Multiple Hopf Bifurcations of Four Coupled van der Pol Oscillators with Delay

Abstract

In this paper, a system of four coupled van der Pol oscillators with delay is studied. Firstly, the conditions for the existence of multiple periodic solutions of the system are given. Secondly, the multiple periodic solutions of spatiotemporal patterns of the system are obtained by using symmetric Hopf bifurcation theory. The normal form of the system on the central manifold and the bifurcation direction of the bifurcating periodic solutions are derived. Finally, numerical simulations are attached to demonstrate our theoretical results.

1. Introduction

Recently, the dynamical behaviors of coupled oscillators have become a hot research topic, receiving a lot of attention. More and more research results show that the coupled oscillators can describe the dynamical behaviors in physics, chemistry, biology and other disciplines [1,2,3,4]. The van der Pol oscillator is an important nonlinear oscillator that has widely been used in various fields of natural science [5,6]. Many scholars pay attention to the dynamical behaviors caused by the interaction between coupled van der Pol oscillators, such as stability, instability and periodicity. For example, in Ref. [7], the circadian rhythm of eyes was studied by using three coupled van der Pol oscillators. In Ref. [8], the control of the dynamics of the van der Pol oscillator coupled to a linear circuit was investigated. In Ref. [9], the occurrence of explosion death was studied by using coupled van der Pol oscillators. Batool et al. [10] studied the limit cycle behavior of the van der Pol model.

Due to the existence of delay for signal transmission and chemical reaction, time delay is inevitable in a coupled system. In addition, time delay is ubiquitous in the coupled system and has an important impact on the dynamical behavior of the system. For example, in Ref. [11], the effect of synchronization of the delayed van der Pol oscillator was discussed. In Refs. [12,13], some dynamical behaviors of delayed van der Pol oscillator were verified.

The symmetric system is also one of the research hot spots. In general, symmetry describes some kind of spatial invariance of the system. In Refs. [14,15], the Hopf bifurcation of three symmetric coupled van der Pol oscillators with delay was analyzed. In the artificial intelligence field, the movement of robots is controlled by imitating the rhythmic movement of animals; for example, in Ref. [16], it is shown that CPG can be controlled by a two-mode van der Pol oscillator under certain conditions.

The van der Pol oscillators are often used in biological modeling as follows:

$$\left\{\begin{array}{c}\dot{x}=y,\hfill \\ \dot{y}=\alpha (p^2-x^2)\dot{x}-w^{2}x,\hfill \end{array}\right.$$

where x is the output signal from the oscillator; parameters $\alpha$, p and w are constants that can influence the characteristics of oscillators. Commonly, the shape of the wave is affected by parameter $\alpha$, and the amplitude of an output depends on parameter p mostly. The output frequency mainly depends on parameter w when amplitude parameter p is fixed. But the alteration of parameter p can lightly change the frequency of the signal, and $\alpha$ also can affect the output frequency.

Letting $w^2=1$, we study the following four coupled van der Pol oscillator systems with time delay:

$${\ddot{x}}_i=\alpha (p^2-x_i^2){\dot{x}}_i-x_i+a{\dot{x}}_i(t-\tau )+b{\dot{x}}_{i+1}(t-\tau )+c{\dot{x}}_{i+2}(t-\tau )+b{\dot{x}}_{i+3}(t-\tau ),$$

(1)

where $i=1,2,3,4\left(\mathrm{mod}4\right)$, and $a,b,c$ and $\tau \ge 0$ are constants. System (1) is symmetrical, and the classical Hopf bifurcation theory cannot be applied since symmetrical systems usually have multiple eigenvalues.

The content of this paper is as follows. In Section 2, the conditions for the occurrence of multiple Hopf bifurcations of system (1) are given. In Section 3, the spatiotemporal patterns of the multiple periodic solutions of the system are obtained by using the results of the symmetric system in Ref. [17]. In Section 4, the normal form of the system on the central manifold is obtained by using the normal form theory in Ref. [18]. Finally, some numerical simulations are carried out to verify the theoretical results.

2. Existence of Equivariant Hopf Bifurcation

Let $\dot{x_i}=y_i,X_i=\left(\begin{array}{c}{x}_i\\ y_i\end{array}\right)\in R^2$ $(i=1,2,3,4(\mathrm{mod}4\left)\right)$. Then, system (1) can be rewritten as

$$\dot{X_i}=D_1{X}_i\left(t\right)+A_1{X}_i(t-\tau )+B_1{X}_{i+1}(t-\tau )+C_1{X}_{i+2}(t-\tau )+B_1{X}_{i+3}(t-\tau )+g\left(X_i\left(t\right)\right),$$

(2)

where

$$D_1=\left(\begin{array}{cc}0& 1\\ -1& \alpha p^2\end{array}\right),A_1=\left(\begin{array}{cc}0& 0\\ 0& a\end{array}\right),B_1=\left(\begin{array}{cc}0& 0\\ 0& b\end{array}\right),$$

$$C_1=\left(\begin{array}{cc}0& 0\\ 0& c\end{array}\right),g\left(\begin{array}{c}{x}_i\\ y_i\end{array}\right)=\left(\begin{array}{c}0\\ -\alpha x_i^2{y}_i\end{array}\right).$$

It is clear that the origin (0,0,0,0,0,0,0,0) is an equilibrium of Equation (2). The linearization of Equation (2) at the origin is

$$\dot{X_i}=D_1{X}_i\left(t\right)+A_1{X}_i(t-\tau )+B_1{X}_{i+1}(t-\tau )+C_1{X}_{i+2}(t-\tau )+B_1{X}_{i+3}(t-\tau ),i=1,2,3,4\left(\mathrm{mod}4\right).$$

(3)

The characteristic matrix $\Delta \left(\lambda \right)$ of the system (3) is given by

$$\begin{array}{c}\hfill \Delta \left(\lambda \right)=\left(\begin{array}{cccc}\lambda I_2-D_1-A_1{e}^{-\lambda \tau }& -B_1{e}^{-\lambda \tau }& -C_1{e}^{-\lambda \tau }& -B_1{e}^{-\lambda \tau }\\ -B_1{e}^{-\lambda \tau }& \lambda I_2-D_1-A_1{e}^{-\lambda \tau }& -B_1{e}^{-\lambda \tau }& -C_1{e}^{-\lambda \tau }\\ -C_1{e}^{-\lambda \tau }& -B_1{e}^{-\lambda \tau }& \lambda I_2-D_1-A_1{e}^{-\lambda \tau }& -B_1{e}^{-\lambda \tau }\\ -B_1{e}^{-\lambda \tau }& -C_1{e}^{-\lambda \tau }& -B_1{e}^{-\lambda \tau }& \lambda I_2-D_1-A_1{e}^{-\lambda \tau }\end{array}\right),\end{array}$$

where $I_2$ is a $2\times 2$ identity matrix.

Let $u_r=e^{\frac{\pi }{2}ri},f\left(u_r\right)=D_1+A_1{e}^{-\lambda \tau }+u_r{B}_1{e}^{-\lambda \tau }+u_r^2{C}_1{e}^{-\lambda \tau }+u_r^3{B}_1{e}^{-\lambda \tau },r=0,1,2,3$. When $r=1,3$, we have $\det (\lambda I_2-f\left(u_1\right))=\det (\lambda I_2-f\left(u_3\right))$, and the characteristic equation of system (3) is

$$\det \Delta \left(\lambda \right)=\prod _{r=0}^3\det (\lambda I_2-f\left(u_r\right))={\Delta }_1{\Delta }_2{\Delta }_3^2=0,$$

(4)

where ${\Delta }_1={\lambda }^2-\lambda (\alpha p^2+(a+2b+c)e^{-\lambda \tau })+1,{\Delta }_2={\lambda }^2-\lambda (\alpha p^2+(a-2b+c)e^{-\lambda \tau })+1$, ${\Delta }_3={\lambda }^2-\lambda (\alpha p^2+(a-c)e^{-\lambda \tau })+1$. Obviously, $\lambda =0$ is not the root of Equation (4).

