Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

Abstract

We investigate the resonant double Hopf bifurcation in a diffusive complex Ginzburg–Landau model with delayed feedback and phase shift. The conditions for the existence of resonant double Hopf bifurcation are obtained by analyzing the roots’ distribution of the characteristic equation, and a general formula to determine the bifurcation point is given. For the cases of 1:2 and 1:3 resonance, we choose time delay, feedback strength and phase shift as bifurcation parameters and derive the normal forms which are proved to be the same as those in non-resonant cases. The impact of cubic terms on the unfolding types is discussed after obtaining the normal form till 3rd order. By fixing phase shift, we find that varying time delay and feedback strength simultaneously can induce the coexistence of two different periodic solutions, the existence of quasi-periodic solutions and strange attractors. Also, the effects on the existence of transient quasi-periodic solution exerted by the phase shift are illustrated.

A Appendix

1.1 A.1 Center manifold reduction

In this section, we choose $\epsilon$, $\tau$ and $\beta$ as bifurcation parameters and derive the normal form near the double Hopf bifurcation point with 1:2 and 1:3 resonance by center manifold theory and normal form method [9, 10, 16], respectively. Re-scale the time by $t\rightarrow t/\tau$, then system (3) can be written as

$$\begin{aligned} \left\{ \begin{aligned} \frac{\partial u_1(x,t)}{\partial t}&=\tau \left( 1-\epsilon \cos \beta \right) u_1(x,t)\\&-\tau \left( \omega _0+\epsilon \sin \beta \right) u_2(x,t)+\tau \epsilon \cos \beta u_1(x,t-1)\\&+\tau \epsilon \sin \beta u_2(x,t-1)-\tau u_1^3(x,t)\\&+\tau \alpha u_2^3(x,t)-\tau u_1(x,t)u_2^2(x,t)\\&+\tau \alpha u_1^2(x,t)u_2(x,t)+\tau \frac{\partial ^2u_1(x,t)}{\partial x^2},\\&~x\in (0,l\pi ),~t>0, \\ \frac{\partial u_2(x,t)}{\partial t}&=\tau \left( \omega _0+\epsilon \sin \beta \right) u_1(x,t)\\&+\tau \left( 1-\epsilon \cos \beta \right) u_2(x,t)-\tau \epsilon \sin \beta u_1(x,t-1)\\&+\tau \epsilon \cos \beta u_2(x,t-1)-\tau \alpha u_1^3(x,t)\\&- \tau u_2^3(x,t)-\tau \alpha u_1(x,t)u_2^2(x,t)\\&- \tau u_1^2(x,t)u_2(x,t)+\tau \frac{\partial ^2u_2(x,t)}{\partial x^2},\\&~~x\in (0,l\pi ),~t>0,\\ \frac{\partial u_1(0,t)}{\partial x}&=\frac{\partial u_2(0,t)}{\partial x}=0, ~\frac{\partial u_1(l\pi ,t)}{\partial x}\\&=\frac{\partial u_2(l\pi ,t)}{\partial x}=0,~t>0. \end{aligned} \right. \end{aligned}$$

(21)

Assume that Eq.(5) has characteristic roots $i\omega _-, i\omega _+$ at bifurcation point $(\epsilon _c,\tau _c,\beta _c)$. Then, the characteristic equation of system (21) at equilibrium (0, 0) has two pairs of purely imaginary roots $\pm i\tau _c \omega _-$ and $\pm i\tau _c \omega _+$.

Following the notation in [6], we define the real-valued Hilbert space

$$\begin{aligned} X:= & {} \left\{ (u_1,u_2)^T\in (H^2(0,l\pi ))^2|\frac{\partial u_i}{\partial x}(0,t)\right. \\= & {} \left. \frac{\partial u_i}{\partial x}(l\pi ,t)=0,i=1,2\right\} , \end{aligned}$$

and the corresponding complexification space

$$\begin{aligned} X_{{\mathbb {C}}}=\{U_1+iU_2|U_i\in X,i=1,2\}, \end{aligned}$$

with the general complex-value $L^2$ inner product

$$\begin{aligned} \langle u,v\rangle =\int _{0}^{l\pi }(\overline{u_1}v_1+u_2\overline{v_2})\mathrm {d}x, \end{aligned}$$

for $u=(u_1,u_2)^T,v=(v_1,v_2)^T\in X_{{\mathbb {C}}}$. Fix phase space $C=C([-1,0], X_{{\mathbb {C}}})$ with the supremum norm. Write $u_t=u(t+\theta ), -1\le \theta \le 0$, and denote $\epsilon =\epsilon _c+\mu _1$, $\tau =\tau _c+\mu _2,\beta =\beta _c+\mu _3$, $\mu =(\mu _1,\mu _2,\mu _3)$, then Eq. (21) can be written as

$$\begin{aligned} \frac{\mathrm {d}U(t)}{\mathrm {d}t}=D(\mu )\varDelta U(t)+L(\mu )U_t+F(U_t,\mu ), \end{aligned}$$

