Multiple time delay induced Hopf bifurcation of a cortex - basal ganglia model for Parkinson’s Disease

Abstract

Exploring the origin of beta - band oscillation in the cortex - basal ganglia model plays an important role in understanding the mechanism of Parkinson’s disease. In this paper, we investigate the effect of three synaptic transmission time delays in the subthalamic nucleus(STN) - the globus pallidus external segment(GPe) loop, the excitatory neurons in the cortex(EXN) - the inhibitory neurons in the cortex(INN) loop and EXN - STN loop on critical conditions of occurrence of beta - band oscillation through Hopf bifurcation theory including linear stability analysis, center manifold theorem and normal form analysis. Our results reveal that suitable transmission time delay can induce beta - band oscillation through Hopf bifurcation, and the critical condition under which Hopf bifurcation occurs is more sensitive to the transmission time delay in EXN - STN loop $T_3$, where if $T_3>0.00185$, beta - band oscillation always occurs for any transmission time delay in STN - GPe, EXN - INN loops. Our theoretical analyses provide some ideas for the future research of Parkinson’s disease.

Introduction

Parkinson’s disease(PD) is the second most common neurodegenerative disorder with motor dysfunction including rest tremor, motor retardation and nonmotor impairment such as mood and sleep disorders, cognitive decline and incontinence, which seriously affect the patients’s normal life and put great burdens on families and society (Parkinson 2002; Aarsland et al. 2021). These symptoms of PD are related to the decrease in the dopamine level in striatum, which caused by the loss of dopaminergic neurons in the substantia nigra pars compacta (SNc) in midbrain (Jankovic 2008). The lack of dopamine (DA) in the SNc induce the excessive beta - band (13 Hz - 35 Hz) oscillation activity in the cortex and basal ganglia of Parkinson’s patients with severe motor dysfunction symptoms (Leventhal et al. 2012; McGregor and Nelson 2019), which can be alleviated by suppressing these oscillations via drugs and surgical treatments (Holt and Netoff 2014). Therefore, analyzing the origin of beta - band oscillation in PD plays an important role in understanding PD pathogenesis and has attracted more attention of theoretical researchers, who carry out dynamical analyses on the mathematical models.

A number of mathematical models are established to explore the conditions for generating beta - band oscillation in PD (Hu et al. (2022); Muddapu and Chakravarthy (2020); Shouno et al. (2017); Muddapu et al. (2019)). The excitatory - inhibitory loop between the subthalamic nucleus(STN) and external segment of the globus pallidus(GPe) is a core one in these models, in which STN consisting of the glutamatergic neurons sends excitatory projection to GPe while GPe composing the gammaaminobutyric acid neurons exerts inhibitory projections to STN (Holgado et al. 2010; Pavlides et al. 2012). The mean - field models consisting of STN - GPe networks are developed to obtain the conditions of beta - band oscillation (Holgado et al. 2010). In addition to STN and GPe, the cortex neurons projecting to STN play an important role in regulating beta - band oscillation according to experiments results (Bevan et al. 2006). A cortex - basal ganglia model with excitatory and inhibitory neurons in the cortex is proposed by Pavlides et al. to verify the beta - band oscillation in the experimental observation in Ref. Tachibana et al. (2011). These models have been put forward to obtain the boundary conditions of the beta - band oscillation through analyzing the conditions under which Hopf bifurcation occurs in these models.

Analyzing the conditions under which Hopf bifurcation occurs in cortex - basal ganglia models play an important role in understanding the mechanism of beta - band oscillation in PD. Hopf bifurcation may make a stable steady state lose stability and induce the appearance of a stable limit cycle corresponding to oscillation state. In the aspect of numerical simulations, Ref. Nevado-Holgado et al. (2011) obtains the conditions for the onset of Hopf bifurcation when the connection weights in STN - GPe are linearly increased from the healthy to the parkinsonian parameters and the effect of the connection weights, synaptic transmission time delay and time constant on Hopf bifurcation are explored in GPe - Gpe model in Ref. Chen et al. (2023). On the theoretical side, the conditions for the occurrence of Hopf bifurcation are analyzed with identical synaptic transmission time delays in a cortex - basal ganglia network through normal form theory and central manifold theorem (Chen et al. 2020) . Besides, critical conditions of Hopf bifurcation induced by two heterogeneous synaptic time delays in EXN - INN and STN - GPe loops are derived in an improved cortex - basal ganglia network with additional self - feedback projections in the EXN and INN (Wang et al. 2022). Also, the critical stability boundaries separating stable and oscillatory neural firing are obtained in terms of connection weights and two synaptic transmission time delays in a STN - GP(globus pallidus) network simplifying from a STN - GP one with three different synaptic time delays through time - shift transformation (Rahman et al. 2018). The synaptic transmission is a biological process in which a neuron sends projection to a target neuron across synapse with different time delays between different neurons (Rahman et al. 2018). The effect of different time delays between different neuronal populations in cortex - basal ganglia on the mechanism of beta - band oscillation are worth further exploration through analyzing the conditions for generating Hopf bifurcation.

Motivated by the above idea, in this paper, we explore the effect of three synaptic transmission time delays in the connections between GPe and STN($T_1$), between EXN and INN($T_2$) and between EXN in the cortex and basal ganglia($T_3$) on Hopf bifurcation for four cases. In each case, the conditions for the beta - band oscillations in the model are obtained using stability theory through analyzing the characteristic equation of the linearized system. The direction of Hopf bifurcation is judged through the center manifold theorem and normal form method. These theoretical results are further confirmed by numerical simulations. Our results reveal the important role of multiple time delays on beta - band oscillation and enhance the understanding of beta-band oscillation dynamics. This research may help us provide some clue for the treatment of PD.

This paper is organized as follows. The cortex - basal ganglia model is given in Sect. “The model”. Sect.Hopf bifurcation induced by multiple time    delay through theoretical analyses” presents the sufficient conditions for the existence of Hopf bifurcation in four cases. Then numerical simulations verify these theoretical results in Sect. “Hopf bifurcation induced by multiple time delay through numerical simulations”. Finally, we conclude the paper in Sect. “Conclusion”.

The model

Figure 1 shows the schematic framework of the cortex - basal ganglia model in Ref. Pavlides et al. (2015); Wang et al. (2023) with EXN and INN in the cortex and STN and GPe in basal ganglia. The excitatory and inhibitory projections between these neurons are represented by the black lines ended with the arrow and the bar, respectively. EXN exerts excitatory projections to the INN and STN, while INN and GPe exert inhibitory projections to EXN and STN, respectively. STN projects a direct excitatory connection to GPe and an indirect inhibitory connection to EXN via thalamus (Wang et al. 2023). C denotes a constant excitatory input to the cortex and Str represents inhibitory projection from the striatum. ${\omega }_{\mathit{EI}}$, ${\omega }_{\mathit{IE}}$, ${\omega }_{\mathit{CS}}$, ${\omega }_{\mathit{SC}}$, ${\omega }_{\mathit{GS}}$, ${\omega }_{\mathit{SG}}$ are connection weights between these neuron populations and $T_{\mathit{EI}}$, $T_{\mathit{IE}}$, $T_{\mathit{CS}}$, $T_{\mathit{SC}}$, $T_{\mathit{GS}}$, $T_{\mathit{SG}}$ are synaptic transmission time delays.

Fig. 1.

