Double Hopf bifurcation in nonlocal reaction-diffusion systems with spatial average kernel¹¹1This research is supported by National Natural Science Foundation of China (No 11771109).

Reaction-diffusion equations have been proposed to model the complex phenomenon in cell biology, neural network, virus dynamics, biochemical reaction, etc., see [35, 52] and references therein. However, individuals of a species at different locations may compete for common resource or communicate by chemical means [7, 16, 21], and nonlocal interactions should be considered. In 1989, Britton [7] proposed a single population model with nonlocal effect, where the nonlocal term takes the following form:

$$g*u=\int_{\Omega}g(x,y)u(y,t)\mathrm{d}y.$$

(1.1)

Here $\Omega$ is the region where the population lives, $u(x,t)$ represents the density of the species at location $x$ and time $t$. The model is based on the following two assumptions:

1. (i)

   individuals in grouping together can reduce the risk of predation, which is referred to as the aggregation mechanism;

2. (ii)

   the intraspecific competition at a certain point depends on not only the density at this point but also a weighted average in the neighborhood of this point.

For unbounded one spatial dimension domain $\Omega=(-\infty,\infty)$, Britton [7] also considered the nonlocal effects on two species competition model, and it was shown that the aggregation may lead to the coexistence of the two species. For bounded domain $\Omega$, a typical scenario of nonlocal dispersal is the “spatial average kernel”, that is,

$$g(x,y)\equiv\dfrac{1}{\mathrm{vol}\,\Omega}.$$

(1.2)

Furter and Grinfeld [21] obtained that this average kernel can induce spatial nonhomogeneous patterns even for single population model, see [51] for more general models.

There have been extensive results on the nonlocal effects, including existence and stability of solutions, traveling wave solutions, pattern formation, bifurcation analysis, etc., see [5, 6, 22, 24, 17, 38, 16, 20, 23, 43, 10, 12, 13, 11] and references therein. For unbounded one spatial dimension domain $\Omega=(-\infty,\infty)$, Merchant and Nagata [43] chose different types of kernel $g(x,y)$, and showed that the nonlocal competition could induce complex spatiotemporal patterns, see also [3, 49]. Motivated by [43], Chen et al. [13, 11] considered the case that the spatial domain $\Omega=(0,L)$, and chose the spatial average kernel, i.e., $g(x,y)\equiv 1/L$. They found that, for the nonlocal Rosenzweig-MacArthur (RM) and Holling-Tanner predator-prey models, the constant positive steady state could lose the stability when the given parameter passes through some Hopf bifurcation values, and the bifurcating periodic solutions near such values can be spatially nonhomogeneous. It is well known that Hopf bifurcation has been used to illustrate the periodic phenomena in the natural world, such as regular changes in population size, the periodic outbreak of infectious diseases, and chemical oscillations of some autocatalytic reactions [32, 44, 31]. However, for reaction-diffusion system without nonlocal effect, the bifurcating periodic solutions near the threshold are always spatially homogeneous, while nonhomogeneous periodic solutions can also occur through Hopf bifurcation, but they are always unstable, and consequently hard to simulate. Note that, nonlocal effect could also induced spatially nonhomogeneous steady states through steady state bifurcation[21, 51]. Therefore, nonlocal effect could be used as a new mechanism for pattern formation.

We point out that, for reaction-diffusion equations with nonlocal delay effect, the Hopf bifurcation has also been studied by many researchers, see [10, 12, 29, 30, 28, 55, 56, 25, 26] and references therein. For the homogeneous Neumann boundary condition, Gourley and So [25] showed the existence of Hopf bifurcation for a food-limited population model, see also [38] for the model with age structure. For the homogeneous Dirichlet boundary condition, Chen and Shi [10] developed the method proposed by Busenberg and Huang [9], and studied the existence of Hopf bifurcation near the positive spatially nonhomogeneous steady state. Different from [9, 10], Guo [29] proposed the method of Lyapunov-Schmidt reduction to show the existence and even the multiplicity of the spatially nonhomogeneous steady state, and the associated Hopf bifurcation. There are also other results on Hopf bifurcation for reaction-diffusion models with nonlocal delay, see e.g. [26, 33, 55].