We make the following assumptions:

Hypothesis 1

(H1). $\alpha p^2+(a+2b+c)<0$

Hypothesis 2

(H2). $\alpha p^2+(a-2b+c)<0$

Hypothesis 3

(H3). $\alpha p^2+(a-c)<0$

Hypothesis 4

(H4). ${\alpha }^2{p}^4-{(a+2b+c)}^2-2>0$

Hypothesis 5

(H5). ${\alpha }^2{p}^4-{(a-2b+c)}^2-2>0$

Hypothesis 6

(H6). ${\alpha }^2{p}^4-{(a-c)}^2<0$

Lemma 1.

If H1 and H4 are satisfied, then when $\tau \ge 0$, all roots of the equation ${\Delta }_1=0$ have negative real parts.

Proof. 

When $\tau =0$, the equation ${\Delta }_1=0$ becomes ${\lambda }^2-\lambda (\alpha p^2+(a+2b+c))+1=0$, and the solution is obtained as follows:

$$\lambda =\frac{(\alpha p^2+(a+2b+c))\pm \sqrt{{(\alpha p^2+(a+2b+c))}^2-4}}{2}.$$

By H1, the roots of the equation ${\Delta }_1=0$ have negative real parts. When $\tau >0$, let $\lambda =i\omega (\omega >0)$ be the root of the equation ${\Delta }_1=0$, and then we have

$$-{\omega }^2-i\omega \left(\alpha p^2\right)-\left(i\omega \right)(a+2b+c)(cos\omega \tau -isin\omega \tau )+1=0.$$

Separating the real and imaginary parts, we have

$$\left\{\begin{array}{c}1-{\omega }^2=\omega (a+2b+c)sin\left(\omega \tau \right),\\ -\omega \alpha p^2=\omega (a+2b+c)cos\left(\omega \tau \right),\end{array}\right.$$

which implies

$${\omega }^4+{\omega }^2({\alpha }^2{p}^4-{(a+2b+c)}^2-2)+1=0,$$

and

$${\omega }^2=\frac{-({\alpha }^2{p}^4-{(a+2b+c)}^2-2)\pm \sqrt{{({\alpha }^2{p}^4-{(a+2b+c)}^2-2)}^2-4}}{2}.$$

If H4 holds, the above formula does not hold. So the lemma holds. □

A similar lemma is as follows.

Lemma 2.

If H2 and H5 are satisfied, then when $\tau \ge 0$, all roots of the equation ${\Delta }_2=0$ have negative real parts.

Now we consider the equation as follows:

$${\Delta }_3^2={[{\lambda }^2-\lambda (\alpha p^2+(a-c)e^{-\lambda \tau })+1]}^2=0.$$

When $\tau =0$, the equation ${\Delta }_3=0$ becomes ${\lambda }^2-\lambda (\alpha p^2+(a-c))+1=0$, and the solution is obtained as follows:

$$\lambda =\frac{(\alpha p^2+(a-c))\pm \sqrt{{(\alpha p^2+(a-c))}^2-4}}{2}.$$

Let $\lambda =i\beta (\beta >0)$ be a root of ${\Delta }_3^2=0$. Substituting $i\beta$ into ${\Delta }_3^2=0$, we have

$$-{\beta }^2-i\beta (\alpha p^2+(a-c)e^{-i\beta \tau })+1=0.$$

Separating the real and imaginary parts of the above equation, we obtain

$$\left\{\begin{array}{c}1-{\beta }^2=\beta (a-c)sin\left(\beta \tau \right),\\ -\alpha p^2\beta =\beta (a-c)cos\left(\beta \tau \right).\end{array}\right.$$

Solving the above equation, we have

$${\beta }^{\pm }=\sqrt{\frac{-({\alpha }^2{p}^4-2-{(a-c)}^2)\pm \sqrt{{({\alpha }^2{p}^4-2-{(a-c)}^2)}^2-4}}{2}}.$$

From the above analysis, we have the following lemma.

Lemma 3.

For the equation ${\Delta }_3^2=0$, we have the following results:

(1) 

If H3 holds, then all roots of the equation ${\Delta }_3^2=0$ have negative real parts when $\tau =0$.

(2) 

If H6 holds, then there exist ${\tau }_k^{\pm }$ such that ${\Delta }_3^2(\pm i{\beta }^{\pm })=0$ holds when $\tau ={\tau }_k^{\pm }(k=0,1,2,\dots )$, where

$${\tau }_k^{\pm }=\left\{\begin{array}{c}(1/{\beta }^{\pm })(arccos\left(\frac{-\alpha p^2}{a-c}\right)+2k\pi )a-c<0,\\ (1/{\beta }^{\pm })(2\pi -arccos\left(\frac{-\alpha p^2}{a-c}\right)+2k\pi )a-c>0,\end{array}\right.k=0,1,2\dots .$$

From Equation (4), we know $i{\beta }^{\pm }$ is the dual pure imaginary root of Equation (4) when $b\ne c$. Let $\lambda \left(\tau \right)=\alpha \left(\tau \right)+i\beta \left(\tau \right)$ be the root of Equation (4), satisfying $\alpha \left({\tau }_k^{\pm }\right)=0$ and $\beta \left({\tau }_k^{\pm }\right)={\beta }^{\pm }$. Taking the derivative of the equation ${\Delta }_3=0$ with respect to $\tau$, we can obtain

$${\left(\frac{d\lambda }{d\tau }\right)}^{-1}=\frac{2\lambda -\alpha p^2}{-{\lambda }^2(a-c)e^{-\lambda \tau }}+\frac{1}{{\lambda }^2}-\frac{1}{\lambda }\tau ,$$

when $\lambda =i{\beta }^{\pm }$ and $\tau ={\tau }_k^{\pm }$, we have

$$\begin{gathered} Re{\left(\frac{d\lambda }{d\tau }\right)}^{-1}{|}_{\lambda =i{\beta }^{\pm },\tau ={\tau }_k^{\pm }}=\frac{{\left({\beta }^{\pm }\right)}^2({\left({\beta }^{\pm }\right)}^2+1)({\left({\beta }^{\pm }\right)}^2-1)}{{({\left({\beta }^{\pm }\right)}^4-{\left({\beta }^{\pm }\right)}^2)}^2+{\left({\left({\beta }^{\pm }\right)}^3\alpha p^2\right)}^2}, \\ Re{\left(\frac{d\lambda }{d\tau }\right)}^{-1}{{|}_{\lambda =i{\beta }^{+},\tau ={\tau }_k^{+}}>0,Re{\left(\frac{d\lambda }{d\tau }\right)}^{-1}|}_{\lambda =i{\beta }^{-},\tau ={\tau }_k^{-}}<0, \end{gathered}$$

(5)

which means that the transversality condition is satisfied at ${\tau }_k^{\pm }(k=0,1,2,\dots )$.