The improved cortex - basal ganglia model

The mathematical model describing the dynamics of the firing rate of STN, GPe, EXN, INN are given as follows,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\tau }_S\frac{dS(t)}{\mathit{dt}}=F_S(-{\omega }_{\mathit{GS}}G(t-T_{\mathit{GS}})+{\omega }_{\mathit{CS}}E(t-T_{\mathit{CS}}))-S(t),\hfill \\ \hfill & {\tau }_G\frac{dG(t)}{\mathit{dt}}=F_G({\omega }_{\mathit{SG}}S(t-T_{\mathit{SG}})-Str)-G(t),\hfill \\ \hfill & {\tau }_E\frac{dE(t)}{\mathit{dt}}=F_E(-{\omega }_{\mathit{SC}}S(t-T_{\mathit{SC}})-{\omega }_{\mathit{IE}}I(t-T_{\mathit{IE}})+C)-E(t),\hfill \\ \hfill & {\tau }_I\frac{dI(t)}{\mathit{dt}}=F_I({\omega }_{\mathit{EI}}E(t-T_{\mathit{EI}}))-I(t).\hfill \end{array}\right)\end{array}$$ 1

where S(t), G(t), E(t), I(t) are the firing rates of STN, GPe, EXI, INN. ${\tau }_X(X=S,G,E,I)$ represent the membrane time constants for neuron population X. ${\omega }_{\mathit{XY}}$ and $T_{\mathit{XY}}$ represent the connection weights and transmission time delays from neuron populations X to neuron populations Y, respectively. C represents the constant excitatory input to cortical excitatory neurons and Str denotes the constant inhibitory input to GPe from striatum. The activation function $F_X(·)(X=S,G,E,I)$ characterizing the effect of synaptic input on firing rate of the neuron populations X are given as follows,

$$\begin{array}{c}\hfill F_X(in)=\frac{M_X}{1+(\frac{M_X-B_X}{B_X})\mathit{exp}(\frac{-4in}{M_X})},(X=S,G,E,I)\end{array}$$ 2

Where $M_X$ represents the maximum firing rate of the neuron populations X and $B_X$ represents the firing rate of the neuron populations X without external input.

The value of all parameters in this paper are shown in Table 1, which are adapted from Ref. Chen et al. (2020); Yan and Wang (2017); Wang et al. (2023, 2022). Here, ${\tau }_X(X=S,G,E,I)$ are the same for simplicity.

Table 1.

Parameter values used in the paper

Parameter	Value	Parameter	Value
${\tau }_S$	10 ms	$M_S$	300 spk/s
${\tau }_G$	10 ms	$B_S$	17 spk/s
${\tau }_E$	10 ms	$M_G$	400 spk/s
${\tau }_I$	10 ms	$B_G$	75 spk/s
${\omega }_{\mathit{GS}}$	3.22	$M_E$	71.77 spk/s
${\omega }_{\mathit{CS}}$	6.6	$B_E$	3.62 spk/s
${\omega }_{\mathit{SG}}$	2.56	$M_I$	277.39 spk/s
${\omega }_{\mathit{EI}}$	1.56	$B_I$	9.87 spk/s
${\omega }_{\mathit{IE}}$	1.56	Str	40.51 spk/s
${\omega }_{\mathit{SC}}$	4	C	172.18 spk/s

Based on the fact that the time delays $T_{\mathit{EI}}$, $T_{\mathit{IE}}$, $T_{\mathit{CS}}$, $T_{\mathit{SC}}$, $T_{\mathit{GS}}$, $T_{\mathit{SG}}$ play crucial roles in generating beta - band oscillations in the cortex - basal ganglia circuit, we will explore the effect of these transmission time delays on beta - band oscillations through Hopf bifurcation theory in the following section.

Hopf bifurcation induced by multiple time delay through theoretical analyses

In this section, the critical condition under which Hopf bifurcation occurs in the cortex - basal ganglia model (1) with multiple time delay are analyzed through quasilinearization method, center manifold and normal form (Wang et al. 2022; Eugeni et al. 2018). Here, the cortex - basal ganglia model includes three excitatory - inhibitory loop STN - EXN, STN - GPE and EXN - INN, transmission time delays between these neurons in each loop are assumed to be the same $T_{\mathit{GS}}=T_{\mathit{SG}}=T_1$, $T_{\mathit{EI}}=T_{\mathit{IE}}=T_2$, $T_{\mathit{CS}}=T_{\mathit{SC}}=T_3$ for facilitating mathematical analysis. Then the model (1) becomes the following form,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& \tau \frac{dS(t)}{\mathit{dt}}=F_S(-{\omega }_{\mathit{GS}}G(t-T_1)+{\omega }_{\mathit{CS}}E(t-T_3))-S(t),\hfill \\ \hfill & \tau \frac{dG(t)}{\mathit{dt}}=F_G({\omega }_{\mathit{SG}}S(t-T_1)-Str)-G(t),\hfill \\ \hfill & \tau \frac{dE(t)}{\mathit{dt}}=F_E(-{\omega }_{\mathit{SC}}S(t-T_3)-{\omega }_{\mathit{IE}}I(t-T_2)+C)-E(t),\hfill \\ \hfill & \tau \frac{dI(t)}{\mathit{dt}}=F_I({\omega }_{\mathit{EI}}E(t-T_2))-I(t).\hfill \end{array}\right)\end{array}$$ 3

Let $u_0=(S^{\ast },G^{\ast },E^{\ast },I^{\ast })$ be the equilibrium of the system (3) and expand the system (3) at the equilibrium $u_0$ as follows,

$$\begin{array}{cc}\hfill \frac{du(t)}{\mathit{dt}}=& C_{1}u(t)+C_{2}u(t-T_1)+C_{3}u(t-T_2)+C_{4}u(t-T_3)+f(u(t-T_1),\hfill \\ \hfill & u(t-T_2),u(t-T_3)),\hfill \end{array}$$ 4

where $u(t)=(S(t)-S^{\ast },G(t)-G^{\ast },E(t)-E^{\ast },I(t)-I^{\ast })$,

$$\begin{array}{cc}& C_1=\left[\begin{array}{cccc}-\frac{1}{\tau }& 0& 0& 0\\ 0& -\frac{1}{\tau }& 0& 0\\ 0& 0& -\frac{1}{\tau }& 0\\ 0& 0& 0& -\frac{1}{\tau }\end{array}\right],C_2=\left[\begin{array}{cccc}0& a_{12}& 0& 0\\ a_{21}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],\hfill \\ \hfill & C_3=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& a_{34}\\ 0& 0& a_{43}& 0\end{array}\right],C_4=\left[\begin{array}{cccc}0& 0& a_{13}& 0\\ 0& 0& 0& 0\\ a_{31}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],\hfill \end{array}$$ 5

with

$$\begin{array}{cc}& a_{12}=\frac{4{\omega }_{\mathit{GS}}{S}^{\ast 2}}{\tau M_S^2}-\frac{4{\omega }_{\mathit{GS}}{S}^{\ast }}{\tau M_S},\hfill \\ \hfill & a_{13}=-\frac{4{\omega }_{\mathit{CS}}{S}^{\ast 2}}{\tau M_S^2}+\frac{4{\omega }_{\mathit{CS}}{S}^{\ast }}{\tau M_S},\hfill \\ \hfill & a_{21}=-\frac{4{\omega }_{\mathit{SG}}{G}^{\ast 2}}{\tau M_G^2}+\frac{4{\omega }_{\mathit{SG}}{G}^{\ast }}{\tau M_G},\hfill \\ \hfill & a_{31}=\frac{4{\omega }_{\mathit{SC}}{E}^{\ast 2}}{\tau M_E^2}-\frac{4{\omega }_{\mathit{SC}}{E}^{\ast }}{\tau M_E},\hfill \\ \hfill & a_{34}=\frac{4{\omega }_{\mathit{IE}}{E}^{\ast 2}}{\tau M_E^2}-\frac{4{\omega }_{\mathit{IE}}{E}^{\ast }}{\tau M_E},\hfill \\ \hfill & a_{43}=-\frac{4{\omega }_{\mathit{EI}}{I}^{\ast 2}}{\tau M_I^2}+\frac{4{\omega }_{\mathit{EI}}{I}^{\ast }}{\tau M_I},\hfill \end{array}$$ 6

where $f(u(t-T_1),u(t-T_2),u(t-T_3))$ is a higher order term.