(22)

where

$$\begin{aligned}&U(t)=\left( \begin{matrix} u_1(t)\\ u_2(t) \end{matrix} \right) , \nonumber \\&U_t=\left( \begin{matrix} u_{1t}\\ u_{2t} \end{matrix} \right) ,\nonumber \\&D(\mu )=(\tau _c+\mu _1)\left( \begin{matrix} 1&{}0\\ 0&{}1 \end{matrix} \right) ,\nonumber \\&L(\mu )U_t=(\tau _c+\mu _2)\nonumber \\&\qquad \qquad \qquad \left( \begin{aligned}&1-(\epsilon _c+\mu _1)\cos (\beta _c+\mu _3)\\&\quad -(\omega _0+(\epsilon _c+\mu _1)\sin (\beta _c+\mu _3))\\&\omega _0+(\epsilon _c+\mu _1)\sin (\beta _c+\mu _3)\\&\quad 1-(\epsilon _c+\mu _1)\cos (\beta _c+\mu _3) \end{aligned} \right) \nonumber \\&\qquad \qquad \qquad \left( \begin{aligned}&u_{1t}(0)\\&u_{2t}(0) \end{aligned} \right) \nonumber \\&\qquad \qquad \qquad + (\tau _c+\mu _2)(\epsilon _c+\mu _1)\nonumber \\&\qquad \qquad \qquad \left( \begin{aligned}&\cos (\beta _c+\mu _3)&\sin (\beta _c+\mu _3)\\&-\sin (\beta _c+\mu _3)&\cos (\beta _c+\mu _3) \end{aligned} \right) \nonumber \\&\qquad \qquad \qquad \left( \begin{aligned}&u_{1t}(-1)\\&u_{2t}(-1) \end{aligned} \right) ,\nonumber \\&F(U_t,\mu )=(\tau _c+\mu _2) \nonumber \\&\qquad \qquad \qquad \left( \begin{aligned}&-u_{1t}^3(0)+\alpha u_{2t}^3(0)-\\&\quad u_{1t}(0)u_{2t}^2(0)+\alpha u_{1t}^2(0)u_{2t}(0)\\&-\alpha u_{1t}^3(0)-u_{2t}^3(0)\\&\quad -\alpha u_{1t}(0)u_{2t}^2(0)-u_{1t}^2(0)u_{2t}(0) \end{aligned} \right) .\nonumber \\ \end{aligned}$$

(23)

Let

$$\begin{aligned}&\gamma _{n}(x)=\frac{\cos \frac{n}{l}x}{\Vert \cos \frac{n}{l}x\Vert _{L^2}}=\left\{ \begin{aligned}&\sqrt{\frac{1}{l\pi }},n=0,\\&\sqrt{\frac{2}{l\pi }}\cos \frac{n}{l},n\ge 1, \end{aligned} \right. \end{aligned}$$

and $\beta _{n}^j(x)=\gamma _n(x)e_j$, where $e_j$ is the $j-$th unit coordinate vector of ${\mathbb {R}}^2$. Then $\{\beta _{n}^j\}_{n\ge 0}$ are eigenfunctions of $\varDelta$ with eigenvalues $-\frac{n^2}{l^2}$ under homogeneous Neumann boundary condition. The subspace $B_n$ of C is defined as

$$\begin{aligned} B_{n}:={\text {span}}\left\{ \left\langle v(\cdot ), \beta _{n}^{j}\right\rangle \beta _{n}^{j} \mid v \in C,j=1,2\right\} . \end{aligned}$$

Define the enlarged phase space BC as

$$\begin{aligned} BC:= & {} \{\varphi :[-1,0]\rightarrow X_{{\mathbb {C}}}|\varphi \text{ is } \text{ continuous } \text{ on } [-1,0), \\&\exists \lim _{\theta \rightarrow 0^-}\varphi (\theta )\in X_{{\mathbb {C}}} \} \end{aligned}$$

As shown in [16], the infinitesimal generator A of the solution map of the linearization of (24) is defined by

$$\begin{aligned}&A: C_0^{1} \rightarrow B C, \quad A u=\dot{u}+X_{0}\left[ D_0\varDelta u(0)+L_{0} u-\dot{u}(0)\right] .\\&\quad X_0(\theta )=\left\{ \begin{aligned}&0,\quad -1\le \theta <0,\\&Id,\quad \theta =0, \end{aligned} \right. \end{aligned}$$

with $C_0^1=\{\varphi \in C:{\dot{\varphi }}\in C,\varphi (0)\in {\text {dom}}(\varDelta )\}$. In BC, Eq. (21) can be written as an abstract ordinary differential equation

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t} u=A u+X_{0} {\tilde{F}}, \end{aligned}$$

(24)

where

$$\begin{aligned} {\tilde{F}}(u, \mu ){=}\left[ D(\mu ){-}D_0\right] \varDelta u{+}\left[ L(\mu ){-}L_{0}\right] u+F(u, \mu ), \end{aligned}$$

(25)

with

$$\begin{aligned} \begin{aligned}&D_0=\tau _{c}\left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \\ \end{array} \right) ,\\&L_0(U_t)=\tau _c \left( \begin{aligned}&1-\epsilon _c\cos \beta _c&-(\omega _0+\epsilon _c\sin \beta _c)\\&\omega _0+\epsilon _c\sin \beta _c&1-\epsilon _c\cos \beta _c \end{aligned} \right) \\&\qquad \left( \begin{aligned}&u_{1t}(0)\\&u_{2t}(0) \end{aligned} \right) \\&\qquad + \tau _c\epsilon _c\left( \begin{aligned}&\cos \beta _c&\sin \beta _c\\&-\sin \beta _c&\cos \beta _c \end{aligned} \right) \left( \begin{aligned}&u_{1t}(-1)\\&u_{2t}(-1) \end{aligned} \right) ,\\ \end{aligned} \end{aligned}$$

and $D(\mu )$, $L(\mu )$ and $F(u,\mu )$ are defined in (23).