Let $C=C([-\tau ,0],R^8)$, $z_t\in C,z_t\left(\theta \right)=z(t+\theta )$ for $-\tau \le \theta \le 0$, then Equation (2) can be rewritten as

$$\dot{z}=L{z}_t+F{z}_t,$$

(6)

where

$$L\phi =[\left(\begin{array}{cccc}{D}_1& 0& 0& 0\\ 0& D_1& 0& 0\\ 0& 0& D_1& 0\\ 0& 0& 0& D_1\end{array}\right)\phi \left(0\right)+\left(\begin{array}{cccc}{A}_1& B_1& C_1& B_1\\ B_1& A_1& B_1& C_1\\ C_1& B_1& A_1& B_1\\ B_1& C_1& B_1& A_1\end{array}\right)\phi (-\tau )],$$

$$F\phi ={\left(\begin{array}{cccccccc}0,& -\alpha {\phi }_1^2{\phi }_2,& 0,& -\alpha {\phi }_3^2{\phi }_4,& 0,& -\alpha {\phi }_5^2{\phi }_6,& 0,& -\alpha {\phi }_7^2{\phi }_8\end{array}\right)}^T,$$

$$\phi ={({\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4,{\phi }_5,{\phi }_6,{\phi }_7,{\phi }_8)}^T\in C.$$

Compact Lie groups can be considered as follows: the cycle group $Z_4=<\rho >$ of order 4, and the dihedral group $D_4$ of order 8, which is generated by $Z_4$ together with the flip k of order 2. $D_4$ acts on $R^8$ by

$${\left(\rho M\right)}_i=M_{i+1},$$

$${\left(kM\right)}_i=M_{6-i},$$

where $M_i\in R^2,i\left(\mathrm{mod}4\right)$. Then, we have $L\left(\rho \phi \right)=\rho L\left(\phi \right),F\left(\rho \phi \right)=\rho F\left(\phi \right),L\left(k\phi \right)=kL\left(\phi \right),F\left(k\phi \right)=kF\left(\phi \right)$. So, we obtain the following lemma.

Lemma 4.

System (2) is $D_4$ equivariant.

By Lemmas 1–4 and (5), the following theorem is obtained.

Theorem 1.

Suppose H1–H6 are satisfied.

(1) 

All roots of Equation (4) have negative real parts for $0\le \tau <{\tau }_0^{\pm }$ and at least a pair of roots with positive real parts for $\tau >{\tau }_0^{\pm }$.

(2) 

Zero equilibrium of system (2) is asymptotically stable for $0\le \tau <{\tau }_0^{\pm }$ and unstable for $\tau >{\tau }_0^{\pm }$.

(3) 

If $b\ne \pm c$, system (2) exhibits an equivariant Hopf bifurcation at $\tau ={\tau }_k^{\pm }(k=0,1,2\dots )$.

3. Spatiotemporal Patterns of Bifurcating Periodic Solutions

In this section, we study the periodic solution when $\beta ={\beta }^{+}$. The case of $\beta ={\beta }^{-}$ can be considered similarly. The infinitesimal generator $A\left(\tau \right)$ of the $C_0$-semigroup generated by the linear system (3) has a pair of purely imaginary eigenvalues $\pm i{\beta }^{+}$ at the critical value ${\tau }_k^{+}(k=0,1,2\dots )$. The corresponding eigenvectors can be chosen as

$$v_1\left(\theta \right)=V_1{e}^{i{\beta }^{+}\theta },v_2\left(\theta \right)=V_2{e}^{i{\beta }^{+}\theta },$$

where $V_1=\left(\begin{array}{c}q\\ iq\\ -q\\ -iq\end{array}\right),V_2=\left(\begin{array}{c}q\\ -iq\\ -q\\ iq\end{array}\right),q=\left(\begin{array}{c}1\\ i{\beta }^{+}\end{array}\right).$

Denoting the action of $\Gamma =D_4$ on $R^2$ by

$$\rho \left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{c}-x_2\\ x_1\end{array}\right),k\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)=\left(\begin{array}{c}{x}_1\\ -x_2\end{array}\right),$$

where $\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)\in R^2$.

Lemma 5.

$R^2$ is an absolutely irreducible representation of Γ, and $\mathrm{Ker}\Delta ({\tau }_k^{+},i{\beta }^{+})$ is isomorphic to $R^2\oplus R^2$.

Proof. 

It can be easily proved that $R^2$ is an absolutely irreducible representation of $\Gamma$.

Let $\mathrm{Ker}\Delta ({\tau }_k^{+},i{\beta }^{+})=\{(c_1+d_{1}i)V_1+(c_2+d_{2}i)V_2|c_1,c_2,d_1,d_2\in R\}$. Defining a mapping J from $\mathrm{Ker}\Delta ({\tau }_k^{+},i{\beta }^{+})$ to $R^4$ by

$$J((c_1+d_{1}i)V_1+(c_2+d_{2}i)V_2)={(c_1+c_2,d_1-d_2,d_1+d_2,c_2-c_1)}^T.$$

Clearly, J is a linear isomorphism from $\mathrm{Ker}\Delta ({\tau }_k^{+},i{\beta }^{+})$ to $R^4$. Note that

$$\begin{array}{cc}\hfill \rho ((c_1+d_{1}i)V_1+(c_2+d_{2}i)V_2)& =(c_1+d_{1}i)\rho \left(V_1\right)+(c_2+d_{2}i)\rho \left(V_2\right)\hfill \\ & =(c_1+d_{1}i)e^{\frac{\pi }{2}i}{V}_1+(c_2+d_{2}i)e^{-\frac{\pi }{2}i}{V}_2\hfill \\ & =(-d_1+c_{1}i)V_1+(d_2-c_{2}i)V_2,\hfill \\ \hfill k((c_1+d_{1}i)V_1+(c_2+d_{2}i)V_2)& =(c_1+d_{1}i)V_2+(c_2+d_{2}i)V_1.\hfill \end{array}$$

Then

$$\begin{array}{cc}\hfill J\left(\rho ((c_1+d_{1}i)V_1+(c_2+d_{2}i)V_2)\right)& =J((-d_1+c_{1}i)V_1+(d_2-c_{2}i)V_2)\hfill \\ & ={(d_2-d_1,c_1+c_2,c_1-c_2,d_2+d_1)}^T\hfill \\ & =\rho \left(J((c_1+d_{1}i)V_1+(c_2+d_{2}i)V_2)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill J\left(k((c_1+d_{1}i)V_1+(c_2+d_{2}i)V_2)\right)& =J((c_1+d_{1}i)V_2+(c_2+d_{2}i)V_1)\hfill \\ & ={(c_2+c_1,d_2-d_1,d_2+d_1,c_1-c_2)}^T\hfill \\ & =k\left(J((c_1+d_{1}i)V_1+(c_2+d_{2}i)V_2)\right).\hfill \end{array}$$

This completes the proof. □

Lemma 6.

