi ^cw^2= & {} \left( x_+P_0(x^c,\gamma )+x_-P_{2n_0}(x^c,\gamma )\right) e_{n_0}\\&+\,\left( x_+P_{-2n_0}(x^c,\gamma )+x_-P_0(x^c,\gamma )\right) e_{-n_0}+O((\Vert x^c\Vert +|\gamma -\gamma _0|)^4). \end{aligned}$$

Next, one has

$$\begin{aligned} w^3=(x_- e_{-n_0} +x_+ e_{n_0})^3+O((\Vert x^c\Vert +|\gamma -\gamma _0|)^4), \end{aligned}$$

so that we get

$$\begin{aligned} \Pi ^cw^3=3x_+^2x_-e_{n_0}+3x_+x_-^2e_{-n_0}+O((\Vert x^c\Vert +|\gamma -\gamma _0|)^4). \end{aligned}$$

Coupling the above computations allows us to compute a Taylor expansion up to order 3 for the quantity $\Pi ^c G(x^c+\Psi (x^c,\gamma ),\gamma )$ and more precisely we get

$$\begin{aligned}&\Pi ^c G(x^c+\Psi (x^c,\gamma ),\gamma )\nonumber \\&\quad =\frac{b-\mu }{\gamma _0\mu }\frac{(2\gamma _0-\gamma )(\gamma -\gamma _0)}{\gamma _0^2} \left( \frac{n_0\pi }{L}\right) ^2\left( c_{-n_0}(K) x_- e_{-n_0} + c_{n_0}(K) x_+ e_{n_0}\right) \nonumber \\&\qquad +\,\left( \dfrac{\pi }{L}\right) ^2\left( x_-P_{2n_0}(x^c,\gamma )c_{-n_0}(K) n_0^2\right. \nonumber \\&\qquad \left. -x_-P_{2n_0}(x^c,\gamma )c_{2n_0}(K) 2n_0^2-x_+P_{0}(x^c,\gamma )c_{n_0}(K) n_0^2\right) e_{n_0}\nonumber \\&\qquad +\,\left( \dfrac{\pi }{L}\right) ^2\left( x_+P_{-2n_0}(x^c,\gamma )c_{n_0}(K) n_0^2\right. \\&\qquad \left. -x_+P_{-2n_0}(x^c,\gamma )c_{-2n_0}(K) 2n_0^2-x_-P_{0}(x^c,\gamma )c_{-n_0}(K) n_0^2\right) e_{-n_0}\nonumber \\&\qquad +\frac{\mu ^3\gamma _0}{b^2}(x_+P_0(x^c,\gamma )+x_-P_{2n_0}(x^c,\gamma ))e_{n_0}\nonumber \\&\qquad +\frac{\mu ^3\gamma _0}{b^2}(x_+P_{-2n_0}(x^c,\gamma )+x_-P_0(x^c,\gamma ))e_{-n_0} \nonumber \\&\qquad +\,\frac{\mu ^4\gamma _0^2}{b^3}\left( 3x_+^2x_-e_{n_0}+3x_+x_-^2e_{-n_0}\right) +\,O((\Vert x^c\Vert +|\gamma -\gamma _0|)^4)\text { in }H^0 .\nonumber \end{aligned}$$

(5.3)

Similarly, we also obtain a Taylor expansion for the quantity $\Pi ^s G(x^c+\Psi (x^c,\gamma ),\gamma )$ up to order 2 as follows:

$$\begin{aligned} \Pi ^s G(x^c+\Psi (x^c,\gamma ),\gamma )=\,&2\left( \dfrac{n_0\pi }{L}\right) ^2 \left( c_{n_0}(K) x_+^2e_{2n_0}+c_{-n_0}(K)x_-^2e_{-2n_0}\right) \\&+\,\frac{\mu ^3\gamma _0}{b^2}\left( x_+^2 e_{2n_0}+2 x_+x_- e_{0}+x_-^2 e_{-2n_0}\right) \\&+\,O((\Vert x^c\Vert +|\gamma -\gamma _0|)^3). \end{aligned}$$

We now plug the above Taylor expansion into (5.4) to identify the polynomials $P_n$ needed to obtain a Taylor expansion up to order 3 of the reduced system.