For convenience, we denote

$$\begin{aligned} \left\langle v(\cdot ), \beta _{n}\right\rangle =\left( \left\langle v(\cdot ), \beta _{n}^{(1)}\right\rangle ,\left\langle v(\cdot ), \beta _{n}^{(2)}\right\rangle \right) ^{T}. \end{aligned}$$

Therefore, on $B_n$, system (22) can be presented as

$$\begin{aligned} \frac{\mathrm {d}U(t)}{\mathrm {d}t}=D_0\varDelta U(t)+L_0(U_t)+{\tilde{F}}(U_t,\mu ). \end{aligned}$$

(26)

Thus, the linearized system is

$$\begin{aligned} \frac{\mathrm {d}U(t)}{\mathrm {d}t}=D_0\varDelta U(t)+L_0(U_t). \end{aligned}$$

For $\phi =(\phi _{1}, \phi _{2})^{T}\in C$, define linear operator

$$\begin{aligned} L_{0} \phi =\int _{-1}^{0} \mathrm {d} \eta _k(\theta ) \phi (\theta )+\frac{n_k^2}{l^2}D_0\phi (0), \end{aligned}$$

(27)

where

$$\begin{aligned} \eta _k(\theta )=\left\{ \begin{array}{ll}\tau _{c} A_{1}, &{} \theta =0, \\ 0, &{} \theta \in (-1,0), \\ -\tau _{c} B_{1}, &{} \theta =-1.\end{array}\right. \end{aligned}$$

and

$$\begin{aligned} A_{1}= & {} \left( \begin{array}{cc}1-\epsilon _{c}\cos \beta _c &{} -(\omega _{0}+\epsilon _c\sin \beta _c ) \\ \omega _{0}+\epsilon _c\sin \beta &{} 1-\epsilon _{c}\cos \beta _c \end{array}\right) ,\\ B_{1}= & {} \epsilon _{c} \left( \begin{array}{cc}\cos \beta &{} \sin \beta _c \\ -\sin \beta _c &{} \cos \beta _c \end{array}\right) . \end{aligned}$$

We define the operator $A^*:D(A^*)\subset C^*\rightarrow C^*$ with

$$\begin{aligned} (A^*\alpha )(s)=\left\{ \begin{aligned}&-\alpha '(s)\qquad s\in (0,1],\\&\int _{-1}^{0}\alpha (-\theta )\mathrm {d}\eta (\theta )\qquad s=0. \end{aligned} \right. \end{aligned}$$

(28)

Note that $A^{*}$ is the formal adjoint of the infinitesimal operator A under the following the bilinear form on $C^{*}\times C$

$$\begin{aligned} (\psi , \varphi )_k=\psi (0) \varphi (0)-\int _{-1}^{0} \int _{0}^{\theta } \psi (\xi -\theta ) \mathrm {d} \eta _k(\theta ) \varphi (\xi ) \mathrm {d} \xi , \end{aligned}$$

(29)

where $\varphi (\theta )=\left( \varphi _{1}(\theta ), \varphi _{2}(\theta )\right) \in C, \psi (s)=\left( \psi _{1}(s), \psi _{2}(s)\right) ^{T} \in C^{*}$

Denote the basis of $P_{\Lambda }$ and $P_{\Lambda }^*$ by $\Phi (\theta )$ and $\Psi (s)$, respectively. More precisely,

$$\begin{aligned} \Phi (\theta )= & {} \left( q_{1}(\theta ), {\bar{q}}_{1}(\theta ), q_{2}(\theta ), {\bar{q}}_{2}(\theta )\right) \\= & {} \left( \begin{aligned}&q_{11}(\theta )&\overline{q_{11}}(\theta )&\quad q_{21}(\theta )&\overline{q_{21}}(\theta )&\\&q_{12}(\theta )&\overline{q_{12}}(\theta )&\quad q_{22}(\theta )&\overline{q_{22}}(\theta )&\end{aligned} \right) , \end{aligned}$$

and

$$\begin{aligned} \Psi (s)=\left( q_{1}^{*}(s), {\bar{q}}_{1}^{*}(s), q_{2}^{*}(s), {\bar{q}}_{2}^{*}(s)\right) ^{T}:=(\psi _{i j})_{i=1,2,3,4;j=1,2}. \end{aligned}$$

A direct computation yields that

$$\begin{aligned} \begin{aligned} q_1(\theta )&=(1,-i)^Te^{i\tau _c\omega _+\theta },\qquad q_2(\theta )=(1,-i)^Te^{i\tau _c\omega _-\theta },\\ q_1^*(s)&=\frac{1}{2(1+\epsilon _c\tau _ce^{-i(\tau _c\omega _++\beta _c)})}(1,i),\\ q_2^*(s)&=\frac{1}{2(1+\epsilon _c\tau _ce^{-i(\tau _c\omega _-+\beta _c)})}(1,i). \end{aligned} \end{aligned}$$

(30)

Through the previous discussions, we know that Eq. (5) has two pairs of purely imaginary roots at the double Hopf bifurcation point. And the other roots remain negative real parts. According to [10], BC could be decomposed by $BC=P_{\Lambda } \oplus Q_{\Lambda }$. $P_{\Lambda }$ is a four-dimensional center subspace spanned by the linear operator $L_0$ connected with purely imaginary roots. $Q_{\Lambda }$ is the complementary space. Also, $Q_{\Lambda }$ is the kernel space of projection $\pi :BC\rightarrow P_{\Lambda }$ with

$$\begin{aligned} \pi (\varphi )=\sum _{k=1}^2\Phi _k(\Psi _k,\langle \varphi (\cdot ),\beta _{n_k}\rangle )_k\cdot \beta _{n_k}. \end{aligned}$$

Let $u_{t}(\theta )=\sum \nolimits _{k=1}^2\Phi _k(\theta ) {\tilde{z}}_k(t)+v_t(\theta )$, then Eq. (24) could be decomposed as

$$\begin{aligned} \begin{array}{l}\dot{z}=B z+\Psi (0) \left( \begin{matrix} \langle {\tilde{F}}\left( \sum \nolimits _{k=1}^2(\Phi _k {\tilde{z}}_k)\cdot \beta _{n_k}+v, \mu \right) ,\beta _{n_1} \rangle \\ \langle {\tilde{F}}\left( \sum \nolimits _{k=1}^2(\Phi _k {\tilde{z}}_k)\cdot \beta _{n_k}+v, \mu \right) ,\beta _{n_2} \rangle \\ \end{matrix} \right) ,\\ \dot{v}=A_{Q^{1}} v+(I-\pi ) X_{0} {\tilde{F}}(\sum \limits _{k=1}^2(\Phi _k {\tilde{z}}_k)\cdot \beta _{n_k}+v, \mu ), \end{array} \end{aligned}$$

(31)

$v \in Q \cap C_0^{1}:= Q^{1}\subset {\text {Ker}} \pi$. $A_{Q^1}$ is A restricted in $Q^{1}$.