The generalized eigenspace

$$U_{\pm i{\beta }^{+}}=\left\{\sum _{j=1}^4{y}_j{\epsilon }_j,j=1,2,3,4.y_j\in R\right\},$$

where

$$\begin{array}{c}{\epsilon }_1=cos\left({\beta }^{+}t\right)\mathrm{Re}{\mathrm{V}}_1-sin\left({\beta }^{+}\mathrm{t}\right){\mathrm{ImV}}_1,\\ {\epsilon }_2=sin\left({\beta }^{+}t\right)\mathrm{Re}{\mathrm{V}}_1+cos\left({\beta }^{+}\mathrm{t}\right){\mathrm{ImV}}_1,\\ {\epsilon }_3=cos\left({\beta }^{+}t\right)\mathrm{Re}{\mathrm{V}}_2-sin\left({\beta }^{+}\mathrm{t}\right){\mathrm{ImV}}_2,\\ {\epsilon }_4=sin\left({\beta }^{+}t\right)\mathrm{Re}{\mathrm{V}}_2+cos\left({\beta }^{+}\mathrm{t}\right){\mathrm{ImV}}_2.\end{array}$$

It is easy to obtain $k\left({\mathrm{ReV}}_1\right)={\mathrm{ReV}}_2,$ $k\left({\mathrm{ReV}}_2\right)={\mathrm{ReV}}_1,$ $k\left({\mathrm{ImV}}_1\right)={\mathrm{ImV}}_2,$ $k\left(Im{V}_2\right)=Im{V}_1,$ $\rho \left({\mathrm{ReV}}_1\right)=-{\mathrm{ImV}}_1,$ $\rho \left({\mathrm{ReV}}_2\right)={\mathrm{ImV}}_2,$ $\rho \left({\mathrm{ImV}}_1\right)={\mathrm{ReV}}_1,$ $\rho \left({\mathrm{ImV}}_2\right)=-{\mathrm{ReV}}_2.$

Let $T=\frac{2\pi }{{\beta }^{+}}$ and denote $P_T$ as the Banach space of all continuous $T$-periodic functions $x.$ The role of $D_4\times S^1$ on $P_T$ is

$$(\gamma ,\theta )x\left(t\right)=\gamma x(t+\theta ),(\gamma ,\theta )\in D_4\times S^1,x\in P_T,$$

where $S^1=<e^{i\theta }>$ is the temporal.

Denoting $S{P}_T$ as the subspace of $P_T$ consists of all T-periodic solutions of system (2) with $\tau ={\tau }_k^{+}$. For each subgroup $\sum \subset D_4\times S^1$, $Fix(\sum ,S{P}_T)=\{x\in S{P}_T,(\gamma ,\theta )x=x,forall(\gamma ,\theta )\in \sum \}$ is a subspace.

For the following subgroups of $D_4\times S^1$,

$$\begin{array}{c}\sum _1=\langle (k,1)\rangle ,\sum _2=\langle (k,-1)\rangle ,\\ \sum _3=\langle (\rho ,e^{i\frac{T}{4}})\rangle ,\sum _4=\langle (\rho ,e^{-i\frac{T}{4}})\rangle .\end{array}$$

We have the following lemma.

Lemma 7.

$\mathrm{Fix}({\sum }_j,S{P}_T)(j=1,2,3,4.)$ are all the two-dimensional subspaces of $S{P}_T,$ where

    $\mathrm{Fix}({\sum }_1,S{P}_T)=\left\{y_1({\epsilon }_1+{\epsilon }_3)+y_2({\epsilon }_2+{\epsilon }_4)|y_1,y_2\in R\right\}$,

    $\mathrm{Fix}({\sum }_2,S{P}_T)=\left\{y_1({\epsilon }_1-{\epsilon }_3)+y_2({\epsilon }_2-{\epsilon }_4)|y_1,y_2\in R\right\}$,

    $\mathrm{Fix}({\sum }_3,S{P}_T)=\left\{y_1{\epsilon }_3+y_2{\epsilon }_4|y_1,y_2\in R\right\}$,

    $\mathrm{Fix}({\sum }_4,S{P}_T)=\left\{y_1{\epsilon }_1+y_2{\epsilon }_2|y_1,y_2\in R\right\}$.

Proof. 

Let $x=x_1{\epsilon }_1+x_2{\epsilon }_2+x_3{\epsilon }_3+x_4{\epsilon }_4$.

(i)

It is necessary for us to prove $x\in \mathrm{Fix}({\sum }_1,S{P}_T)$ if and only if $kx=x$. By Lemma 6, we have

$$kx=k(x_1{\epsilon }_1+x_2{\epsilon }_2+x_3{\epsilon }_3+x_4{\epsilon }_4)=x_1{\epsilon }_3+x_2{\epsilon }_4+x_3{\epsilon }_1+x_4{\epsilon }_2.$$

Hence, $kx=x$ if and only if $x_1=x_3$ and $x_2=x_4$. This implies that $\mathrm{Fix}({\sum }_1,S{P}_T)$ is spanned by ${\epsilon }_1+{\epsilon }_3$ and ${\epsilon }_2+{\epsilon }_4$.

(ii)

It is enough to prove that $x\in \mathrm{Fix}({\sum }_2,S{P}_T)$ if and only if $kx\left(t\right)=x(t+\frac{T}{2})$. It is clearly that

$$x(t+\frac{T}{2})=x_1{\epsilon }_1(t+\frac{T}{2})+x_2{\epsilon }_2(t+\frac{T}{2})+x_3{\epsilon }_3(t+\frac{T}{2})+x_4{\epsilon }_4(t+\frac{T}{2}),$$

where

$$\begin{array}{cc}\hfill {\epsilon }_1(t+\frac{T}{2})& =cos\left({\beta }^{+}(t+\frac{T}{2})\right){\mathrm{ReV}}_1-sin\left({\beta }^{+}(\mathrm{t}+\frac{\mathrm{T}}{2})\right){\mathrm{ImV}}_1\hfill \\ & =-cos\left({\beta }^{+}t\right){\mathrm{ReV}}_1+sin\left({\beta }^{+}\mathrm{t}\right){\mathrm{ImV}}_1\hfill \\ & =-{\epsilon }_1.\hfill \end{array}$$

By the same method,

$${\epsilon }_2(t+\frac{T}{2})=-{\epsilon }_2,{\epsilon }_3(t+\frac{T}{2})=-{\epsilon }_3,{\epsilon }_4(t+\frac{T}{2})=-{\epsilon }_4.$$

So

$$x(t+\frac{T}{2})=-x_1{\epsilon }_1-x_2{\epsilon }_2-x_3{\epsilon }_3-x_4{\epsilon }_4.$$

Then we have $kx\left(t\right)=x(t+\frac{T}{2})$ if and only if $-x_1=x_3,x_2=-x_4$, and $\mathrm{Fix}({\sum }_2,S{P}_T)$ is the span of ${\epsilon }_1-{\epsilon }_3$ and ${\epsilon }_2-{\epsilon }_4$.