The above mentioned steady state and Hopf bifurcations are both codimension-one bifucation. A natural question is that whether nonlocal effect could induce codimension-two bifurcation for reaction-diffusion equations. For the reaction-diffusion equation, the interaction of a Turing instability (leading to spatially nonhomogeneous steady states) with a Hopf bifurcation (giving rise to temporal oscillations) has been observed and studied in chemical, biological and physical systems, see [42, 41, 57, 48] and references therein. For example, Rovinsky and Menzinger [48] studied this Turing-Hopf interaction for three models of chemically active media by using Poincaré-Birkhoff method and shown the bistability of spatially nonhomogeneous steady states and homogeneous oscillations. In the framework of amplitude equation formalism, De Wit et al. [57] investigated the bifurcation scenarios near the Turing-Hopf singularity. Recently, to show an accurate dynamic classification at this singularity, Song et al.[54] applied the normal form theory proposed by Faria [18] to a general reaction-diffusion equation, and obtained a series of explicit formulas for calculating the normal forms associated with the Turing-Hopf bifurcation. This spatiotemporal dynamics induced by the Turing-Hopf bifurcation were observed in several reaction-diffusion models [53, 59, 4], see also [1, 50] for the reaction-diffusion system with delay. Another typical codimension-two bifurcation is the double Hopf bifurcation. As in the Turing-Hopf bifurcation, when the parameters vary near the threshold value, the system may exhibit rich dynamics such as periodic orbit, invariant two torus, invariant three-torus, and even chaos, see e.g. [27, 45, 15, 34, 37, 60]. Recently, for a general delayed reaction-diffusion system with time delay, Du et al. [15] also obtained an algorithm for deriving the normal form near a codimension-two double Hopf bifurcation by virtue of the the normal form method proposed by Faria [19, 18].

To our knowledge, compared to the classical reaction-diffusion system, few studies have considered the high codimensional bifurcations in nonlocal reaction-diffusion systems. Recently, Wu and Song [58] studied the dynamical classification of a nonlocal diffusive Rosenzweig-MacArthur model near the Turing-Hopf singularity. A numerical simulation [13] revealed that two Hopf bifurcation curves could intersect in a two-parameter plane. However, there exist no results on double Hopf bifurcation for nonlocal reaction-diffusion systems. In this paper, we aim to consider this problem, and consider the following general reaction-diffusion system

$$\begin{cases}\cfrac{\partial U}{\partial t}=D(\mu)\Delta U+f(\mu,U,\widehat{U}),&x\in(0,\ell\pi),\;t>0,\\ \cfrac{\partial U}{\partial x}(0,t)=\cfrac{\partial U}{\partial x}(\ell\pi,t)=0,&t>0,\\ \end{cases}$$

(1.3)

where $D(\mu)=\text{diag}(d_{1}(\mu),d_{2}(\mu),\cdots,d_{n}(\mu))$ with $d_{i}(\mu)>0$ and $\mu\in\mathbb{R}^{2}$, $U=(u_{1},u_{2},\cdots,u_{n})^{T}\in X$, $\widehat{U}=(\widehat{u}_{1},\widehat{u}_{2},\cdots,\widehat{u}_{n})^{T}$ with $\widehat{u}_{i}=\frac{1}{\ell\pi}\int_{0}^{\ell\pi}u_{i}(y,t)dy$, $i=1,2,\cdots,n$, $f=(f_{1},f_{2},\cdots,f_{n})^{T}$ with $f_{i}$ is $C^{k}(k\geq 3)$ smooth, and $f_{i}(\mu,\mathbf{0},\mathbf{0})=0$. We point out that if $n=1$, then system (1.3) is reduced to a general form in [21]. If $n=2$, then (1.3) becomes a two-component interaction system, which could model the nonlocal intraspecific and interspecific competition for population models, see [43, 2, 3, 49]. The purpose of this paper is to develop an explicit algorithm for computing normal forms on the center manifold near a codimension-two double-Hopf singularity for model (1.3). We should remark that when the ratio of two angular frequencies is some particular value, e.g. 1:2, the corresponding double-Hopf bifurcation may be codimension-three, referred to as the strong resonance case. In this article, we will not consider this case and focus only the codimension-two double Hopf bifurcation. We find that, compared with the traditional reaction-diffusion system, (1.3) is more likely to induce spatial nonhomogeneous patterns, and consequently exhibit rich dynamical behaviors at the corresponding singularity, such as spatially nonhomogeneous periodic solutions, spatially nonhomogeneous quasi-periodic solutions, coexistence of homogeneous/nonhomogeneous oscillations, and so on.