1.2 A.2 Center Manifold Reduction and Normal Form

In this part, we provide the general methods to obtain the normal form on the center manifold. First, the Taylor expansion of ${\tilde{F}}(\varphi ,\mu )$ is represented as

$$\begin{aligned} {\tilde{F}}(\varphi ,\mu )=\sum _{i\ge 2}\frac{1}{i!}\tilde{F_i}(\varphi ,\mu ) \end{aligned}$$

here $\tilde{F_i}$ is the ith Frechet derivation of ${\tilde{F}}$. Then (31) turns to be

$$\begin{aligned} \begin{aligned} \dot{z}&=Bz+\sum _{i\ge 2}\frac{1}{i!}f_{i}^2(z,v,\mu ),\\ \frac{\mathrm {d}v}{\mathrm {d}t}&=A_1v+\sum _{i\ge 2}\frac{1}{i!}f_i^2(z,v,\mu ). \end{aligned} \end{aligned}$$

(32)

where $z=(z_1,z_2,z_3,z_4)^T\in {\mathbb {R}}^4,v\in Q^1$,

$$\begin{aligned} \begin{array}{l}f_i^1(z,v,\mu ){=}\Psi (0) \left( \begin{matrix} \langle \tilde{F_i}(\sum \limits _{k=1}^2(\Phi _k {\tilde{z}}_k)\cdot \beta _{n_k}{+}v, \mu ),\beta _{n_1} \rangle \\ \langle \tilde{F_i}(\sum \limits _{k=1}^2(\Phi _k {\tilde{z}}_k)\cdot \beta _{n_k}{+}v, \mu ),\beta _{n_2} \rangle \\ \end{matrix} \right) ,\\ f_i^2=(I-\pi ) X_{0} \tilde{F_i}(\sum \limits _{k=1}^2(\Phi _k {\tilde{z}}_k)\cdot \beta _{n_k}+v, \mu ).\end{array} \end{aligned}$$

Operator $M_i=(M_i^1,M_i^2),i\ge 2$ is defined by

$$\begin{aligned} \begin{aligned} M_i^1: V_i^7({\mathbb {C}}^4)&\rightarrow V_i^7({\mathbb {C}}^4),\\ (M_i^1p)(z,\mu )&=D_zp(z,\mu )Bz-Bp(z,\mu ),\\ M^2_i:V_i^7(Q_1)&\subset V_i^7({\text {Ker}}\pi )\rightarrow V_i^7({\text {Ker}}\pi ),\\ (M_i^2h)(z,\mu )&=D_zh(z,\mu )Bz-A_1h(z,\mu ). \end{aligned} \end{aligned}$$

where $V_i^7({\mathbb {C}}^4)$ denotes the space of homogeneous polynomials of degree i in 7 variables $z{=}(z_1,z_2,z_3,z_4),\mu =(\mu _1,\mu _2,\mu _3)$ The normal forms are derived by transformations [32] like

$$\begin{aligned} (z, y, \mu )=({\widehat{z}}, {\widehat{y}}, \mu )+\frac{1}{i !}\left( U_{i}^{1}({\widehat{z}}, \mu ), U_{i}^{2}({\widehat{z}}, \mu ), 0\right) , \end{aligned}$$

Through finite transformations, the normal form of (31) must have the form

$$\begin{aligned} \begin{aligned} \dot{z}&=B z+\sum _{i \ge 2} \frac{1}{i !} g_{i}^{1}(z, v, \mu ), \\ \frac{d v}{d t}&=A_{1} v+\sum _{i \ge 2} \frac{1}{i !} g_{i}^{2}(z, v, \mu ), \end{aligned} \end{aligned}$$

where $g_i=(g_i^1, g_i^2), i\ge 2,$ are the terms of order i. We can obtain the terms by

$$\begin{aligned} g_i(z,v,\mu )=\bar{f_i}(z,v,\mu )-M_iU_i(z,\mu ), \end{aligned}$$

and $U_i\in V_i^7({\mathbb {C}}^4)\times V_i^7(Q_1)$ can be obtained by

$$\begin{aligned} U_i(z,\mu )=(M_i)^{-1}{\text {P}}_{{\text {Im}}(M_i^2)\times {\text {Im}}(M_i^2)}\circ \bar{f_i}(z,0,\mu ) \end{aligned}$$

where ${\text {P}}$ is the projection operator. Here $\bar{f_i}=(\bar{f_i}^1,\bar{f_i}^2)$ are the terms of order i in (z, v) obtained after the computation of normal forms till order $i-1$. According to [6], system (31) is finally equivalent

$$\begin{aligned} \begin{aligned} \dot{z}&=B z+\Psi (0)\left( \begin{array}{l}\left\langle {\widetilde{F}}\left( \mu , \Phi _{x}+y\right) , \beta _{n_{1}}\right\rangle \\ \left\langle {\widetilde{F}}\left( \mu , \Phi _{z_{x}}+v\right) , \beta _{n_{2}}\right\rangle \end{array}\right) \\ \frac{\mathrm {d} v}{\mathrm {~d} t}&=A_{1} v+\left( I-\pi _{0}\right) X_{0} {\widetilde{F}}\left( \mu , \Phi z_{x}+v\right) \end{aligned} \end{aligned}$$