The characteristic equation of the Eq. (4) is given as follows,

$$\begin{array}{c}\hfill \begin{array}{cc}& \mathrm{\Delta }(\lambda ,T_1,T_2,T_3)=\left|\begin{array}{cccc}\lambda +\frac{1}{\tau }& -a_{12}{e}^{-\lambda T_1}& -a_{13}{e}^{-\lambda T_3}& 0\\ -a_{21}{e}^{-\lambda T_1}& \lambda +\frac{1}{\tau }& 0& 0\\ -a_{31}{e}^{-\lambda T_3}& 0& \lambda +\frac{1}{\tau }& -a_{34}{e}^{-\lambda T_2}\\ 0& 0& -a_{43}{e}^{-\lambda T_2}& \lambda +\frac{1}{\tau }\end{array}\right|\hfill \\ \hfill & ={(\lambda +\frac{1}{\tau })}^4-{(\lambda +\frac{1}{\tau })}^2(a_{12}{a}_{21}{e}^{-2\lambda T_1}+a_{34}{a}_{43}{e}^{-2\lambda T_2}+a_{13}{a}_{31}{e}^{-2\lambda T_3})\hfill \\ \hfill & +a_{12}{a}_{21}{a}_{34}{a}_{43}{e}^{-2\lambda T_1}{e}^{-2\lambda T_2}=0.\hfill \end{array}\end{array}$$ 7

Here, Lemma1 used in the following analyses is given as follows,

Lemma 1

Ruan and Wei (2003) For the follow exponential polynomial:

$P(\lambda ,e^{-\lambda T_1},...,e^{-\lambda T_m})={\lambda }^n+k_1^0{\lambda }^{n-1}+...+k_{n-1}^0\lambda +k_n^0+[k_1^1{\lambda }^{n-1}+...+k_{n-1}^1\lambda +k_n^1]e^{-\lambda T_1}+...+[k_1^m{\lambda }^{n-1}+...+k_{n-1}^m\lambda +k_n^m]e^{-\lambda T_m}$

where $T_j\ge 0(j=1,2,...,m)$ and $k_i^j(i=1,2,...,n;j=1,2,...,m)$ are constants. As $(T_1,T_2,...,T_m)$ change, the sum of the orders of the zeros of $P(\lambda ,e^{-\lambda T_1},...,e^{-\lambda T_m})$ in the open right half plane can change only if a zero appears on or crosses the imaginary axis.

In order to explore the effect of transmission time delays $T_1$, $T_2$ and $T_3$ on beta - band oscillations, we investigate the local stability of positive equilibrium $u_0=(S^{\ast },G^{\ast },E^{\ast },I^{\ast })$ of the system (3) and the existence of Hopf bifurcation for four case (1) $T_1=T_2=T_3=0$ (2) $T_1=T_2=0$, $T_3>0$ (3) $T_2=0$, $T_1>0$, $T_3>0$ (4) $T_1>0$, $T_3>0$,$T_2>0$.

Case 1 $T_1=T_2=T_3=0$.

For $T_1=T_2=T_3=0$, the characteristic equation (7) becomes

$$\begin{array}{c}\hfill \begin{array}{cc}& \mathrm{\Delta }(\lambda ,0,0,0)\hfill \\ \hfill & ={(\lambda +\frac{1}{\tau })}^4-{(\lambda +\frac{1}{\tau })}^2(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31})+a_{12}{a}_{21}{a}_{34}{a}_{43}\hfill \\ \hfill & ={\lambda }^4+b_1{\lambda }^3+b_2{\lambda }^2+b_3\lambda +b_4\hfill \\ \hfill & =0,\hfill \end{array}\end{array}$$ 8

where

$$\begin{array}{cc}& b_1=\frac{4}{\tau },\hfill \\ \hfill & b_2=\frac{6}{{\tau }^2}-(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}),\hfill \\ \hfill & b_3=\frac{4}{{\tau }^3}-\frac{2}{\tau }(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}),\hfill \\ \hfill & b_4=\frac{1}{{\tau }^4}-\frac{1}{{\tau }^2}(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31})+a_{12}{a}_{21}{a}_{34}{a}_{43}.\hfill \end{array}$$ 9

According to Routh - Hurwitz Lemma (DeJesus and Kaufman (1987)), the conditions of local stability of $u_0=(S^{\ast },G^{\ast },E^{\ast },I^{\ast })$ are given in Theorem 1.

Theorem 1

All roots of Eq. (7) have negative real parts and $u_0=(S^{\ast },G^{\ast },E^{\ast },I^{\ast })$ is locally asymptotically stable if and only if the following conditions ($H_1$) is satisfied. $(H_1)$:

$$\begin{array}{cc}& b_i>0(i=1,...,4),\hfill \\ \hfill & b_1{b}_2>b_3,\hfill \\ \hfill & b_1{b}_2{b}_3>b_1^2{b}_4+b_3^2.\hfill \end{array}$$ 10

Case 2 $T_1=T_2=0,T_3>0$

For $T_1=T_2=0,T_3>0$, the characteristic equation (7) becomes

$$\begin{array}{c}\hfill \begin{array}{cc}& \mathrm{\Delta }(\lambda ,0,0,T_3)\hfill \\ \hfill & ={(\lambda +\frac{1}{\tau })}^4-{(\lambda +\frac{1}{\tau })}^2(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}{e}^{-2\lambda T_3})+a_{12}{a}_{21}{a}_{34}{a}_{43}\hfill \\ \hfill & =0.\hfill \end{array}\end{array}$$ 11

Suppose that $\lambda =i\omega (\omega >0)$ is a root of Eq. (11), so

$$\begin{array}{c}\hfill \begin{array}{cc}& {(i\omega +\frac{1}{\tau })}^4-{(i\omega +\frac{1}{\tau })}^2(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}{e}^{-2i\omega T_3})+a_{12}{a}_{21}{a}_{34}{a}_{43}\hfill \\ \hfill & =(\frac{1}{{\tau }^4}+{\omega }^4-\frac{6{\omega }^2}{{\tau }^2})+i(\frac{4\omega }{{\tau }^3}-\frac{4{\omega }^3}{\tau })\hfill \\ \hfill & -(\frac{1}{{\tau }^2}-{\omega }^2+i\frac{2\omega }{\tau })[a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}(cos(2\omega T_3)-isin(2\omega T_3))]\hfill \\ \hfill & +a_{12}{a}_{21}{a}_{34}{a}_{43}\hfill \\ \hfill & =0.\hfill \end{array}\end{array}$$ 12

Separating the real part and the imaginary part of Eq. (12), we get

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& (\frac{1}{{\tau }^4}+{\omega }^4-\frac{6{\omega }^2}{{\tau }^2})-(\frac{1}{{\tau }^2}-{\omega }^2)(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}cos(2\omega T_3))-a_{13}{a}_{31}\frac{2\omega }{\tau }sin(2\omega T_3)\hfill \\ \hfill & +a_{12}{a}_{21}{a}_{34}{a}_{43}=0,\hfill \\ \hfill & \frac{4\omega }{{\tau }^3}-\frac{4{\omega }^3}{\tau }-\frac{2\omega }{\tau }(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}cos(2\omega T_3))+(\frac{1}{{\tau }^2}-{\omega }^2)a_{13}{a}_{31}sin(2\omega T_3)=0.\hfill \end{array}\right)\end{array}$$ 13