First note that the left-hand side of (5.2) can be rewritten as

$$\begin{aligned} \begin{aligned}&\partial _{x^c}\Psi (x^c,\gamma ) \left[ A^c x^c+ \Pi ^c G(x^c+\Psi (x^c,\gamma ),\gamma )\right] \\&\quad = \partial _{x^c}\Psi (x^c,\gamma ) \left[ \lambda _{n_0}(\gamma _0)x_+e_{n_0}+\lambda _{-n_0}(\gamma _0)x_-e_{-n_0}+h.o.t.\ge 2\right] \\&\quad = \lambda _{n_0}(\gamma _0)x_+ \partial _{x^c}\Psi (x^c,\gamma ) e_{n_0}+ \lambda _{-n_0}(\gamma _0)x_- \partial _{x^c}\Psi (x^c,\gamma ) e_{-n_0}+h.o.t.\ge 3 \end{aligned} \end{aligned}$$

(5.4)

where $h.o.t.\ge 2$ (resp. 3) means those terms with order greater than 2 (resp. 3). And similarly, the right-hand side of (5.2) can be rewritten as

$$\begin{aligned} \begin{aligned}&A^s \Psi (x^c,\gamma )+\Pi ^s G(x^c+\Psi (x^c,\gamma ),\gamma )\\&\quad =\sum _{n\ne \pm n_0}\lambda _n(\gamma _0)P_n(x^c,\gamma )e_n-2\left( \dfrac{n_0\pi }{L}\right) ^2 \left( c_{n_0}(K) x_+^2e_{2n_0}+c_{-n_0}(K)x_-^2e_{-2n_0}\right) \\&\qquad +\,\frac{\mu ^3\gamma _0}{b^2}\left( x_+^2 e_{2n_0}+2 x_+x_- e_{0}+x_-^2 e_{-2n_0}\right) +h.o.t.\ge 3\\&\quad =\sum _{n\ne \pm n_0}\lambda _n(\gamma _0)P_n(x^c,\gamma )e_n+C_0x_+x_-e_0+C_{2n_0} x_+^2e_{2n_0} \\&\qquad +\,C_{-2n_0} x_-^2e_{-2n_0}+h.o.t.\ge 3, \end{aligned} \end{aligned}$$

(5.5)

wherein we have set

$$\begin{aligned} C_0= & {} 2\frac{\mu ^3\gamma _0}{b^2} ,\;C_{2n_0}=-2\left( \dfrac{n_0\pi }{L}\right) ^2 c_{n_0}(K)+\frac{\mu ^3\gamma _0}{b^2},\nonumber \\ C_{-2n_0}= & {} -2\left( \dfrac{n_0\pi }{L}\right) ^2c_{-n_0}(K)+\frac{\mu ^3\gamma _0}{b^2}. \end{aligned}$$

(5.6)

According to (5.3), we only need to compute those terms when $n=0,\pm 2n_0$. Next, since (5.4) and (5.5) are equal, identifying the terms of order 2 yields

$$\begin{aligned}&\lambda _{n_0}(\gamma _0) x_+ \frac{\partial }{\partial x_+}P_n(x^c,\gamma )+\lambda _{-n_0}(\gamma _0)x_- \frac{\partial }{\partial x_-} P_n(x^c,\gamma )\\&\quad =\lambda _n(\gamma _0)P_n(x^c,\gamma )+Q_n(x^c) \text { for }n=0,\pm 2n_0, \end{aligned}$$

where we have defined

$$\begin{aligned} Q_0(x^c)=C_0 x_+x_-,\;Q_{2n_0}(x^c)=C_{2n_0} x_+^2,\;Q_{-2n_0}(x^c)=C_{-2n_0}x_-^2. \end{aligned}$$

Recalling that $P_n$ are homogeneous polynomials of degree 2 with respect to the three variables $x_-$, $x_+$ and $(\gamma -\gamma _0)$, obtains that

$$\begin{aligned} \begin{aligned} P_0(x^c,\gamma )&=-\frac{C_0}{\lambda _0(\gamma _0)}x_+x_-,\\ P_{2n_0}(x^c,\gamma )&=-\frac{C_{2n_0}}{\lambda _{2n_0}(\gamma _0)}x_+^2,\\ P_{-2n_0}(x^c,\gamma )&=-\frac{C_{-2n_0}}{\lambda _{-2n_0}(\gamma _0)}x_-^2, \end{aligned} \end{aligned}$$

where the constants $C_0$, $C_{\pm 2m_0}$ are defined in (5.6). Finally, substituting the above expression into the Taylor expansion of $\Pi ^c G(x^c+\Psi (x^c,\gamma ),\gamma )$ yields the following reduced system up to order 3,

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\hbox {d}x_+(t)}{\hbox {d}t}=\left[ i\omega +a(\gamma )\right] x_++x_- x_+^2 \beta +h.o.t\ge 4,\\ x_-(t)=\overline{x}_+(t),\\ \dfrac{\hbox {d} \gamma (t)}{\hbox {d}t}=0. \end{array}\right. } \end{aligned}$$