The cases with $\omega _-:\omega _+\not = k_1:k_2,$ for $k_1,k_2\in {\mathbb {N}}$, have been discussed by many researchers. Our work focuses on $\omega _-:\omega _+=k_1:k_2,k_1,k_2\in {\mathbb {N}},1\le k_1\le k_2\le 4$. We know that for $z_p\mu ^{\iota }e^{\xi }$, with $z^p=z_1^{p_1}z_2^{p_2}z_3^{p_3}z_4^{p_4},\mu ^\iota =\mu _1^{\iota _1}\mu _2^{\iota _2}\mu _3^{\iota _3},p_1,p_2,p_3,p_4,\iota _1,\iota _2,\iota _3\in {\mathbb {N}}_0,p_1+p_2+p_3+p_4+\iota _1+\iota _2+\iota _3=i$ and $\{e_{\xi }\},\xi =1,2,3,4$ be the orthonormal basis in ${\mathbb {R}}^4$, we have

$$\begin{aligned} \begin{aligned}&M_i^1(z^p\mu ^\iota e_{\xi })=D_z(z^p\mu ^\iota e_{\xi })Bz-Bz^p\mu ^\iota e_{\xi }\\&=\left\{ \begin{array}{l}\left( (p_{1}{-}p_2{+}(-1)^{\xi } )\omega _{+}{+}(p_3{-}p_{4})\omega _{-}\right) iz^{p} \mu ^{\iota } e_{\xi }, \xi {=}1,2, \\ \left( (p_1{-}p_2)\omega _+{+}(p_3{-}p_4{+}(-1)^{\xi })\omega _-\right) i z^{p} \mu ^{\iota } e_\xi , \xi {=}3,4, \end{array} \right. \end{aligned} \end{aligned}$$

From elementary linear algebra, we know that $V_i^7({\mathbb {C}}^4)$ can be (nonuniquely) represented as follows

$$\begin{aligned} V_i^7({\mathbb {C}}^4) = M^1_i(V_i^7({\mathbb {C}}^4))\oplus {\text {Im}}(M_i^1)^c \end{aligned}$$

where ${\text {Im}}(M_i^1)^c$ represents a space complementary to $M^1_i(V_i^7({\mathbb {C}}^4))$. The elements in ${\text {Im}}(M_i^1)^c$ is completely depends on the solution $p_{\xi }=(p_1,p_2,p_3,p_4)$ of the following equations

$$\begin{aligned} \begin{aligned} k_2(p_{1}-p_2+(-1)^{\xi } )+k_1(p_3-p_{4})=0 , \xi =1,2,\\ k_2(p_1-p_2)+k_1(p_3-p_4+(-1)^{\xi })=0, \xi =3,4, \end{aligned} \end{aligned}$$

(33)

for fixed i and $\xi$ when computing the complementary space of $M_i^1$.

1.3 A.3 1:2 resonance

It is easy to verify that when $k_1:k_2 = 1:2$, Eq (33) has the following solution

$$\begin{aligned} \begin{aligned} p_1&=(0,0,2,0),\\ p_2&=(0,0,0,2),\\ p_3&=(1,0,0,1),\\ p_4&=(0,1,1,0). \end{aligned} \end{aligned}$$

Therefore, the complementary space for $k_1:k_2=1:2$ is

$$\begin{aligned}&{\text {Im}}\left( M_{2}^{1}\right) ^{c}\\&={\text {span}}\left\{ \left( \begin{aligned} z_3^2\\ 0\\ 0\\ 0 \end{aligned} \right) , \left( \begin{aligned} 0\\ z_4^2\\ 0\\ 0 \end{aligned} \right) , \left( \begin{aligned} 0\\ 0\\ z_1z_4\\ 0 \end{aligned} \right) ,\right. \\&\left. \left( \begin{aligned} 0\\ 0\\ 0\\ z_2z_3 \end{aligned} \right) , \left( \begin{aligned} \mu _iz_1\\ 0\\ 0\\ 0 \end{aligned} \right) , \left( \begin{aligned} 0\\ \mu _iz_2\\ 0\\ 0 \end{aligned} \right) , \left( \begin{aligned} 0\\ 0\\ \mu _iz_3\\ 0 \end{aligned} \right) , \left( \begin{aligned} 0\\ 0\\ 0\\ \mu _i z_4 \end{aligned} \right) \right\} ,\\&\quad i=1,2,3. \end{aligned}$$

Thus, Eq. (31) on the center manifold has the form

$$\begin{aligned} \dot{z}=B z+\frac{1}{2} g_{2}^{1}(z, 0, \mu )+\text {h.o.t.}. \end{aligned}$$

where $g_{2}^{1}(z,0,\mu )={\text {Proj}}_{{\text {Im}}(M_2^1)^c}f_2^1(z,0,\mu )$. Therefore, the normal form until the second order has the form

$$\begin{aligned} \begin{array}{l} \dot{z}_{1}= i \tau _c \omega _{+} z_{1}+\lambda _{1} z_{1}+a_{33} z_{3}^{2}+ \text{ h.o.t. }, \\ \dot{z}_{2}=-i \tau _c \omega _{+} z_{2}+\bar{\lambda }_{1} z_{2}+a_{44}z_4^2+ \text{ h.o.t. }, \\ \dot{z}_{3}=i \tau _c \omega _{-} z_{3}+\lambda _{2} z_{3}+a_{14}z_1z_4+ \text{ h.o.t. }, \\ \dot{z}_{4}=-i \tau _c \omega _{-} z_{4}+\bar{\lambda }_{2} z_{4}+a_{23}z_2z_3+ \text{ h.o.t. }.\end{array} \end{aligned}$$