(iii)

It is sufficient to prove that $x\in \mathrm{Fix}({\sum }_3,S{P}_T)$ if and only if $\rho x\left(t\right)=x(t-\frac{T}{4})$. By Lemma 6, we can obtain

$$\rho x=\rho (x_1{\epsilon }_1+x_2{\epsilon }_2+x_3{\epsilon }_3+x_4{\epsilon }_4)=-x_1{\epsilon }_2+x_2{\epsilon }_1+x_3{\epsilon }_4-x_4{\epsilon }_3$$

and

$$\begin{array}{cc}\hfill x(t-\frac{T}{4})& =x_1{\epsilon }_1(t-\frac{T}{4})+x_2{\epsilon }_2(t-\frac{T}{4})+x_3{\epsilon }_3(t-\frac{T}{4})+x_4{\epsilon }_4(t-\frac{T}{4})\hfill \\ & =x_1(cos\left({\beta }^{+}(t-\frac{T}{4})\right)\mathrm{Re}{V}_1-sin\left({\beta }^{+}(t-\frac{T}{4})\right)\mathrm{Im}{V}_1)+x_2(sin\left({\beta }^{+}(t-\frac{T}{4})\right)\mathrm{Re}{V}_1\hfill \\ & +cos\left({\beta }^{+}(t-\frac{T}{4})\right)\mathrm{Im}{V}_1)+x_3(cos\left({\beta }^{+}(t-\frac{T}{4})\right)\mathrm{Re}{V}_2-sin\left({\beta }^{+}(t-\frac{T}{4})\right)\mathrm{Im}{V}_2)\hfill \\ & +x_4(sin\left({\beta }^{+}(t-\frac{T}{4})\right)\mathrm{Re}{V}_2+cos\left({\beta }^{+}(t-\frac{T}{4})\right)\mathrm{Im}{V}_2),\hfill \end{array}$$

then

$$cos\left({\beta }^{+}(t-\frac{T}{4})\right)=sin\left({\beta }^{+}t\right),sin\left({\beta }^{+}(t-\frac{T}{4})\right)=-cos\left({\beta }^{+}t\right).$$

Thus,

$$x(t-\frac{T}{4})=x_1{\epsilon }_2-x_2{\epsilon }_1+x_3{\epsilon }_4-x_4{\epsilon }_3.$$

It can be obtained that $\rho x\left(t\right)=x(t-\frac{\pi }{4})$ if and only if $x_1=x_2=0,$ such that $\mathrm{Fix}({\sum }_3,S{P}_T)$ is the span of ${\epsilon }_3$ and ${\epsilon }_4$.

(iv)

It is adequate to prove that $x\in \mathrm{Fix}({\sum }_4,S{P}_T)$ if and only if $\rho x(t-\frac{T}{4})=x\left(t\right)$. Similarly, we have

$$\rho x(t-\frac{T}{4})=\rho (x_1{\epsilon }_2-x_2{\epsilon }_1+x_3{\epsilon }_4-x_4{\epsilon }_3)=x_1{\epsilon }_1+x_2{\epsilon }_2-x_3{\epsilon }_3-x_4{\epsilon }_4.$$

Therefore, $\rho x(t-\frac{T}{4})=x\left(t\right)$ if and only if $x_3=x_4=0$, and $\mathrm{Fix}({\sum }_4,S{P}_T)$ is the span of ${\epsilon }_1$ and ${\epsilon }_2$.

□

By Lemmas 5–7 and Theorem 4.1 of Ref. [17], we can obtain the following theorem.

Theorem 2.

There exist three branches of small-amplitude periodic solutions with period T near $\frac{2\pi }{{\beta }^{+}}$ for Equation (2), bifurcated from the zero equilibrium at the critical values ${\tau }_k^{+}(i=0,1,2\dots )$, and they are:

(i) 

Discrete waves: $x_i\left(t\right)=x_{i+1}(t\pm \frac{T}{4})$ for $i\left(mod4\right)$ and $t\in R$;

(ii) 

Standing waves: $x_i\left(t\right)=x_{4+2j-i}(t+\frac{T}{2})$ for $i\left(mod4\right),j\in \{1,2,3,4\}$ and $t\in R$;

(iii) 

Mirror-reflecting solutions: $x_i\left(t\right)=x_{4+2j-i}\left(t\right)$ for $i\left(mod4\right),j\in \{1,2,3,4\}$ and $t\in R$.

4. Normal Form on Center Manifolds and Bifurcation Direction of Bifurcating Periodic Solutions

In this section, we employ the algorithm and notations of Ref. [18] to derive the normal forms of the Hopf bifurcation of system (2) on the center manifold and the bifurcation direction of the bifurcating periodic solutions.

Let $t\mapsto \frac{t}{\tau }$, then Equation (6) can be rewritten as

$$\dot{z}=L\left(\tau \right)z_t+F(z_t,\tau ),$$

(7)

where

$$L\left(\tau \right)\phi =\tau [\left(\begin{array}{cccc}{D}_1& 0& 0& 0\\ 0& D_1& 0& 0\\ 0& 0& D_1& 0\\ 0& 0& 0& D_1\end{array}\right)\phi \left(0\right)+\left(\begin{array}{cccc}{A}_1& B_1& C_1& B_1\\ B_1& A_1& B_1& C_1\\ C_1& B_1& A_1& B_1\\ B_1& C_1& B_1& A_1\end{array}\right)\phi (-1)],$$

$$F(\phi ,\tau )=\tau {\left(\begin{array}{cccccccc}0,& -\alpha {\phi }_1^2{\phi }_2,& 0,& -\alpha {\phi }_3^2{\phi }_4,& 0,& -\alpha {\phi }_5^2{\phi }_6,& 0,& -\alpha {\phi }_7^2{\phi }_8\end{array}\right)}^T,$$

$$\phi ={({\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4,{\phi }_5,{\phi }_6,{\phi }_7,{\phi }_8)}^T\in C.$$

Let $\mu =\tau -{\tau }_k^{+}$, then

$$\dot{z}=L\left(0\right)z_t+G(z_t,\mu ),$$

(8)

where $G(z_t,\mu )=L\left(\mu \right)z_t-L\left(0\right)z_t+F(z_t,\mu )$.

The base of the center space X at $\tau ={\tau }_k^{+}$ can be taken as $\Phi \left(\theta \right)=({\varphi }_1,{\varphi }_2,{\varphi }_3,{\varphi }_4)$, where

$$\begin{array}{c}{\varphi }_1\left(\theta \right)=e^{i{\beta }^{+}{\tau }_k^{+}\theta }{V}_1,{\varphi }_2\left(\theta \right)=e^{-i{\beta }^{+}{\tau }_k^{+}\theta }{\overline{V}}_1,\\ {\varphi }_3\left(\theta \right)=e^{i{\beta }^{+}{\tau }_k^{+}\theta }{V}_2,{\varphi }_4\left(\theta \right)=e^{-i{\beta }^{+}{\tau }_k^{+}\theta }{\overline{V}}_2,\end{array}-1\le \theta \le 0.$$

It is easy to check that the base for the adjoint space $X^{*}$ is $\Psi \left(s\right)=({\psi }_1\left(s\right),{\psi }_2\left(s\right),{\psi }_3\left(s\right),$${\psi }_4{\left(s\right))}^T,$ where

$$\begin{array}{c}{\psi }_1\left(s\right)=\overline{D}{e}^{-i{\beta }^{+}{\tau }_k^{+}s}{\overline{V}}_1^T,{\psi }_2\left(s\right)=D{e}^{i{\beta }^{+}{\tau }_k^{+}s}{V}_1^T,\\ {\psi }_3\left(s\right)=\overline{D}{e}^{-i{\beta }^{+}{\tau }_k^{+}s}{\overline{V}}_2^T,{\psi }_4\left(s\right)=D{e}^{i{\beta }^{+}{\tau }_k^{+}s}{V}_2^T,\end{array}0\le s\le 1,$$

and $\langle \Psi ,\Phi \rangle =I_4$ for the adjoint bilinear form on $C^{*}\times C$ as

$$\langle \psi ,\varphi \rangle =\psi \left(0\right)\varphi \left(0\right)-{\int }_{-1}^0{\int }_0^{\theta }\psi (\xi -\theta )d\eta (\theta ,0)\varphi \left(\xi \right)d\xi ,$$

we have

$$D=\frac{1}{4}\frac{1}{1+{\left({\beta }^{+}\right)}^2-e^{i{\beta }^{+}{\tau }_k^{+}}{\left({\beta }^{+}\right)}^2(a-c)}.$$