Then, we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& sin(2\omega T_3)=\frac{c_1{\omega }^5+c_2{\omega }^3+c_3\omega }{c_4{\omega }^4+c_5{\omega }^2+c_6},\hfill \\ \hfill & cos(2\omega T_3)=\frac{c_7{\omega }^6+c_8{\omega }^4+c_9{\omega }^2+c_{10}}{c_4{\omega }^4+c_5{\omega }^2+c_6},\hfill \end{array}\right)\end{array}$$ 14

where

$$\begin{array}{cc}& c_1=\frac{2a_{13}{a}_{31}}{\tau },\hfill \\ \hfill & c_2=\frac{4a_{13}{a}_{31}}{{\tau }^3},\hfill \\ \hfill & c_3=\frac{2a_{13}{a}_{31}}{{\tau }^5}-\frac{2a_{12}{a}_{21}{a}_{13}{a}_{31}{a}_{34}{a}_{43}}{\tau },\hfill \\ \hfill & c_4=-a_{13}^2{a}_{31}^2,\hfill \\ \hfill & c_5=\frac{-2a_{13}^2{a}_{31}^2}{{\tau }^2},\hfill \\ \hfill & c_6=\frac{-a_{13}^2{a}_{31}^2}{{\tau }^4},\hfill \\ \hfill & c_7=a_{13}{a}_{31},\hfill \\ \hfill & c_8=\frac{4a_{13}{a}_{31}}{{\tau }^2}+a_{13}{a}_{31}(a_{12}{a}_{21}+a_{34}{a}_{43}),\hfill \\ \hfill & c_9=\frac{2a_{13}{a}_{31}(a_{12}{a}_{21}+a_{34}{a}_{43})}{{\tau }^2}-\frac{a_{13}{a}_{31}}{{\tau }^4}+a_{12}{a}_{21}{a}_{13}{a}_{31}{a}_{34}{a}_{43},\hfill \\ \hfill & c_{10}=\frac{a_{13}{a}_{31}(a_{12}{a}_{21}+a_{34}{a}_{43})}{{\tau }^4}-\frac{a_{13}{a}_{31}}{{\tau }^6}-\frac{a_{12}{a}_{21}{a}_{13}{a}_{31}{a}_{34}{a}_{43}}{{\tau }^2}.\hfill \end{array}$$ 15

According to Eq. (14) and ${sin}^2(2\omega T_3)+{cos}^2(2\omega T_3)=1$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill d_1{\omega }^{12}+d_2{\omega }^{10}+d_3{\omega }^8+d_4{\omega }^6+d_5{\omega }^4+d_6{\omega }^2+d_7=0,\end{array}\end{array}$$ 16

where

$$\begin{array}{cc}& d_1=c_7^2,\hfill \\ \hfill & d_2=c_1^2+2c_7{c}_8,\hfill \\ \hfill & d_3=2c_1{c}_2+2c_7{c}_9+c_8^2-c_4^2,\hfill \\ \hfill & d_4=2c_1{c}_3+c_2^2+2c_7{c}_{10}+2c_8{c}_9-2c_4{c}_5,\hfill \\ \hfill & d_5=2c_2{c}_3+2c_8{c}_9+c_9^2-2c_4{c}_6-c_5^2,\hfill \\ \hfill & d_6=c_3^2+2c_9{c}_{10}-2c_5{c}_6,\hfill \\ \hfill & d_7=c_{10}^2-c_6^2.\hfill \end{array}$$ 17

Let $z={\omega }^2$, then Eq. (16) becomes

$$\begin{array}{c}\hfill \begin{array}{c}\hfill d_1{z}^6+d_2{z}^5+d_3{z}^4+d_4{z}^3+d_5{z}^2+d_{6}z+d_7=0.\end{array}\end{array}$$ 18

Here, we put forward the assumption

($H_2$): Eq. (18) has at least one positive root.

Without loss of generality, we assume that Eq. (18) has six positive root $z_k(k=1,...,N,N\le 6)$. Then the roots of Eq. (16) can be expressed as ${\omega }_k=\sqrt{z_k}$. According to Eq. (14), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_{3k}^n& =\frac{1}{2{\omega }_k}arccos\frac{c_7{\omega }_k^6+c_8{\omega }_k^4+c_9{\omega }_k^2+c_{10}}{c_4{\omega }_k^4+c_5{\omega }_k^2+c_6}+\frac{n\pi }{{\omega }_k},\hfill \\ \hfill & (k=1,...,N;n=0,1,...).\hfill \end{array}\end{array}$$ 19

Let $T_3^0=\underset{k=1,...,N}{\mathit{min}}\{T_{3k}^n\}$ be the critical transmission time delay corresponding to pure imaginary root $\pm i{\omega }_0^{(3)}$.

Next, we will prove the transversality condition with respect to time delay $T_3$.

Differentiating the two sides of Eq. (11) about time delay $T_3$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {(\frac{\text{d}\lambda }{\text{d}{T}_3})}^{-1}\hfill \\ \hfill & =-\frac{4{(\lambda +\frac{1}{\tau })}^3-2(\lambda +\frac{1}{\tau })(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}{e}^{-2\lambda T_3})+2a_{13}{a}_{31}{T}_3{(\lambda +\frac{1}{\tau })}^2{e}^{-2\lambda T_3}}{2a_{13}{a}_{31}\lambda {(\lambda +\frac{1}{\tau })}^2{e}^{-2\lambda T_3}}\hfill \\ \hfill & =-\frac{P_1}{Q_1}.\hfill \end{array}\end{array}$$ 20

Therefore, $Re\{{(\frac{\text{d}\lambda }{\text{d}{T}_3})}_{\lambda =i{\omega }_0^{(3)}}^{-1}\}=-\frac{P_{1R}{Q}_{1R}+P_{1I}{Q}_{1I}}{Q_{1R}^2+Q_{1I}^2}$ , where $P_{1R}(Q_{1R})$ and $P_{1I}(Q_{1I})$ represent the real and imaginary parts of $P_1(Q_1)$, respectively.

$$\begin{array}{cc}& P_{1R}=4(\frac{1}{{\tau }^3}-\frac{3{({\omega }_0^{(3)})}^2}{\tau })-2[\frac{a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}cos(2{\omega }_0^{(3)}{T}_3)}{\tau }+a_{13}{a}_{31}{\omega }_0^{(3)}sin(2{\omega }_0^{(3)}{T}_3)]\hfill \\ \hfill & +2a_{13}{a}_{31}{T}_3[(\frac{1}{{\tau }^2}-{({\omega }_0^{(3)})}^2)cos(2{\omega }_0^{(3)}{T}_3)+\frac{2{\omega }_0^{(3)}}{\tau }sin(2{\omega }_0^{(3)}{T}_3)],\hfill \\ \hfill & P_{1I}=4(\frac{3{\omega }_0^{(3)}}{{\tau }^2}-{({\omega }_0^{(3)})}^3)-2[{\omega }_0^{(3)}(a_{12}{a}_{21}+a_{34}{a}_{43}+a_{13}{a}_{31}cos(2{\omega }_0^{(3)}{T}_3))-\frac{a_{13}{a}_{31}sin(2{\omega }_0^{(3)}{T}_3)}{\tau }]\hfill \\ \hfill & +2a_{13}{a}_{31}{T}_3[\frac{2{\omega }_0^{(3)}}{\tau }cos(2{\omega }_0^{(3)}{T}_3)+({({\omega }_0^{(3)})}^2-\frac{1}{{\tau }^2})sin(2{\omega }_0^{(3)}{T}_3)],\hfill \\ \hfill & Q_{1R}=2a_{13}{a}_{31}{\omega }_0^{(3)}[(\frac{1}{{\tau }^2}-{({\omega }_0^{(3)})}^2)sin(2{\omega }_0^{(3)}{T}_3)-\frac{2{\omega }_0^{(3)}}{\tau }cos(2{\omega }_0^{(3)}{T}_3)],\hfill \\ \hfill & Q_{1I}=2a_{13}{a}_{31}{\omega }_0^{(3)}[(\frac{1}{{\tau }^2}-{({\omega }_0^{(3)})}^2)cos(2{\omega }_0^{(3)}{T}_3)+2\frac{2{\omega }_0^{(3)}}{\tau }sin(2{\omega }_0^{(3)}{T}_3)].\hfill \end{array}$$ 21

If the condition ($H_3$): $P_{1R}{Q}_{1R}+P_{1I}{Q}_{1I}\ne 0$ holds, $Re\{{(\frac{\text{d}\lambda }{\text{d}{T}_3})}_{\lambda =i{\omega }_0^{(3)}}^{-1}\}\ne 0$ , so the transversality condition of Hopf bifurcation is satisfied.