(34)

To find the third-order term in normal form, we only need to fix $i=3$ and solve equation (33) again. Omitting the high-order terms of perturbation parameters, we could derive the complementary space

$$\begin{aligned} {\text {Im}}(M^{1}_{3})^{c}={\text {span}}\left( \begin{array}{cccccccc} \left( \begin{array}{c} z^{2}_{1}z_{2} \\ 0 \\ 0 \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ z^{2}_{2}z_{1} \\ 0 \\ 0 \\ \end{array} \right) , &{}\left( \begin{array}{c} z_{1}z_{3}z_{4} \\ 0 \\ 0 \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ z_{2}z_{3}z_{4} \\ 0 \\ 0 \\ \end{array} \right) ,\\ &{}\quad \left( \begin{array}{c} 0 \\ 0 \\ z_{1}z_{2}z_{3} \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ z_{1}z_{2}z_{4} \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ 0 \\ z^{2}_{3}z_{4} \\ 0 \\ \end{array} \right) , &{}\left( \begin{array}{c} 0 \\ 0 \\ 0 \\ z_{3}z_{4}^{2} \\ \end{array} \right) \\ \end{array} \right) \end{aligned}$$

Thus, the normal form is supposed to be

$$\begin{aligned} \begin{array}{l} \dot{z}_{1}{=} i \tau _c \omega _{+} z_{1}{+}\lambda _{1} z_{1}{+}b_{33}z_3^2{+}a_{11} z_{1}^{2} z_{2}{+}a_{12} z_{1} z_{3} z_{4}{+} \text{ h.o.t. }, \\ \dot{z}_{2}{=}-i \tau _c \omega _{+}z_2{+}\bar{\lambda }_{1} z_{2}{+}b_{44}z_{4}^{2}{+}{\bar{a}}_{11} z_{1} z_{2}^{2}{+}{\bar{a}}_{12} z_{2} z_{3} z_{4}{+} \text{ h.o.t. }, \\ \dot{z}_{3}{=}i \tau _c \omega _{-} z_{3}{+}\lambda _{2} z_{3}{+}b_{14}z_1z_4{+}a_{21} z_{3}^{2} z_{4}{+}a_{22} z_{1} z_{2} z_{3}{+} \text{ h.o.t. }, \\ \dot{z}_{4}{=}-i \tau _c \omega _{-} z_{4}{+}\bar{\lambda }_{2} z_{4}{+}b_{23}z_2z_3{+}{\bar{a}}_{21} z_{3} z_{4}^{2}{+}{\bar{a}}_{22} z_{1} z_{2} z_{4}{+} \text{ h.o.t. }.\end{array} \end{aligned}$$

(35)

Here, we can easily see that $\lambda _i(i=1,2)$ must be the complex linear combination of $\mu _1,\mu _2,\mu _3$ and other coefficients are all constants.

1.4 A.4 1:3 resonance

As we had done for 1:2 resonance, omit the high-order term of parameters, we immediately obtain the complementary space for $M_2^1$ and $M_3^1$

$$\begin{aligned}&{\text {Im}}\left( M_{2}^{1}\right) ^{c}={\text {span}}\nonumber \\&\quad \left\{ \left( \begin{aligned} \mu _iz_1\\ 0\\ 0\\ 0 \end{aligned} \right) , \left( \begin{aligned} 0\\ \mu _iz_2\\ 0\\ 0 \end{aligned} \right) ,\right. \\&\left. \left( \begin{aligned} 0\\ 0\\ \mu _iz_3\\ 0 \end{aligned} \right) , \left( \begin{aligned} 0\\ 0\\ 0\\ \mu _i z_4 \end{aligned} \right) \right\} , i=1,2. \end{aligned}$$

and

$$\begin{aligned}&{\text {Im}}(M^{1}_{3})^{c}={\text {span}}\\&\quad \left( \begin{array}{cccccccccccc} \left( \begin{array}{c} z_{3}^3 \\ 0 \\ 0 \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ z_4^3 \\ 0 \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ 0 \\ z_1z_4^2 \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ z_2z_3^2 \\ \end{array} \right) ,\\ \left( \begin{array}{c} z^{2}_{1}z_{2} \\ 0 \\ 0 \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ z^{2}_{2}z_{1} \\ 0 \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} z_{1}z_{3}z_{4} \\ 0 \\ 0 \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ z_{2}z_{3}z_{4} \\ 0 \\ 0 \\ \end{array} \right) ,\\ \left( \begin{array}{c} 0 \\ 0 \\ z_{1}z_{2}z_{3} \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ z_{1}z_{2}z_{4} \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ 0 \\ z^{2}_{3}z_{4} \\ 0 \\ \end{array} \right) , &{} \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ z_{3}z_{4}^{2} \\ \end{array} \right) \\ \end{array} \right) . \end{aligned}$$

Therefore, the normal form until third order has the form

$$\begin{aligned} \begin{array}{l} \dot{z}_{1}= i \tau _c \omega _{+} z_{1}{+}\lambda _{1} z_{1}{+}a_{11} z_{1}^{2} z_{2}{+}a_{12} z_{1} z_{3} z_{4}{+}b_{333}z_3^3{+}\text { h.o.t.}, \\ \dot{z}_{2}=-i \tau _c \omega _{{+}} z_{2}{+}\bar{\lambda }_{1} z_{2}{+}{\bar{a}}_{11} z_{1} z_{2}^{2}{+}{\bar{a}}_{12} z_{2} z_{3} z_{4}{+}b_{444}z_4^3{+} \text{ h.o.t. }, \\ \dot{z}_{3}=i \tau _c \omega _{-} z_{3}{+}\lambda _{2} z_{3}{+}a_{21} z_{3}^{2} z_{4}{+}a_{22} z_{1} z_{2} z_{3}{+} b_{133}z_1z_4^2{+} \text{ h.o.t. }, \\ \dot{z}_{4}=-i \tau _c \omega _{-} z_{4}{+}\bar{\lambda }_{2} z_{4}{+}{\bar{a}}_{21} z_{3} z_{4}^{2}{+}{\bar{a}}_{22} z_{1} z_{2} z_{4}{+}b_{233}z_2z_3^2{+} \text{ h.o.t. }. \end{array} \end{aligned}$$