Here, we have set $\lambda _{n_0}(\gamma _0)=i\omega$,

$$\begin{aligned} a(\gamma )=\left( \dfrac{n_0\pi }{L}\right) ^2 \dfrac{ b-\mu }{\gamma _0 \mu }c_{n_0}(K)\dfrac{(2\gamma _0-\gamma )(\gamma -\gamma _0)}{\gamma _0^2}, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \beta =&\frac{3 \gamma _0^2 \mu ^4}{b^3}+\frac{2 \gamma _0 \mu ^3 \left( \pi ^2 b^2 n_0^2 c_{n_0}(K)-\gamma _0 \mu ^3 L^2\right) }{b^4 L^2 \lambda _0\left( \gamma _0\right) }\\&+\,\frac{\left( 2 \pi ^2 b^2 n_0^2 c_{n_0}(K)-\gamma _0 \mu ^3 L^2\right) \left( \pi ^2 b^2 n_0^2 \left( c_{-n_0}(K)-2 c_{2 n_0}(K)\right) +\gamma _0 \mu ^3 L^2\right) }{b^4 L^4 \lambda _{2 n_0}\left( \gamma _0\right) }. \end{aligned} \end{aligned}$$

The first equation in the above system turns out to be the Poincaré normal form. It allows us to study the stability of the bifurcated periodic solution. To that aim observe that

$$\begin{aligned} \mathrm {Re} (a (\gamma ))=\left( \frac{n_0\pi }{L}\right) ^2\varepsilon _0 \dfrac{(2\gamma _0-\gamma )(\gamma -\gamma _0)}{\gamma _0^2}, \end{aligned}$$

so that $\mathrm {Re} (a (\gamma ))>0$ for $\gamma >\gamma _0$ and negative for $\gamma <\gamma _0$ but close to $\gamma _0$. The stability of the bifurcating period orbit is fully by the sign of the real part of the $\beta$. However, we are not able to conclude about this sign. To summarize the Hopf bifurcation at $\gamma _0$ is:

• 1. supercritical if $\mathrm{Re}\,\beta <0$, namely the origin is stable for $\gamma <\gamma _0$ and unstable for $\gamma >\gamma _0$. Moreover, when $\gamma >\gamma _0$ the system has a stable limit cycle. Here, the circular limit cycle has a radius proportional to $\sqrt{\gamma -\gamma _0}$.

• 2. subcritical if $\mathrm{Re}\,\beta >0$, namely the origin is stable for $\gamma <\gamma _0$ and unstable when $\gamma >\gamma _0$. Moreover, when $\gamma <\gamma _0$ the system has an unstable limit cycle, with a radius proportional to $\sqrt{\gamma _0-\gamma }$.

1.2 Numerical Scheme

The numerical method used is based on upwind scheme. We refer to Engquist and Osher (1981), Leveque (2002) for more results about this subject. We briefly illustrate our numerical scheme in this section: the approximation of the convolution term is as follows:

$$\begin{aligned} (K\circ u(t,\cdot ))(x) =\int _{\left[ -L,L\right] } u(t,y)K(x-y)\hbox {d}y \approx \sum _{j}K(x-x_j)u(t,x_j)\Delta x. \end{aligned}$$

In addition, we define

$$\begin{aligned} l_i^n:=\sum _{j}K(x_i-x_j)u(t_n,x_j)\Delta x, \end{aligned}$$

(5.7)

for $i=1,2,\ldots ,M,\ n=0,1,2,\ldots ,N$.

We use the numerical scheme as illustrated in Hillen et al. (2007) to deal with the nonlocal convection term and the scheme reads as follows:

$$\begin{aligned} \frac{u^{n+1}_i-u^n_i}{\Delta t}&=\varepsilon \frac{u_{i+1}^{n+1}-2u_i^{n+1}+u_{i-1}^{n+1}}{\Delta x^2}+ \frac{F_{i+\frac{1}{2}}^n-F_{i-\frac{1}{2}}^n}{\Delta x},\\\nonumber i&=1,2,\ldots ,M,\ n=0,1,2,\ldots ,N \end{aligned}$$

(5.8)

where

$$\begin{aligned} F_{i+\frac{1}{2}}^n=\left\{ \begin{array}{cc} g_{i+\frac{1}{2}}^n u_i^{n+1},&{}\text {if}\ g_{i+\frac{1}{2}}^n\ge 0\\ g_{i+\frac{1}{2}}^n u_{i+1}^{n+1},&{}\text {if}\ g_{i+\frac{1}{2}}^n< 0, \end{array}\right. \ i=0,1,2,\ldots ,M. \end{aligned}$$

(5.9)

with

$$\begin{aligned} g_{i+\frac{1}{2}}^n=\dfrac{l_{i+1}^n-l_i^n}{\Delta x},\ i=0,1,2,\ldots ,M. \end{aligned}$$

By the periodic boundary condition, one has $g_{\frac{1}{2}}^n=g_{M+\frac{1}{2}}^n$ and $u_0^n=u_M^n, u_1^n=u_{M+1}^n$, therefore,

$$\begin{aligned} F_{M+\frac{1}{2}}^n=F_{\frac{1}{2}}^n=\left\{ \begin{array}{cc} g_{\frac{1}{2}}^n u_0^{n+1},&{}\text {if}\ g_{\frac{1}{2}}^n\ge 0\\ g_{\frac{1}{2}}^n u_1^{n+1},&{}\text {if}\ g_{\frac{1}{2}}^n< 0. \end{array}\right. \end{aligned}$$