Based on the above analyses, we get the following Theorem.

Theorem 2

For $T_3^0$ as the minimum value that makes the characteristic equation (11) has pure imaginary roots, if the conditions ($H_1$) - ($H_3$) are true, combining with Lemma 1, we can draw the following conclusions,

1. For $T_1=T_2=0$, all roots of the characteristic equation (11) have negative real parts for any $T_3\in (0,T_3^0)$ , the system (3) is asymptotically stable at the equilibrium $u_0$ for any $T_3\in (0,T_3^0)$ and is unstable for $T_3>T_3^0$.

2. The system (3) experiences Hopf bifurcation at the equilibrium $u_0$ when $T_3$ crosses the critical value $T_3^0$.

Case 3 $T_2=0,T_3\in (0,T_3^0),T_1>0$

For $T_2=0,T_3\in (0,T_3^0),T_1>0$, the characteristic equation (7) becomes

$$\begin{array}{c}\hfill \begin{array}{cc}& \mathrm{\Delta }(\lambda ,T_1,0,T_3)\hfill \\ \hfill & ={(\lambda +\frac{1}{\tau })}^4-{(\lambda +\frac{1}{\tau })}^2(a_{12}{a}_{21}{e}^{-2\lambda T_1}+a_{34}{a}_{43}+a_{13}{a}_{31}{e}^{-2\lambda T_3})+a_{12}{a}_{21}{a}_{34}{a}_{43}{e}^{-2\lambda T_1}\hfill \\ \hfill & =0.\hfill \end{array}\end{array}$$ 22

We suppose that $\lambda =i\omega (\omega >0)$ is a root of Eq. (22),

$$\begin{array}{c}\hfill \begin{array}{cc}& {(i\omega +\frac{1}{\tau })}^4-{(i\omega +\frac{1}{\tau })}^2(a_{12}{a}_{21}{e}^{-2i\omega T_1}+a_{34}{a}_{43}+a_{13}{a}_{31}{e}^{-2i\omega T_3})+a_{12}{a}_{21}{a}_{34}{a}_{43}{e}^{-2i\omega T_1}\hfill \\ \hfill & ={(i\omega +\frac{1}{\tau })}^4+[a_{12}{a}_{21}{a}_{34}{a}_{43}-a_{12}{a}_{21}{(i\omega +\frac{1}{\tau })}^2]e^{-2i\omega T_1}-{(i\omega +\frac{1}{\tau })}^2(a_{34}{a}_{43}+a_{13}{a}_{31}{e}^{-2i\omega T_3})\hfill \\ \hfill & =[(\frac{1}{{\tau }^4}+{\omega }^4-\frac{6{\omega }^2}{{\tau }^2})+i(\frac{4\omega }{{\tau }^3}-\frac{4{\omega }^3}{\tau })]+[(a_{12}{a}_{21}{a}_{34}{a}_{43}+a_{12}{a}_{21}({\omega }^2-\frac{1}{{\tau }^2}))\hfill \\ \hfill & -i\frac{2a_{12}{a}_{21}\omega }{\tau }](cos(2\omega T_1)-isin(2\omega T_1))\hfill \\ \hfill & +[({\omega }^2-\frac{1}{{\tau }^2})-i\frac{2\omega }{\tau }][(a_{34}{a}_{43}+a_{13}{a}_{31}cos(2\omega T_3))-i{a}_{13}{a}_{31}sin(2\omega T_3]\hfill \\ \hfill & =0.\hfill \end{array}\end{array}$$ 23

Separating the real part and the imaginary part of Eq.(23), we can get

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& (\frac{1}{{\tau }^4}+{\omega }^4-\frac{6{\omega }^2}{{\tau }^2})+[a_{12}{a}_{21}{a}_{34}{a}_{43}+a_{12}{a}_{21}({\omega }^2-\frac{1}{{\tau }^2})]cos(2\omega T_1)-\frac{2a_{12}{a}_{21}\omega }{\tau }sin(2\omega T_1)\hfill \\ \hfill & +({\omega }^2-\frac{1}{{\tau }^2})(a_{34}{a}_{43}+a_{13}{a}_{31}cos(2\omega T_3))-\frac{2a_{13}{a}_{31}\omega }{\tau }sin(2\omega T_3)=0,\hfill \\ \hfill & \frac{4\omega }{{\tau }^3}-\frac{4{\omega }^3}{\tau }-[a_{12}{a}_{21}{a}_{34}{a}_{43}+a_{12}{a}_{21}({\omega }^2-\frac{1}{{\tau }^2})]sin(2\omega T_1)-\frac{2a_{12}{a}_{21}\omega }{\tau }cos(2\omega T_1)\hfill \\ \hfill & +(\frac{1}{{\tau }^2}-{\omega }^2)a_{13}{a}_{31}sin(2\omega T_3)-\frac{2\omega }{\tau }(a_{34}{a}_{43}+a_{13}{a}_{31}cos(2\omega T_3))=0.\hfill \end{array}\right)\end{array}$$ 24

Then, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& sin(2\omega T_1)=-\frac{e_1{\omega }^5+e_2{\omega }^4+e_3{\omega }^3+e_4{\omega }^2+e_5\omega +e_6}{e_7{\omega }^4+e_8{\omega }^2+e_9},\hfill \\ \hfill & cos(2\omega T_1)=\frac{e_{10}{\omega }^6+e_{11}{\omega }^4+e_{12}{\omega }^2+e_{13}\omega +e_{14}}{e_7{\omega }^4+e_8{\omega }^2+e_9}.\hfill \end{array}\right)\end{array}$$ 25

where

$$\begin{array}{cc}& e_1=\frac{2a_{12}{a}_{21}}{\tau },\hfill \\ \hfill & e_2=a_{12}{a}_{21}{a}_{13}{a}_{31}sin(2\omega T_3),\hfill \\ \hfill & e_3=\frac{4a_{12}{a}_{21}}{\tau }(a_{34}{a}_{43}+\frac{1}{{\tau }^2}),\hfill \\ \hfill & e_4=\frac{2a_{12}{a}_{21}{a}_{13}{a}_{31}sin(2\omega T_3)}{{\tau }^2}+a_{12}{a}_{21}{a}_{13}{a}_{31}{a}_{34}{a}_{43}sin(2\omega T_3),\hfill \\ \hfill & e_5=\frac{2a_{12}{a}_{21}}{{\tau }^5}-\frac{2a_{12}{a}_{21}{a}_{34}{a}_{43}}{\tau }(\frac{2}{{\tau }^2}-a_{34}{a}_{43}-a_{13}{a}_{31}cos(2\omega T_3)),\hfill \\ \hfill & e_6=(\frac{1}{{\tau }^2}-a_{34}{a}_{43})\frac{a_{12}{a}_{21}{a}_{13}{a}_{31}sin(2\omega T_3)}{{\tau }^2},\hfill \\ \hfill & e_7={(a_{12}{a}_{21})}^2,\hfill \\ \hfill & e_8=2{(a_{12}{a}_{21})}^2(\frac{1}{{\tau }^2}+a_{34}{a}_{43}),\hfill \\ \hfill & e_9={(a_{12}{a}_{21})}^2{(a_{34}{a}_{43}-\frac{1}{{\tau }^2})}^2,\hfill \\ \hfill & e_{10}=-a_{12}{a}_{21},\hfill \\ \hfill & e_{11}=a_{12}{a}_{21}(\frac{1}{{\tau }^2}+2a_{34}{a}_{43}+a_{13}{a}_{31}cos(2\omega T_3)),\hfill \\ \hfill & e_{12}=\frac{a_{12}{a}_{21}}{{\tau }^2}(\frac{1}{{\tau }^2}-2a_{34}{a}_{43}-2a_{13}{a}_{31}cos(2\omega T_3))\hfill \\ \hfill & -a_{12}{a}_{21}{a}_{34}{a}_{43}(a_{34}{a}_{43}+a_{13}{a}_{31}cos(2\omega T_3)-\frac{6}{{\tau }^2}),\hfill \\ \hfill & e_{13}=\frac{2a_{12}{a}_{21}{a}_{13}{a}_{31}{a}_{34}{a}_{43}sin(2\omega T_3)}{\tau },\hfill \\ \hfill & e_{14}=a_{12}{a}_{21}(\frac{1}{{\tau }^2}-a_{34}{a}_{43})[\frac{1}{{\tau }^4}-\frac{1}{{\tau }^2}(a_{34}{a}_{43}+a_{13}{a}_{31}cos(2\omega T_3))].\hfill \end{array}$$ 26