(36)

1.5 A.5 Application to CGLE

We apply our results above to the CGLE. When $n_1=n_2=0$, then Eq. (31) is turned out to be

$$\begin{aligned} \begin{array}{l}\dot{z}_{1}=i \tau _c\omega _{+} z_{1}+\sum \limits _{j=1}^{2} \psi _{1 j}\left( F_{2}^{j}+F_{3}^{j}\right) + \text{ h.o.t. } , \\ \dot{z}_{2}=-i \tau _c\omega _{+} z_{2}+\sum \limits _{j=1}^{2} \psi _{2 j}\left( F_{2}^{j}+F_{3}^{j}\right) + \text{ h.o.t. } , \\ \dot{z}_{3}=i \tau _c\omega _{-} z_{3}+\sum \limits _{j=1}^{2} \psi _{3 j}\left( F_{2}^{j}+F_{3}^{j}\right) + \text{ h.o.t. } , \\ \dot{z}_{4}=-i \tau _c \omega _{-} z_{4}+\sum \limits _{j=1}^{2} \psi _{4 j}\left( F_{2}^{j}+F_{3}^{j}\right) + \text{ h.o.t. } , \\ \dot{v}=A_{Q^{1}} v+(I-\pi ) X_{0} {\tilde{F}}(\Phi z+v, \mu ). \end{array} \end{aligned}$$

(37)

where

$$\begin{aligned} F_2^1= & {} \left( -\mu _1\tau _c\cos \beta _c+\mu _2(1-\epsilon _c\cos \beta _c)\right. \\&\left. +\mu _3\tau _c\epsilon _c\sin \beta _c\right) (z_1+z_2+z_3+z_4+v_{1t}(0))\\&-\left( \mu _1\tau _c\sin \beta _c+\mu _2(\omega _0+\epsilon _c\sin \beta _c)\right. \\&\left. +\mu _3\tau _c\epsilon _c\cos \beta _c\right) (-iz_1+z_2-iz_3+iz_4+v_{2t}(0))\\&+(\mu _1\tau _c\cos \beta _c+\mu _2\epsilon _c\cos \beta _c-\mu _3\tau _c\epsilon _c\sin \beta _c)\\&\times (e^{-i \tau _c\omega _{+}}z_1+e^{i \tau _c\omega _{+}}z_2\\&+e^{-i \tau _c\omega _{-}}z_3+e^{i \tau _c\omega _{-}}z_4 +v_{1t}(-1))\\&+(\mu _1\tau _c\sin \beta +\mu _2\epsilon _c\sin \beta +\mu _3\tau _c\epsilon _c\cos \beta _c)\\&\times (-ie^{-i \tau _c\omega _{+}}z_1+ie^{i \tau _c\omega _{+}}z_2\\&-ie^{-i \tau _c\omega _{-}}z_3+ie^{i \tau _c\omega _{-}}z_4+v_{2t}(-1)),\\ F_2^2= & {} \left( \mu _1\tau _c\sin \beta _c+\mu _2(\omega _0+\epsilon _c\sin \beta _c)\right. \\&\left. +\mu _3\tau _c\epsilon _c\cos \beta _c\right) (z_1+z_2+z_3+z_4+v_{1t}(0))\\&+\left( -\mu _1\tau _c\cos \beta _c+\mu _2(1-\epsilon _c\cos \beta _c)\right. \\&\left. +\mu _3\tau _c\epsilon _c\sin \beta _c\right) (-iz_1+z_2-iz_3+iz_4+v_{2t}(0))\\&-(\mu _1\tau _c\sin \beta +\mu _2\epsilon _c\sin \beta +\mu _3\tau _c\epsilon _c\cos \beta _c)\\&\times (e^{-i \tau _c\omega _{+}}z_1+e^{i \tau _c\omega _{+}}z_2\\&+e^{-i \tau _c\omega _{-}}z_3+e^{i \tau _c\omega _{-}}z_4 +v_{1t}(-1))\\&+(\mu _1\tau _c\cos \beta _c+\mu _2\epsilon _c\cos \beta _c-\mu _3\tau _c\epsilon _c\sin \beta _c)\\&\times (-ie^{-i \tau _c\omega _{+}}z_1+ie^{i \tau _c\omega _{+}}z_2\\&-ie^{-i \tau _c\omega _{-}}z_3+ie^{i \tau _c\omega _{-}}z_4+v_{2t}(-1)),\\ F_3^1= & {} \tau _c[-(z_{1}+z_{2}+z_{3}+z_{4}+v_{1 t}(0))^3\\&+\alpha (-i z_{1}+iz_{2}-iz_{3}+iz_{4}+v_{2 t}(0))^3\\&-(z_{1}+z_{2}+z_{3}+z_{4}+v_{1 t}(0))\\&(-i z_{1}+iz_{2}-iz_{3}+iz_{4}+v_{2 t}(0))^2\\&+\alpha (z_{1}+z_{2}+z_{3}+z_{4}+v_{1 t}(0))^2\\&(-i z_{1}+iz_{2}-iz_{3}+iz_{4}+v_{2 t}(0)) ],\\ F_3^2= & {} \tau _c[-\alpha (z_{1}+z_{2}+z_{3}+z_{4}\\&+v_{1 t}(0))^3-(-i z_{1}+iz_{2}-iz_{3}+iz_{4}+v_{2 t}(0))^3\\&-\alpha (z_{1}+z_{2}+z_{3}+z_{4}+v_{1 t}(0))(-i z_{1}+iz_{2}\\&-iz_{3}+iz_{4}+v_{2 t}(0))^2\\&-(z_{1}+z_{2}+z_{3}+z_{4}+v_{1 t}(0))^2(-i z_{1}\\&+iz_{2}-iz_{3}+iz_{4}+v_{2 t}(0)) ]. \end{aligned}$$