For $x\in C^4$, we have

$$\Phi \left(0\right)x=(V_1,\overline{V_1},V_2,\overline{V_2})x=x_1{V}_1+x_2{\overline{V}}_1+x_3{V}_2+x_4{\overline{V}}_2,$$

$$\Phi (-1)x=x_1{e}^{-i{\beta }^{+}{\tau }_k^{+}}{V}_1+x_2{e}^{i{\beta }^{+}{\tau }_k^{+}}{\overline{V}}_1+x_3{e}^{-i{\beta }^{+}{\tau }_k^{+}}{V}_2+x_4{e}^{i{\beta }^{+}{\tau }_k^{+}}{\overline{V}}_2,$$

$$\Psi \left(0\right)={(\overline{D}{\overline{V}}_1^T,D{V}_1^T,\overline{D}{\overline{V}}_2^T,D{V}_2^T)}^T.$$

Define a $4\times 4$ matrix

$$\tilde{B}=\mathrm{diag}(i{\beta }^{+}{\tau }_k^{+},-i{\beta }^{+}{\tau }_k^{+},i{\beta }^{+}{\tau }_k^{+},-i{\beta }^{+}{\tau }_k^{+}).$$

Using the decomposition of $z_t=\Phi x+y$, system (8) can be decomposed as

$$\left\{\begin{array}{c}\dot{x}=\tilde{B}x+\Psi \left(0\right)G(\Phi x+y,\mu ),\hfill \\ \dot{y}=A_{Q_1}y+(I-\pi )X_{0}G(\Phi x+y,\mu ),\hfill \end{array}\right.$$

where $x\in C^4,y\in Q_1$. We have the Taylor expansion as follows:

$$\dot{x}=\tilde{B}x+\sum _{j\ge 2}\frac{1}{j!}{f}_j^1(x,y,\mu ),$$

where $f_j^1(x,y,\mu )$ are homogeneous polynomials of degree j in $(x,y,\mu )$ with coefficients in $C^4$. Then, the normal form of (8) on the center manifold at the origin as for $\mu =0$ is given by

$$\dot{x}=\tilde{B}x+\frac{1}{2!}{g}_2^1(x,0,\mu )+\frac{1}{3!}{g}_3^1(x,0,\mu )+h.o.t,$$

(9)

where $g_2^1,g_3^1$ is calculated in the following:

$$\begin{gathered} \begin{array}{cc}\hfill \frac{1}{2!}{f}_2^1(x,0,\mu )& =\Psi \left(0\right)L\left(\mu \right)(\Phi x)\hfill \\ \hfill & =\Psi \left(0\right)\mu [\left(\begin{array}{cccc}{D}_1& 0& 0& 0\\ 0& D_1& 0& 0\\ 0& 0& D_1& 0\\ 0& 0& 0& D_1\end{array}\right)\Phi \left(0\right)x+\left(\begin{array}{cccc}{A}_1& B_1& C_1& B_1\\ B_1& A_1& B_1& C_1\\ C_1& B_1& A_1& B_1\\ B_1& C_1& B_1& A_1\end{array}\right)\Phi (-1)x]\hfill \end{array} \\ =4\mu \left(\begin{array}{c}\overline{D}((m_1+n_1)x_1+(m_2+n_2)x_4)\\ D(({\overline{m}}_1+{\overline{n}}_1)x_2+({\overline{m}}_2+{\overline{n}}_2)x_3)\\ \overline{D}((m_1+n_1)x_3+(m_2+n_2)x_2)\\ D(({\overline{m}}_1+{\overline{n}}_1)x_4+({\overline{m}}_2+{\overline{n}}_2)x_1)\end{array}\right), \end{gathered}$$

(10)

where

$$m_1=\alpha p^2{\left({\beta }^{+}\right)}^2+2i{\beta }^{+},n_1={\left({\beta }^{+}\right)}^2(a-c)e^{-i{\beta }^{+}{\tau }_k^{+}},$$

$$m_2=-\alpha p^2{\left({\beta }^{+}\right)}^2,n_2=-{\left({\beta }^{+}\right)}^2(a-c)e^{i{\beta }^{+}{\tau }_k^{+}}.$$

Since $M_j^{1}p(x,\mu )=D_{x}p(x,\mu )\tilde{B}x-\tilde{B}p(x,\mu ),j\ge 2$, and

$$\begin{array}{cc}\hfill & \mathrm{Ker}\left(M_2^1\right)\cap \mathrm{Span}\{\mu x^q{e}_l:|q|=1,l=1,2,3,4\}\hfill \\ \hfill & =\mathrm{Span}\{\mu x_1{e}_1,\mu x_3{e}_1,\mu x_2{e}_2,\mu x_4{e}_2,\mu x_1{e}_3,\mu x_3{e}_3,\mu x_2{e}_4,\mu x_4{e}_4\},\hfill \end{array}$$

where e₁, e₂, e₃, e₄ is the canonical basis for C⁴. We have

$$\frac{1}{2!}{g}_2^1(x,0,\mu )=\frac{1}{2!}{\mathrm{Proj}}_{\ker \left(M_2^1\right)}{f}_2^1(x,0,\mu )=4\left(\begin{array}{c}\overline{n}\mu x_1\\ n\mu x_2\\ \overline{n}\mu x_3\\ n\mu x_4\end{array}\right),$$

(11)

where $n=D(\overline{m_1}+\overline{n_1}).$

Then we need to compute

$$g_3^1(x,0,\mu )={\mathrm{Proj}}_{\mathrm{Ker}\left(M_3^1\right)}{\tilde{f}}_3^1(x,0,\mu )={\mathrm{Proj}}_{\mathrm{Ker}\left(M_3^1\right)}{\tilde{f}}_3^1(x,0,0)+O({\mu }^2|x|),$$

with

$${\tilde{f}}_3^1(x,0,\mu )=f_3^1(x,0,\mu )+\frac{3}{2}[\left(D_x{f}_2^1\right)U_2^1-D_x{U}_2^1{g}_2^1](x,0,\mu )+\frac{3}{2}\left[\left(D_y{f}_2^1\right)h\right](x,0,\mu ),$$

where $U_2^1$ is the change in variables associated with the transformation from $f_2^1$ to $g_2^1$, and h is such that $M_2^2\left(h\right)=g_2^2$. It is easy to see from (10) and (11) that $f_2^1(x,0,0)=g_2^1(x,0,0)=0$ for $\mu =0$. Therefore, we have $\frac{1}{3!}{\tilde{f}}_3^1(x,0,0)=\frac{1}{3!}{f}_3^1(x,0,0)$ and