According to Eq.(25) and ${sin}^2(2\omega T_1)+{cos}^2(2\omega T_1)=1$, we have

$$\begin{array}{cc}\hfill & f_1{\omega }^{12}+f_2{\omega }^{10}+f_3{\omega }^9+f_4{\omega }^8+f_5{\omega }^7+f_6{\omega }^6+f_7{\omega }^5+f_8{\omega }^4\hfill \\ \hfill & +f_9{\omega }^3+f_{10}{\omega }^2+f_{11}\omega +f_{12}=0,\hfill \end{array}$$ 27

where

$$\begin{array}{cc}& f_1=e_{10}^2,\hfill \\ \hfill & f_2=e_1^2+2e_{10}{e}_{11},\hfill \\ \hfill & f_3=2e_1{e}_2,\hfill \\ \hfill & f_4=2e_1{e}_3+e_2^2+2e_{10}{e}_{12}+e_{11}^2-e_7^2,\hfill \\ \hfill & f_5=2e_1{e}_4+2e_2{e}_3+2e_{10}{e}_{13},\hfill \\ \hfill & f_6=2e_1{e}_5+2e_2{e}_4+e_3^2+2e_{10}{e}_{14}+2e_{11}{e}_{12}-2e_7{e}_8,\hfill \\ \hfill & f_7=2e_1{e}_6+2e_2{e}_5+2e_3{e}_4+2e_{11}{e}_{13},\hfill \\ \hfill & f_8=2e_2{e}_6+2e_3{e}_5+e_4^2+2e_{11}{e}_{14}+e_{12}^2-2e_7{e}_9-e_8^2,\hfill \\ \hfill & f_9=2e_3{e}_6+2e_4{e}_5+2e_{12}{e}_{13},\hfill \\ \hfill & f_{10}=2e_4{e}_6+e_5^2+2e_{12}{e}_{14}+e_{13}^2-2e_8{e}_9,\hfill \\ \hfill & f_{11}=2e_5{e}_6+2e_{13}{e}_{14},\hfill \\ \hfill & f_{12}=e_6^2+e_{14}^2-e_9^2.\hfill \end{array}$$ 28

Firstly, we give the assumption

$(H_4)$: Eq. (27) has at most twelve positive roots ${\omega }_k(k=1,...,N,N\le 12)$.

According to Eq.(25), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill T_{1k}^n& =\frac{1}{2{\omega }_k}arccos\frac{e_{10}{\omega }_k^6+e_{11}{\omega }_k^4+e_{12}{\omega }_k^2+e_{13}{\omega }_k+e_{14}}{e_7{\omega }_k^4+e_8{\omega }_k^2+e_9}+\frac{n\pi }{{\omega }_k},\hfill \\ \hfill & (k=1,...,N;n=0,1,...)\hfill \end{array}\end{array}$$ 29

Let $T_1^0=\underset{k=1,...,N}{\mathit{min}}\{T_{1k}^n\}$ is the critical value of $T_1$ corresponding to pure imaginary root of Eq.(22) $\pm i{\omega }_0^{(1)}$.

Next, we prove the transversal condition about time delay $T_1$. Differentiating the two sides of Eq.(22) about time delay $T_1$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left(\frac{\text{d}\lambda }{\text{d}{T}_1}\right)}^{-1}=\frac{P_2}{Q_2},\hfill \end{array}\end{array}$$ 30

where

$$\begin{array}{cc}& P_2=2{(\lambda +\frac{1}{\tau })}^3-(\lambda +\frac{1}{\tau })(a_{12}{a}_{21}{e}^{-2\lambda T_1}+a_{34}{a}_{43}+a_{13}{a}_{31}{e}^{-2\lambda T_3})\hfill \\ \hfill & +{(\lambda +\frac{1}{\tau })}^2(a_{12}{a}_{21}{T}_1{e}^{-2\lambda T_1}+a_{13}{a}_{31}{T}_3{e}^{-2\lambda T_3}),\hfill \\ \hfill & Q_2=a_{12}{a}_{21}{a}_{34}{a}_{43}\lambda e^{-2\lambda T_1}-a_{12}{a}_{21}\lambda {(\lambda +\frac{1}{\tau })}^2{e}^{-2\lambda T_1}.\hfill \end{array}$$ 31

Therefore, $Re\{{(\frac{\text{d}\lambda }{\text{d}{T}_1})}_{\lambda =i{\omega }_0^{(1)}}^{-1}\}=\frac{P_{2R}{Q}_{2R}+P_{2I}{Q}_{2I}}{Q_{2R}^2+Q_{2I}^2}$,

where $P_{2R}(Q_{2R})$ and $P_{2I}(Q_{2I})$ represent the real and imaginary parts of $P_2(Q_2)$, respectively.