We also adopt the framework of [18] to compute the normal forms on the center manifold of system (1.3) at the codimension-two double Hopf singularity. In summary, we first rewrite system (1.3) into an abstract form, and by decomposing the phase space into center subspace and its complementary space, we obtain the equivalent system on the center manifold. Then a recursive transformation of variables is used to derive the four-dimensional normal forms. During this process, we construct a Boolean function to deal with the impact of nonlocal terms on the computation, which is the innovation. Particularly, for the case of $n=2$, we list some additional formulas in Appendix A which could help to obtain all the coefficient vectors that appear in the process of computing normal forms.

The rest of the paper is organized as follows. The decomposition of phase space and some preliminaries are given in Section 2. The computation of normal forms associated with the codimension-two double Hopf bifurcation is presented in Section 3. In Section 4, we apply our theoretical results in Section 3 to a diffusive Holling-Tanner system with spatial average kernel in prey and obtain the normal forms near the duoble-Hopf singularity. Some periodic oscillations and quasi-periodic quasi-periodic oscillations are also derived numerically in this section. Finally, we give some discussion and conclusion for this paper, and in the Appendix, we collect the details of the coefficient vectors that appear in Section 3 when $n=2$. Throughout the paper, we denote by $\mathbb{N}$ the set of positive integers, and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$ the set of non-negative integers.

In this section, we adopt the framework of [18] to compute the normal forms of the double Hopf bifurcation. To use the center manifold theory for reduction[39, 27, 32], we need rewrite system (1.3) into an abstract form and decompose the phase space.

We first define the following real-value Sobolev space

$$X:=\Big{\{}(u_{1},u_{2},\cdots,u_{n})^{{}^{T}}\in\big{[}H^{2}(0,\ell\pi)\big{]}^{n}|~{}{\partial_{x}}u_{i}(0,t)=\partial_{x}u_{i}(\ell\pi,t)=0,i=1,2,\cdots,n\Big{\}},$$

and then the linear map $u\rightarrow\frac{1}{\ell\pi}\int_{0}^{\ell\pi}u(y,t)dy$ is smooth from $H^{2}(0,\ell\pi)$ to $H^{2}(0,\ell\pi)$. Denote

$$\mathcal{F}_{i}:(\mu,U)\rightarrow f_{i}(\mu,U,\widehat{U}),~{}i=1,2,\cdots,n.\\$$

It follows from Appendix C of [32] that $\mathcal{F}_{i}$ is also smooth from $\mathbb{R}^{2}\times\left[H^{2}(0,\ell\pi)\right]^{n}$ to $H^{2}(0,\ell\pi)$. Hence, system (1.3) can be written as the following abstract form

$$\dfrac{dU(t)}{dt}=D(\mu)\Delta U+\mathcal{F}(\mu,U),$$

(2.1)

where

$$\mathcal{F}(\mu,U)=\left(\begin{array}[]{cc}\mathcal{F}_{1}(\mu,U)\\ \mathcal{F}_{2}(\mu,U)\\ ~{}~{}\vdots\\ \mathcal{F}_{n}(\mu,U)\end{array}\right)=\left(\begin{array}[]{cc}f_{1}(\mu,U,\widehat{U})\\ f_{2}(\mu,U,\widehat{U})\\ ~{}~{}\vdots\\ f_{n}(\mu,U,\widehat{U})\end{array}\right).$$

Let

$$\mathscr{L}(\mu)=D(\mu)\Delta+D_{U}\mathcal{F}(\mu,0),$$

(2.2)

where $D_{U}\mathcal{F}(\mu,0)$ stands for the Fréchet derivative of $\mathcal{F}(\mu,U)$ with respect to $U$ at $U=\mathbf{0}$. To figure out the double Hopf bifurcation with two pairs of purely imaginary eigenvalues, we define the following complexification space of $X$:

$$X_{\mathbb{C}}:=X\oplus iX=\{x_{1}+ix_{2}|~{}x_{1},x_{2}\in X\},$$

with the complex-valued $L^{2}$ inner product $\langle\cdot,\cdot\rangle$, defined by

$$\langle U,V\rangle=\int_{0}^{\ell\pi}(\bar{u}_{1}v_{1}+\bar{u}_{2}v_{2}+\cdots+\bar{u}_{n}v_{n})dx,$$

where $U=(u_{1},u_{2},\cdots,u_{n})^{T}\in X_{\mathbb{C}}$, $V=(v_{1},v_{2},\cdots,v_{n})^{T}\in X_{\mathbb{C}}$.