$$\begin{array}{cc}\hfill \frac{1}{3!}{f}_3^1(x,0,0)& =\Psi \left(0\right){\tau }_k^{+}F(\Phi x,0)\hfill \\ & =\Psi \left(0\right){\tau }_k^{+}\left(\begin{array}{c}0\\ -\alpha {(x_1+x_3+x_2+x_4)}^2(x_1+x_3-x_2-x_4)i{\beta }^{+}\\ 0\\ \alpha {(x_1-x_3-x_2+x_4)}^2(-x_1+x_3-x_2+x_4){\beta }^{+}\\ 0\\ -\alpha {(x_1+x_3+x_2+x_4)}^2(-x_1-x_3+x_2+x_4)i{\beta }^{+}\\ 0\\ \alpha {(-x_1+x_3+x_2-x_4)}^2(x_1-x_3+x_2-x_4){\beta }^{+}\end{array}\right)\hfill \\ & =2\alpha {\tau }_k^{+}{\left({\beta }^{+}\right)}^2\left(\begin{array}{c}\overline{D}({(x_1+x_3+x_2+x_4)}^2(-x_1-x_3+x_2+x_4)+\\ {(x_1-x_3-x_2+x_4)}^2(x_1-x_3+x_2-x_4)),\\ D({(x_1+x_3+x_2+x_4)}^2(x_1+x_3-x_2-x_4)+\\ {(x_1-x_3-x_2+x_4)}^2(x_1-x_3+x_2-x_4)),\\ \overline{D}({(x_1+x_3+x_2+x_4)}^2(-x_1-x_3+x_2+x_4)+\\ {(x_1-x_3-x_2+x_4)}^2(-x_1+x_3-x_2+x_4)),\\ D({(x_1+x_3+x_2+x_4)}^2(x_1+x_3-x_2-x_4)+\\ {(x_1-x_3-x_2+x_4)}^2(-x_1+x_3-x_2+x_4)).\end{array}\right),\hfill \end{array}$$

$$\begin{array}{cc}& \mathrm{Ker}\left(M_3^1\right)\cap \mathrm{Span}\{\mu x^q{e}_l:|q|=3,l=1,2,3,4\}\hfill \\ & =\mathrm{Span}\{x_1{x}_2{x}_3{e}_1,x_1{x}_3{x}_4{e}_1,x_1^2{x}_2{e}_1,x_1^2{x}_4{e}_1,x_3^2{x}_4{e}_1,x_3^2{x}_2{e}_1,\hfill \\ & x_1{x}_2{x}_4{e}_2,x_2{x}_3{x}_4{e}_2,x_2^2{x}_1{e}_2,x_2^2{x}_3{e}_2,x_4^2{x}_1{e}_2,x_4^2{x}_3{e}_2,\hfill \\ & x_1{x}_2{x}_3{e}_3,x_1{x}_3{x}_4{e}_3,x_1^2{x}_2{e}_3,x_1^2{x}_4{e}_3,x_3^2{x}_4{e}_3,x_3^2{x}_2{e}_3,\hfill \\ & x_1{x}_2{x}_4{e}_4,x_2{x}_3{x}_4{e}_4,x_2^2{x}_1{e}_4,x_2^2{x}_3{e}_4,x_4^2{x}_1{e}_4,x_4^2{x}_3{e}_4\},\hfill \end{array}$$

then

$$\frac{1}{3!}{g}_3^1(x,0,0)=4\alpha {\tau }_k^{+}{\left({\beta }^{+}\right)}^2\left(\begin{array}{c}\overline{D}(-2x_1{x}_3{x}_4-x_1^2{x}_2-x_3^2{x}_2)\\ D(-2x_2{x}_3{x}_4-x_2^2{x}_1-x_4^2{x}_1)\\ \overline{D}(-2x_1{x}_2{x}_3-x_1^2{x}_4-x_3^2{x}_4)\\ D(-2x_1{x}_2{x}_4-x_2^2{x}_3-x_4^2{x}_3)\end{array}\right).$$

The normal form on the center manifold becomes

$$\dot{x}=\left(\begin{array}{cccc}i{\beta }^{+}{\tau }_k^{+}& & & \\ & -i{\beta }^{+}{\tau }_k^{+}& & \\ & & i{\beta }^{+}{\tau }_k^{+}& \\ & & & -i{\beta }^{+}{\tau }_k^{+}\end{array}\right)x+4\left(\begin{array}{c}\overline{n}\mu x_1\\ n\mu x_2\\ \overline{n}\mu x_3\\ n\mu x_4\end{array}\right)+\frac{1}{3!}{g}_3^1(x,0,0)+O({\mu }^2|x|),$$

(12)

for $x\in C^4$. By the following coordinates transformation,

$$\left\{\begin{array}{c}{x}_1=w_1-i{w}_2,\\ x_2=w_1+i{w}_2,\\ x_3=w_3-i{w}_4,\\ x_4=w_3+i{w}_4.\end{array}\right.$$

Let ${\rho }_1=w_1^2+w_2^2+w_3^2+w_4^2,{\rho }_2=2(w_1{w}_3+w_2{w}_4)$. Equation (12) becomes

$$\begin{array}{cc}\hfill & \dot{w_1}={\beta }^{+}{\tau }_k^{+}{w}_2+4\mu (\mathrm{Re}\left(n\right)w_1-\mathrm{Im}\left(n\right)w_2)-4\alpha {\tau }_k^{+}{\left({\beta }^{+}\right)}^2[{\rho }_1(\mathrm{Re}\left(D\right)w_1-\mathrm{Im}\left(D\right)w_2)\hfill \\ \hfill & +{\rho }_2(\mathrm{Re}\left(D\right)w_3-\mathrm{Im}\left(D\right)w_4)]+O({\mu }^2|w|+{|w|}^4),\hfill \\ \hfill & \dot{w_2}=-{\beta }^{+}{\tau }_k^{+}{w}_1+4\mu (\mathrm{Im}\left(n\right)w_1+\mathrm{Re}\left(n\right)w_2)-4\alpha {\tau }_k^{+}{\left({\beta }^{+}\right)}^2[{\rho }_1(\mathrm{Im}\left(D\right)w_1+\mathrm{Re}\left(D\right)w_2)+\hfill \\ \hfill & {\rho }_2(\mathrm{Im}\left(D\right)w_3+\mathrm{Re}\left(D\right)w_4)]+O({\mu }^2|w|+{|w|}^4),\hfill \\ \hfill & \dot{w_3}={\beta }^{+}{\tau }_k^{+}{w}_4+4\mu (\mathrm{Re}\left(n\right)w_3-\mathrm{Im}\left(n\right)w_4)-4\alpha {\tau }_k^{+}{\left({\beta }^{+}\right)}^2[{\rho }_2(\mathrm{Re}\left(D\right)w_1-\mathrm{Im}\left(D\right)w_2)\hfill \\ \hfill & +{\rho }_1(\mathrm{Re}\left(D\right)w_3-\mathrm{Im}\left(D\right)w_4)]+O({\mu }^2|w|+{|w|}^4),\hfill \\ & \dot{w_4}=-{\beta }^{+}{\tau }_k^{+}{w}_3+4\mu (\mathrm{Im}\left(n\right)w_3+\mathrm{Re}\left(n\right)w_4)-4\alpha {\tau }_k^{+}{\left({\beta }^{+}\right)}^2[{\rho }_2(\mathrm{Im}\left(D\right)w_1+\mathrm{Re}\left(D\right)w_2)\hfill \\ \hfill & +{\rho }_1(\mathrm{Im}\left(D\right)w_3+\mathrm{Re}\left(D\right)w_4)]+O({\mu }^2|w|+{|w|}^4).\hfill \end{array}$$

(13)

Let

$$z_1\left(t\right)=w_1\left(s\right)+i{w}_2\left(s\right),$$

$$z_2\left(t\right)=w_3\left(s\right)+i{w}_4\left(s\right),$$

$$s=\frac{t}{(1+\sigma ){\beta }^{+}{\tau }_k^{+}},$$

where $\sigma$ is a periodic-scaling parameter, then it follows that

$$\begin{array}{c}(1+\sigma )\dot{z_1}=-i{z}_1+\frac{4\mu n}{{\beta }^{+}{\tau }_k^{+}}{z}_1-4\alpha {\beta }^{+}D\left[\right(|z_1{|}^2+|z_2{|}^2)z_1+(z_1\overline{z_2}+\overline{z_1}{z}_2)z_2],\\ (1+\sigma )\dot{z_2}=-i{z}_2+\frac{4\mu n}{{\beta }^{+}{\tau }_k^{+}}{z}_2-4\alpha {\beta }^{+}D[(z_1\overline{z_2}+\overline{z_1}{z}_2)z_1+(|z_1{|}^2+|z_2{|}^2\left)z_2\right].\end{array}$$