$$\begin{array}{cc}& P_{2R}=2(\frac{1}{{\tau }^3}-\frac{3{({\omega }_0^{(1)})}^2}{\tau })\hfill \\ \hfill & -[\frac{a_{12}{a}_{21}cos(2{\omega }_0^{(1)}{T}_1)+a_{13}{a}_{31}cos(2{\omega }_0^{(1)}{T}_3)+a_{34}{a}_{43}}{\tau }\hfill \\ \hfill & +(a_{12}{a}_{21}sin(2{\omega }_0^{(1)}{T}_1)+a_{13}{a}_{31}sin(2{\omega }_0^{(1)}{T}_3)){\omega }_0^{(1)}]\hfill \\ \hfill & +(\frac{1}{{\tau }^2}-{({\omega }_0^{(1)})}^2)(a_{12}{a}_{21}{T}_{1}cos(2{\omega }_0^{(1)}{T}_1)+a_{13}{a}_{31}{T}_{3}cos(2{\omega }_0^{(1)}{T}_3))\hfill \\ \hfill & +\frac{2{\omega }_0^{(1)}}{\tau }(a_{12}{a}_{21}{T}_{1}sin(2{\omega }_0^{(1)}{T}_1)+a_{13}{a}_{31}{T}_{3}sin(2{\omega }_0^{(1)}{T}_3)),\hfill \\ \hfill & P_{2I}=2(\frac{3{\omega }_0^{(1)}}{{\tau }^2}-{({\omega }_0^{(1)})}^3)\hfill \\ \hfill & -[(a_{12}{a}_{21}cos(2{\omega }_0^{(1)}{T}_1)+a_{13}{a}_{31}cos(2{\omega }_0^{(1)}{T}_3)+a_{34}{a}_{43}){\omega }_0^{(1)}\hfill \\ \hfill & -\frac{a_{12}{a}_{21}sin(2{\omega }_0^{(1)}{T}_1)+a_{13}{a}_{31}sin(2{\omega }_0^{(1)}{T}_3)}{\tau }]\hfill \\ \hfill & +\frac{2{\omega }_0^{(1)}}{\tau }(a_{12}{a}_{21}{T}_{1}cos(2{\omega }_0^{(1)}{T}_1)+a_{13}{a}_{31}{T}_{3}cos(2{\omega }_0^{(1)}{T}_3))\hfill \\ \hfill & -(\frac{1}{{\tau }^2}-{({\omega }_0^{(1)})}^2)(a_{12}{a}_{21}{T}_{1}sin(2{\omega }_0^{(1)}{T}_1)+a_{13}{a}_{31}{T}_{3}sin(2{\omega }_0^{(1)}{T}_3)),\hfill \\ \hfill & Q_{2R}=a_{12}{a}_{21}{a}_{34}{a}_{43}{\omega }_0^{(1)}sin(2{\omega }_0^{(1)}{T}_1)-a_{12}{a}_{21}{\omega }_0^{(1)}\hfill \\ \hfill & [(\frac{1}{{\tau }^2}-{({\omega }_0^{(1)})}^2)sin(2{\omega }_0^{(1)}{T}_1)\hfill \\ \hfill & -\frac{2{\omega }_0^{(1)}}{\tau }cos(2{\omega }_0^{(1)}{T}_1)],\hfill \\ \hfill & Q_{2I}=a_{12}{a}_{21}{a}_{34}{a}_{43}{\omega }_0^{(1)}cos(2{\omega }_0^{(1)}{T}_1)-a_{12}{a}_{21}{\omega }_0^{(1)}\hfill \\ \hfill & [(\frac{1}{{\tau }^2}-{({\omega }_0^{(1)})}^2)cos(2{\omega }_0^{(1)}{T}_1)\hfill \\ \hfill & +\frac{2{\omega }_0^{(1)}}{\tau }sin(2{\omega }_0^{(1)}{T}_1)].\hfill \end{array}$$ 32

Then, we give the hypothesis

($H_5$): $P_{2R}{Q}_{2R}+P_{2I}{Q}_{2I}\ne 0$.

Therefore, if the condition ($H_5$) holds, $Re\{{(\frac{\text{d}\lambda }{\text{d}{T}_1})}_{\lambda =i{\omega }_0^{(1)}}^{-1}\}\ne 0$ is true, then the transversal condition of Hopf bifurcation is satisfied.

From the above analyses, we get the following Theorem.

Theorem 3

For $T_1^0$ as the minimum value that makes the characteristic equation (22) has pure imaginary roots, if the conditions $H_1,H_4-H_5$ are true, combining with Lemma 1, we can draw the following conclusions,

1. For $T_2=0,T_3\in (0,T_3^0)$, all the roots of the characteristic equation (22) have negative real parts for $T_1\in (0,T_1^0)$, the system (3) is asymptotically stable at the equilibrium point $u_0$ for $T_1\in (0,T_1^0)$ and is unstable for $T_1>T_1^0$.

2. The system (3) experiences Hopf bifurcation at the equilibrium $u_0$ when $T_1$ crosses the critical value $T_1^0$.

Case 4 $T_1\in (0,T_1^0),T_3\in (0,T_3^0),T_2>0$.

For $T_1\in (0,T_1^0),T_3\in (0,T_3^0),T_2>0$, the characteristic equation (7) becomes

$$\begin{array}{c}\hfill \begin{array}{cc}& \mathrm{\Delta }(\lambda ,T_1,T_2,T_3)\hfill \\ \hfill & ={(\lambda +\frac{1}{\tau })}^4-{(\lambda +\frac{1}{\tau })}^2(a_{12}{a}_{21}{e}^{-2\lambda T_1}+a_{34}{a}_{43}{e}^{-2\lambda T_2}\hfill \\ \hfill & +a_{13}{a}_{31}{e}^{-2\lambda T_3})\hfill \\ \hfill & +a_{12}{a}_{21}{a}_{34}{a}_{43}{e}^{-2\lambda T_1}{e}^{-2\lambda T_2}=0.\hfill \end{array}\end{array}$$ 33

We suppose that $\lambda =i\omega (\omega >0)$ is a root of Eq. (33), then

$$\begin{array}{c}\hfill \begin{array}{cc}& {(i\omega +\frac{1}{\tau })}^4-{(i\omega +\frac{1}{\tau })}^2(a_{12}{a}_{21}{e}^{-2i\omega T_1}+a_{34}{a}_{43}{e}^{-2i\omega T_2}\hfill \\ \hfill & +a_{13}{a}_{31}{e}^{-2i\omega T_3})\hfill \\ \hfill & +a_{12}{a}_{21}{a}_{34}{a}_{43}{e}^{-2i\omega T_1}{e}^{-2i\omega T_2}\hfill \\ \hfill & ={(i\omega +\frac{1}{\tau })}^4+[a_{12}{a}_{21}{a}_{34}{a}_{43}{e}^{-2i\omega T_1}-a_{34}{a}_{43}{(i\omega +\frac{1}{\tau })}^2]e^{-2i\omega T_2}\hfill \\ \hfill & -{(i\omega +\frac{1}{\tau })}^2(a_{12}{a}_{21}{e}^{-2i\omega T_1}+a_{13}{a}_{31}{e}^{-2i\omega T_3})\hfill \\ \hfill & =[(\frac{1}{{\tau }^4}+{\omega }^4-\frac{6{\omega }^2}{{\tau }^2})+i(\frac{4\omega }{{\tau }^3}-\frac{4{\omega }^3}{\tau })]+[a_{12}{a}_{21}{a}_{34}{a}_{43}cos(2\omega T_1)+a_{34}{a}_{43}({\omega }^2-\frac{1}{{\tau }^2}))\hfill \\ \hfill & -i(a_{12}{a}_{21}{a}_{34}{a}_{43}sin(2\omega T_1)+\frac{2a_{34}{a}_{43}\omega }{\tau })]\hfill \\ \hfill & (cos(2\omega T_2)-isin(2\omega T_2))\hfill \\ \hfill & -(\frac{1}{{\tau }^2}-{\omega }^2+i\frac{2\omega }{\tau })[(a_{12}{a}_{21}cos(2\omega T_1)+a_{13}{a}_{31}cos(2\omega T_3)\hfill \\ \hfill & -i(a_{12}{a}_{21}sin(2\omega T_1)+a_{13}{a}_{31}sin(2\omega T_3))]\hfill \\ \hfill & =0.\hfill \end{array}\end{array}$$ 34