Considering the perturbation caused by the nonlocal terms, we rewrite system (2.1) in a intuitive form

$$\dfrac{dU(t)}{dt}=D(\mu)\Delta U+L(\mu)U+\widehat{L}(\mu)\widehat{U}+F(U,\widehat{U},\mu),$$

(2.3)

where $L$ and $\widehat{L}$ are bounded linear operators from $\mathbb{R}^{2}\times X_{\mathbb{C}}$ to $X_{\mathbb{C}}$, and $F:X_{\mathbb{C}}\times X_{\mathbb{C}}\times\mathbb{R}^{2}\rightarrow X_{\mathbb{C}}$ is a $C^{k}(k\geq 3)$ function such that $F(0,0,\mu)=0$ and $D_{U}F(0,0,\mu)=D_{\widehat{U}}F(0,0,\mu)=0$.

Then the linearization of system (2.3) at $\mathbf{0}$ takes the following form

$$\dfrac{dU(t)}{dt}=D(\mu)\Delta U+L(\mu)U+\widehat{L}(\mu)\widehat{U}.$$

(2.4)

It is well known that the eigenvalue problem

$$-\Delta\xi=\sigma\xi,~{}x\in(0,\ell\pi),~{}\xi^{\prime}(0)=\xi^{\prime}(\ell\pi)=0$$

has eigenvalues $\sigma_{n}=\frac{n^{2}}{\ell^{2}}$ ($n\in\mathbb{N}_{0}$), and the corresponding normalized eigenfunctions

$$\xi_{n}(x)=\cfrac{\cos{\frac{n}{\ell}x}}{\parallel\cos{\frac{n}{\ell}x}\parallel}=\begin{cases}\sqrt{\frac{1}{\ell\pi}},&n=0,\\ \sqrt{\frac{2}{\ell\pi}}\cos{\frac{n}{\ell}x},&n\in\mathbb{N}.\end{cases}$$

(2.5)

Letting $\beta_{n}^{i}(x)=\xi_{n}(x)e_{i}$, $i=1,2,\cdots,n$, where $e_{i}$ is the $i$th unit coordinate vector of $\mathbb{R}^{n}$, we see that $\{\beta_{n}^{i}\}_{n\in\mathbb{N}_{0}}$ are eigenfunctions of $-D(\mu)\Delta$ with corresponding eigenvalues $d_{i}(\mu)\frac{n^{2}}{\ell^{2}}$, and $\{\beta_{n}^{i}\}_{n\in\mathbb{N}_{0}}$ form an orthonormal basis of $X_{\mathbb{C}}$.

For $U\in X_{\mathbb{C}}$ and $\beta_{k}=(\beta_{k}^{1},\beta_{k}^{2},\cdots,\beta_{k}^{n})$, we define $\langle\beta_{k},U\rangle=\left(\langle\beta_{k}^{1},U\rangle,\langle\beta_{k}^{2},U\rangle,\cdots,\langle\beta_{k}^{n},U\rangle\right)^{{}^{T}}$, and denote

$$\mathcal{B}_{k}:=\text{span}\left\{~{}\langle\beta_{k}^{j},U\rangle\beta_{k}^{j}~{}|~{}U\in X_{\mathbb{C}},j=1,2,\cdots,n~{}\right\}.$$

Let

$$\varphi=\sum^{\infty}_{k=0}\xi_{k}(x)\left(\begin{array}[]{c}a_{k}^{1}\\ a_{k}^{2}\\ ~{}\vdots\\ a_{k}^{n}\end{array}\right)$$

be the eigenfunction with respect to eigenvalue $\lambda(\mu)$. Then

$$\lambda(\mu)\varphi-D(\mu)\Delta\varphi-L(\mu)\varphi-\widehat{L}(\mu)\widehat{\varphi}=0,$$

(2.6)

where $\widehat{\varphi}=\frac{1}{\ell\pi}\int_{0}^{\ell\pi}\varphi dx$. Note that

$$\dfrac{1}{\ell\pi}\int_{0}^{\ell\pi}\xi_{n}(x)dx=\begin{cases}\dfrac{1}{\sqrt{\ell\pi}},~{}&n=0,\\ ~{}~{}0,~{}&n\in\mathbb{N}.\end{cases}$$

(2.7)