(14)

Let us denote $g(z,\mu )$ as follows:

$$g(z,\mu )=\left(\begin{array}{c}i{z}_1-\frac{4\mu n}{{\beta }^{+}{\tau }_k^{+}}{z}_1+4\alpha {\beta }^{+}D\left[\right(|z_1{|}^2+|z_2{|}^2)z_1+(z_1\overline{z_2}+\overline{z_1}{z}_2)z_2]\\ i{z}_2-\frac{4\mu n}{{\beta }^{+}{\tau }_k^{+}}{z}_2+4\alpha {\beta }^{+}D[(z_1\overline{z_2}+\overline{z_1}{z}_2)z_1+(|z_1{|}^2+|z_2{|}^2\left)z_2\right]\end{array}\right),$$

then Equation (14) can be written as

$$(1+\sigma )\dot{z}+g(z,\mu )=0.$$

(15)

According to [19] (pp. 296–297, Theorems 6.3 and 6.5), the bifurcations of the small-amplitude periodic solution of (15) are completely determined by the zero point of the equation

$$-i(1+\sigma )z+g(z,\mu )=0,$$

(16)

and (16) can be written as

$$A\left(\begin{array}{c}{z}_1\\ z_2\end{array}\right)+B\left(\begin{array}{c}{z}_1^2{\overline{z}}_1\\ z_2^2{\overline{z}}_2\end{array}\right)+C\left(\begin{array}{c}{\overline{z}}_1{z}_2^2\\ {\overline{z}}_2{z}_1^2\end{array}\right)=0,$$

where

$$A=A_0+A_N(|z_1{|}^2+|z_2{|}^2),A_0=-\frac{4\mu n}{{\beta }^{+}{\tau }_k^{+}}-i\sigma ,$$

$$A_N=8\alpha {\beta }^{+}D,B=B_0=-4\alpha {\beta }^{+}D,C=4\alpha {\beta }^{+}D.$$

From the expressions of D and those above, we obtain the following expressions

$$\mathrm{Re}\left(D\right)=\frac{1}{4}\frac{1+{\left({\beta }^{+}\right)}^2+\alpha p^2{\left({\beta }^{+}\right)}^2}{{(1+{\left({\beta }^{+}\right)}^2+\alpha p^2{\left({\beta }^{+}\right)}^2)}^2+{(1-{\left({\beta }^{+}\right)}^2)}^2{\left({\beta }^{+}\right)}^2},$$

$$\mathrm{Re}(A_N+B)=4\alpha {\beta }^{+}\mathrm{Re}\left(D\right),$$

$$\mathrm{Re}(2A_N+B+C)=16\alpha {\beta }^{+}\mathrm{Re}\left(D\right),$$

$$\mathrm{Re}(2A_N+B-C)=8\alpha {\beta }^{+}\mathrm{Re}\left(D\right),$$

where $\mathrm{Re}(A_N+B),\mathrm{Re}(2A_N+B+C),\mathrm{Re}(2A_N+B-C)$ are positive when $\alpha >0$ or $\alpha <\frac{-1-{\left({\beta }^{+}\right)}^2}{p^2{\left({\beta }^{+}\right)}^2}$, and $\mathrm{Re}(A_N+B),\mathrm{Re}(2A_N+B+C),\mathrm{Re}(2A_N+B-C)$ are negative when $\frac{-1-{\left({\beta }^{+}\right)}^2}{p^2{\left({\beta }^{+}\right)}^2}<\alpha <0$.

We have the following theorem by the result of [19] (p. 383, Theorem 3.1).

Theorem 3.

The following statements are true.

(i) 

If $\alpha >0$ or $\alpha <\frac{-1-{\left({\beta }^{+}\right)}^2}{p^2{\left({\beta }^{+}\right)}^2}$, then the bifurcations are supercritical at the critical values $\tau ={\tau }_k^{+}(k=0,1,2,\dots )$.

(ii) 

If $\frac{-1-{\left({\beta }^{+}\right)}^2}{p^2{\left({\beta }^{+}\right)}^2}<\alpha <0$, then the bifurcations are subcritical at the critical values $\tau ={\tau }_k^{+}(k=0,1,2,\dots )$.

5. Numerical Simulations

In this section, we consider system (2) with parameters $\alpha =-2.5$, $p=-1,w=1$, $a=-1,b=\frac{1}{2},c=2$, which is equivalent to system (1). According to the calculation, we obtain ${\tau }_0^{+}=1.2010$, ${\beta }^{+}=2.1282$ and $\frac{-1-{\left({\beta }^{+}\right)}^2}{p^2{\left({\beta }^{+}\right)}^2}=-1.2208$. On the basis of Theorem 1, we know the zero equilibrium is asymptotically stable when $\tau <{\tau }_0^{+}$ (see Figure 1a, Figure 2a, Figure 3a and Figure 4a). When $\tau >{\tau }_0^{+}$, it is unstable, and some periodic solutions (standing waves (Figure 1b and Figure 2b) and mirror-reflecting waves (Figure 3b and Figure 4b)) appear from the zero equilibrium. Figure 1c and Figure 2c show that the standing waves of the system (2) are unstable. Figure 3c and Figure 4c show that the mirror-reflecting waves of the system (2) are unstable. Since $\alpha <\frac{-1-{\left({\beta }^{+}\right)}^2}{p^2{\left({\beta }^{+}\right)}^2}$, according to Theorem 3, the bifurcated periodic solutions are supercritical.

6. Conclusions

In this paper, system (1) with four coupled van der Pol oscillators is investigated. The system (1) is symmetrical and $D_4$ equivariant. Hence, we discuss the equivariant Hopf bifurcation of system (1). By analyzing the characteristic equation of system (1), the conditions and critical value ${\tau }_k^{\pm }(k=0,1,2\dots )$ of losing the stability of the zero equilibrium of system (1) are obtained. Near ${\tau }_k^{\pm }(k=0,1,2\dots )$, there exist three branches of small-amplitude periodic solutions of system (1) with a period near $\frac{2\pi }{{\beta }^{\pm }}$: discrete waves, mirror-emitted waves and standing waves. Then the normal form of system (1) on the central manifold is obtained, as well as exploring the Hopf bifurcation direction of the bifurcating periodic solutions, that is, when $\mathrm{Re}(A_N+B)>0,\mathrm{Re}(2A_N+B+C)>0$ and $\mathrm{Re}(2A_N+B-C)>0$ are established, the bifurcations are supercritical at the critical values $\tau ={\tau }_k^{+}(k=0,1,2,\dots )$, and when $\mathrm{Re}(A_N+B)<0,\mathrm{Re}(2A_N+B+C)<0$ and $\mathrm{Re}(2A_N+B-C)<0$ are established, the bifurcations are subcritical at the critical values $\tau ={\tau }_k^{+}(k=0,1,2,\dots )$. Finally, the theoretical results are illustrated by using numerical simulation. In addition, standing waves and mirror-reflecting waves are unstable by numerical simulation. Unfortunately, the discrete wave of the system is not simulated.