First, we compute the normal form until the second order. We only need to compute the coefficients obtained in the previous section. Note that $F_2^j$ and $F_3^j$ $(j=1,2)$ here do not have terms like $z_i^2(i=1,2,3,4)$ and $z_iz_j~(i,j=1,2,3,4)$, then we have

$$\begin{aligned} \lambda _1= & {} \frac{2}{2 + 2\epsilon _c\tau _c e^{-i(\tau _c\omega _++\beta _c)}}[(\tau _c e^{-i(\tau _c\omega _++\beta )}\nonumber \\&-\tau _c\cos \beta _c+i\tau _c\sin \beta _c)\mu _1\nonumber \\&+(\epsilon _c e^{-i(\tau _c\omega _++\beta _c)}+1\nonumber \\&-\epsilon _c\cos \beta _c)c+i(\omega _0+\epsilon _c\sin \beta _c))\mu _2\nonumber \\&+\tau _c\epsilon _c(-ie^{-i(\tau _c\omega _++\beta _c)}+\sin \beta _c+i\cos \beta _c)\mu _3],\nonumber \\ \lambda _2= & {} \frac{2}{2 + 2\epsilon _c\tau _c e^{-i(\tau _c\omega _+-\beta _c)}}[(\tau _c e^{-i(\tau _c\omega _-+\beta _c)}\nonumber \\&-\tau _c\cos \beta _c+i\tau _c\sin \beta _c)\mu _1\nonumber \\&+(\epsilon _c e^{-i(\tau _c\omega _-+\beta _c)}+1\nonumber \\&-\epsilon _c\cos \beta _c+i(\omega _0+\epsilon _c\sin \beta _c))\mu _2\nonumber \\&+\tau _c\epsilon _c(-ie^{-i(\tau _c\omega _-+\beta _c)}\nonumber \\&+\sin \beta _c+i\cos \beta _c)\mu _3]. \end{aligned}$$

(38)

together with

$$\begin{aligned} \begin{aligned} a_{11}&=-\frac{8}{2 + 2\epsilon _c\tau _c e^{-i(\tau _c\omega _++\beta _c)}}\tau _c(1+i\alpha ),~\\ a_{12}&=-\frac{16}{2 + 2\epsilon _c\tau _c e^{-(\tau _c\omega _++\beta )}}\tau _c(1+i\alpha ),\\ a_{21}&=-\frac{8}{2 + 2\epsilon _c\tau _c e^{-i(\tau _c\omega _-+\beta _c)}}\tau _c(1+i\alpha ),~\\ a_{22}&=-\frac{16}{2 + 2\epsilon _c\tau _c e^{-i(\tau _c\omega _-+\beta _c)}}\tau _c(1+i\alpha ),\\ b_{33}&= b_{44} = b_{14} = b_{23} = 0. \end{aligned} \end{aligned}$$

(39)

Make the following polar coordinates transformation

$$\begin{aligned} \begin{array}{l}z_{1}=r_{1} \cos \theta _{1}+i r_{1} \sin \theta _{1}, \\ z_{2}=r_{1} \cos \theta _{1}-i r_{1} \sin \theta _{1}, \\ z_{3}=r_{2} \cos \theta _{2}+i r_{2} \sin \theta _{2}, \\ z_{4}=r_{2} \cos \theta _{2}-i r_{2} \sin \theta _{2}.\end{array} \end{aligned}$$

Let $\overline{r_1}=r_1\sqrt{|{\text {Re}}a_{11}|}$, $\overline{r_2}=r_2\sqrt{|{\text {Re}}a_{22}|}$, ${\overline{t}}=t\epsilon _1$ with $\epsilon _1={\text {Sign}}({\text {Re}}a_{11})$, $\epsilon _2={\text {Sign}}({\text {Re}}a_{21})$, and drop the overlines, then system (35) becomes

$$\begin{aligned} \begin{array}{l}\dot{r}_{1}=r_{1}\left( c_{1}+r_{1}^{2}+b_{0} r_{2}^{2}\right) , \\ \dot{r}_{2}=r_{2}\left( c_{2}+c_{0} r_{1}^{2}+d_{0} r_{2}^{2}\right) , \end{array} \end{aligned}$$

(40)

where

$$\begin{aligned} \begin{array}{l}c_{1}{=}\epsilon _{1}{\text {Re}}\lambda _{1},~ c_{2}{=}\epsilon _{1}{\text {Re}}\lambda _2,~ b_{0}{=}\frac{\epsilon _{1} \epsilon _{2} {\text {Re}} a_{12}}{{\text {Re}} a_{21}}, ~ c_{0}{=}\frac{{\text {Re}} a_{22}}{{\text {Re}} a_{11}}, ~ d_{0}{=}\epsilon _{1} \epsilon _{2}. \end{array} \end{aligned}$$

(41)

Since the CGLE model only has cubic nonlinearity, the normal form for 1:3 resonance is same as 1:2 resonance, which can finally turn to the system(40).