Then (2.6) is equivalent to a sequence of characteristic equations:

$$\begin{cases}\text{det}\Big{(}\lambda(\mu)I-L(\mu)-\widehat{L}(\mu)\Big{)}=0,&n=0,\\ \text{det}\Big{(}\lambda(\mu)I+\dfrac{n^{2}}{\ell^{2}}D(\mu)-L(\mu)\Big{)}=0,&n\in\mathbb{N}.\\ \end{cases}$$

(2.8)

To consider the double Hopf bifurcation, we assume that there exists $\mu_{0}\in\mathbb{R}^{2}$ such that the following conditions hold:

• $\mathbf{(H_{1})}$

  There exist a neighborhood $\mathscr{N}$ of $\mu_{0}$ and $n_{1},n_{2}\in\mathbb{N}_{0}$ such that, for $\mu\in\mathscr{N}$, the linear system (2.4) has two pairs of complex simple eigenvalues $\alpha_{1}(\mu)\pm\omega_{1}(\mu)$ and $\alpha_{2}(\mu)\pm\omega_{2}(\mu)$, which are both continuously differentiable in $\mu$ with $\alpha_{1}(\mu_{0})=0$, $\omega_{1}(\mu_{0})=\omega_{1}>0$, $\alpha_{2}(\mu_{0})=0$, $\omega_{2}(\mu_{0})=\omega_{2}>0$, and all other eigenvalues of (2.4) have non-zero real parts for $\mu\in\mathscr{N}$.

• $\mathbf{(H_{2})}$

  Assume that $\omega_{1}:\omega_{2}\neq i:j$ for $i,j\in\mathbb{N}$ and $1\leq i\leq j\leq 4$, i.e., we only consider the codimension-two double Hopf bifurcation of non-resonance and weak resonance instead of the codimension-three of strongly resonant case.

• $\mathbf{(H_{3})}$

  The conjugate eigenvalues $\alpha_{k}(\mu)\pm\omega_{k}(\mu)$ are obtained by (2.8)${}_{n_{k}}$, and the corresponding eigenvalues belong to $\mathcal{B}_{n_{k}}$ for $k=1,2$. Without lose of generality, we assume $n_{1}\leq n_{2}$.

Let $\mu=\mu_{0}+\alpha$, where $\alpha=(\alpha_{1},\alpha_{2})\in\mathbb{R}^{2}$, and then (2.3) can be transformed as

$$\dfrac{dU(t)}{dt}=D_{0}\Delta U+L_{0}U+\widehat{L}_{0}\widehat{U}+\widetilde{F}(U,\widehat{U},\alpha),$$

(2.9)

where $D_{0}=D(\mu_{0})$, $L_{0}=L(\mu_{0})$, $\widehat{L}_{0}=\widehat{L}(\mu_{0})$, and $\widetilde{F}(U,\widehat{U},\alpha)=[D(\alpha+\mu_{0})-D_{0}]\Delta U+[L(\alpha+\mu_{0})-L_{0}]U+[\widehat{L}(\alpha+\mu_{0})-\widehat{L}_{0}]\widehat{U}+F(U,\widehat{U},\alpha+\mu_{0})$. Then the linear system of (2.9) on $\mathcal{B}_{n_{k}}$ is equivalent to the following ODEs on $\mathbb{C}^{n}$:

$$\dot{z}(t)=A_{n_{k}}z(t),$$

(2.10)

where $A_{n_{k}}$ is an $n\times n$ matrix, and

$$A_{n_{k}}y(t)=\begin{cases}~{}L_{0}y(t)+\widehat{L}_{0}y(t),&n_{k}=0,\\ -\dfrac{n_{k}^{2}}{\ell^{2}}D_{0}y(t)+L_{0}y(t),&n_{k}\in\mathbb{N}.\end{cases}$$

Denote by $A_{n_{k}}^{*}$ the formal adjoint of $A_{n_{k}}$ under the scalar product on $\mathbb{C}^{n}$:

$$(\psi,\phi)_{{}_{\mathbb{C}^{n}}}=\overline{\psi}\,\phi,~{}\text{for}~{}\psi^{{}^{T}},\phi\in\mathbb{C}^{n}.$$

Let $\Lambda=\{\pm i\omega_{1},\pm i\omega_{2}\}$ and let

$$\begin{array}[]{ll}\Phi_{1}=(\phi_{1},\phi_{2}),~{}\Phi_{2}=(\phi_{3},\phi_{4}),~{}\Psi_{1}=\left(\begin{array}[]{cc}\psi_{1}\\ \psi_{2}\end{array}\right),~{}\Psi_{2}=\left(\begin{array}[]{cc}\psi_{3}\\ \psi_{4}\end{array}\right),\end{array}$$

be the basis of the generalized eigenspace of $A_{n_{k}}$ and $A_{n_{k}}^{*}$ corresponding to the eigenvalues $\Lambda$, respectively. Then

$$A_{n_{k}}\Phi_{k}=\Phi_{k}B_{k},~{}A_{n_{k}}^{*}\Psi_{k}=\bar{B}_{k}\Psi_{k},~{}(\Psi_{k},\Phi_{k})_{{}_{\mathbb{C}^{n}}}=I_{2},~{}k=1,2,$$

(2.11)

where $B_{1}=\text{diag}(i\omega_{1},-i\omega_{1})$, $B_{2}=\text{diag}(i\omega_{2},-i\omega_{2})$, and $I_{2}$ is an $2\times 2$ identity matrix. Then we can decompose the phase space $X_{\mathbb{C}}$:

$$X_{\mathbb{C}}=P\oplus\text{Ker}\pi,$$

(2.12)

where $P=\text{Im}\pi$, and $\pi:X_{\mathbb{C}}\rightarrow P$ is the projection, defined by

$$\pi(U)=\sum_{k=1}^{2}\Phi_{k}\left(\Psi_{k},\langle\beta_{n_{k}},U\rangle\right)_{{}_{\mathbb{C}^{n}}}\xi_{n_{k}}.$$

Therefore, $U\in X_{\mathbb{C}}$ can be rewritten in the following form:

$$\begin{array}[]{ll}U&=\sum_{k=1}^{2}(\Phi_{k}\tilde{z}_{k}(t))\xi_{n_{k}}+w\\ &=\Phi z^{x}+w,\end{array}$$

(2.13)

where $\tilde{z}_{k}(t)=\left(\Psi_{k},\langle\beta_{n_{k}},U\rangle\right)_{{}_{\mathbb{C}^{n}}}\in\mathbb{C}^{2}$, $\Phi=(\Phi_{1},\Phi_{2})$, $z^{x}=(z_{1}\xi_{n_{1}},z_{2}\xi_{n_{1}},z_{3}\xi_{n_{2}},z_{4}\xi_{n_{2}})^{T}$, and $w\in\text{Ker}\pi$. For simplification of notations, we denote $z(t)=\text{col}(\tilde{z}_{1}(t),\tilde{z}_{2}(t))=(z_{1}(t),z_{2}(t),z_{3}(t),z_{4}(t))^{{}^{T}}\in\mathbb{C}^{4}$ and

$$\widetilde{F}(z,w,\widehat{w},\alpha)=\widetilde{F}\left(\sum_{k=1}^{2}(\Phi_{k}\tilde{z}_{k}(t))\xi_{n_{k}}+w,\frac{1}{\ell\pi}\int_{0}^{\ell\pi}\left(\sum_{k=1}^{2}(\Phi_{k}\tilde{z}_{k}(t))\xi_{n_{k}}+w\right)dx,\alpha\right).$$

(2.14)

In the following, we will also use the symbol $(z,w,\widehat{w},\alpha)$ instead of $(U,\widehat{U},\alpha)$. Now system (2.9) is equivalent to the following abstract ODEs in $\mathbb{C}^{4}\times\mathrm{Ker}\pi$

$$\begin{cases}~{}\dot{z}~{}=~{}Bz+\bar{\Psi}\left(\begin{array}[]{cc}\big{\langle}\beta_{n_{1}},\widetilde{F}(z,w,\widehat{w},\alpha)\big{\rangle}\\ \big{\langle}\beta_{n_{2}},\widetilde{F}(z,w,\widehat{w},\alpha)\big{\rangle}\end{array}\right),\\ \dfrac{d}{dt}w=\mathscr{L}_{1}w+(I-\pi)\widetilde{F}(z,w,\widehat{w},\alpha),\end{cases}$$

(2.15)

where $B=\text{diag}(B_{1},B_{2})$, $\Psi=\text{diag}(\Psi_{1},\Psi_{2})$, and $\mathscr{L}_{1}$ is the restriction of $\mathscr{L}(\mu_{0})$ on $\text{Ker}\pi$.