Separating the real part and the imaginary part of Eq.(34), we can get

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& (\frac{1}{{\tau }^4}+{\omega }^4-\frac{6{\omega }^2}{{\tau }^2})+[a_{12}{a}_{21}{a}_{34}{a}_{43}cos(2\omega T_1)+a_{34}{a}_{43}({\omega }^2-\frac{1}{{\tau }^2})]cos(2\omega T_2)\hfill \\ \hfill & +({\omega }^2-\frac{1}{{\tau }^2})[a_{12}{a}_{21}cos(2\omega T_1)-a_{12}{a}_{21}{a}_{34}{a}_{43}sin(2\omega T_1)+\frac{2a_{34}{a}_{43}\omega }{\tau }sin(2\omega T_2)\hfill \\ \hfill & +a_{13}{a}_{31}cos(2\omega T_3)]-\frac{2\omega }{\tau }[a_{12}{a}_{21}sin(2\omega T_1)+a_{13}{a}_{31}sin(2\omega T_3)]=0,\hfill \\ \hfill & \frac{4\omega }{{\tau }^3}-\frac{4{\omega }^3}{\tau }-[a_{12}{a}_{21}{a}_{34}{a}_{43}cos(2\omega T_1)+a_{34}{a}_{43}({\omega }^2-\frac{1}{{\tau }^2})]sin(2\omega T_2)\hfill \\ \hfill & -[a_{12}{a}_{21}{a}_{34}{a}_{43}sin(2\omega T_1)+\frac{2a_{34}{a}_{43}}{\tau }]cos(2\omega T_2)+(\frac{1}{{\tau }^2}-{\omega }^2)[a_{12}{a}_{21}sin(2\omega T_1)\hfill \\ \hfill & +a_{13}{a}_{31}sin(2\omega T_3)]-\frac{2\omega }{\tau }[a_{12}{a}_{21}cos(2\omega T_1)+a_{13}{a}_{31}cos(2\omega T_3)]=0.\hfill \end{array}\right)\end{array}$$ 35

Simplifying Eq.(35), we get

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& (k_1\omega +k_2)sin(2\omega T_2)+(k_3{\omega }^2+k_4)cos(2\omega T_2)={\omega }^4+k_5{\omega }^2+k_6\omega +k_7,\hfill \\ \hfill & (k_8{\omega }^2+k_9)sin(2\omega T_2)+(k_{10}{\omega }^2+k_{11})cos(2\omega T_2)=k_{12}{\omega }^3+k_{13}{\omega }^2+k_{14}\omega +k_{15},\hfill \end{array}\right)\end{array}$$ 36

where

$$\begin{array}{cc}& k_1=\frac{2a_{34}{a}_{43}}{\tau },\hfill \\ \hfill & k_2=a_{12}{a}_{21}{a}_{34}{a}_{43}sin(2\omega T_1),\hfill \\ \hfill & k_3=-a_{34}{a}_{43},\hfill \\ \hfill & k_4=\frac{a_{34}{a}_{43}}{{\tau }^2}-a_{12}{a}_{21}{a}_{34}{a}_{43}cos(2\omega T_1),\hfill \\ \hfill & k_5=a_{12}{a}_{21}cos(2\omega T_1)+a_{13}{a}_{31}cos(2\omega T_3)-\frac{6}{{\tau }^2},\hfill \\ \hfill & k_6=-\frac{2}{\tau }(a_{12}{a}_{21}sin(2\omega T_1)+a_{13}{a}_{31}sin(2\omega T_3)),\hfill \\ \hfill & k_7=\frac{1}{{\tau }^4}-\frac{1}{{\tau }^2}(a_{12}{a}_{21}cos(2\omega T_1)+a_{13}{a}_{31}cos(2\omega T_3)),\hfill \\ \hfill & k_8=a_{34}{a}_{43},\hfill \\ \hfill & k_9=a_{12}{a}_{21}{a}_{34}{a}_{43}cos(2\omega T_1)-\frac{a_{34}{a}_{43}}{{\tau }^2},\hfill \\ \hfill & k_{10}=\frac{2a_{34}{a}_{43}}{\tau },\hfill \\ \hfill & k_{11}=a_{12}{a}_{21}{a}_{34}{a}_{43}sin(2\omega T_1),\hfill \\ \hfill & k_{12}=-\frac{4}{\tau },\hfill \\ \hfill & k_{13}=-(a_{12}{a}_{21}sin(2\omega T_1)+a_{13}{a}_{31}sin(2\omega T_3)),\hfill \\ \hfill & k_{14}=\frac{2}{\tau }(\frac{2}{{\tau }^2}-a_{12}{a}_{21}cos(2\omega T_1)-a_{13}{a}_{31}cos(2\omega T_3)),\hfill \\ \hfill & k_{15}=\frac{1}{{\tau }^2}(a_{12}{a}_{21}sin(2\omega T_1)+a_{13}{a}_{31}sin(2\omega T_3)),\hfill \end{array}$$ 37

Then, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& sin(2\omega T_2)=\frac{g_1{\omega }^5+g_2{\omega }^4+g_3{\omega }^3+g_4{\omega }^2+g_5\omega +g_6}{g_7{\omega }^4+g_8{\omega }^2+g_9\omega +g_{10}},\hfill \\ \hfill & cos(2\omega T_2)=\frac{g_{11}{\omega }^6+g_{12}{\omega }^4+g_{13}{\omega }^3+g_{14}{\omega }^2+g_{15}\omega +g_{16}}{g_7{\omega }^4+g_8{\omega }^2+g_9\omega +g_{10}},\hfill \end{array}\right)\end{array}$$ 38

where

$$\begin{array}{cc}& g_1=k_{10}-k_3{k}_{12},\hfill \\ \hfill & g_2=k_{11}-k_3{k}_{13},\hfill \\ \hfill & g_3=k_5{k}_{10}-k_3{k}_{14}-k_4{k}_{12},\hfill \\ \hfill & g_4=k_6{k}_{10}+k_5{k}_{11}-k_3{k}_{15}-k_4{k}_{13},\hfill \\ \hfill & g_5=k_7{k}_{10}+k_6{k}_{11}-k_4{k}_{14},\hfill \\ \hfill & g_6=k_2{k}_{11}-k_4{k}_9,\hfill \\ \hfill & g_7=-k_3{k}_8,\hfill \\ \hfill & g_8=k_1{k}_{10}-k_3{k}_9-k_4{k}_8,\hfill \\ \hfill & g_9=k_1{k}_{11}+k_2{k}_{10},\hfill \\ \hfill & g_{10}=k_2{k}_{11}-k_4{k}_9,\hfill \\ \hfill & g_{11}=-k_8,\hfill \\ \hfill & g_{12}=k_1{k}_{12}-k_5{k}_8-k_9,\hfill \\ \hfill & g_{13}=k_1{k}_{13}+k_2{k}_{12}-k_6{k}_8,\hfill \\ \hfill & g_{14}=k_1{k}_{14}+k_2{k}_{13}-k_7{k}_8-k_5{k}_9,\hfill \\ \hfill & g_{15}=k_1{k}_{15}+k_2{k}_{14}-k_6{k}_9,\hfill \\ \hfill & g_{16}=k_2{k}_{15}-k_7{k}_9,\hfill \end{array}$$ 39

According to Eq.(38) and ${sin}^2(2\omega T_2)+{cos}^2(2\omega T_2)=1$, we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill h_1{\omega }^{12}+h_2{\omega }^{10}+h_3{\omega }^9+h_4{\omega }^8+h_5{\omega }^7+h_6{\omega }^6+h_7{\omega }^5+h_8{\omega }^4+h_9{\omega }^3+h_{10}{\omega }^2+h_{11}\omega +h_{12}=0,\end{array}\end{array}$$ 40

where

$$\begin{array}{cc}& h_1=g_{11}^2,\hfill \\ \hfill & h_2=g_1^2+2g_{11}{g}_{12},\hfill \\ \hfill & h_3=2g_1{g}_2+2g_{11}{g}_{13},\hfill \\ \hfill & h_4=2g_1{g}_3+g_2^2+2g_{11}{g}_{14}+g_{12}^2-g_7^2,\hfill \\ \hfill & h_5=2g_1{g}_4+2g_2{g}_3+2g_{11}{g}_{15}+2g_{12}{g}_{13},\hfill \\ \hfill & h_6=2g_1{g}_5+2g_2{g}_4+g_3^2+2g_{11}{g}_{16}+2g_{12}{g}_{14}+g_{13}^2,\hfill \\ \hfill & h_7=2g_1{g}_6+2g_2{g}_5+2g_3{g}_4+2g_{12}{g}_{15}+2g_{13}{g}_{14}-2g_7{g}_9,\hfill \\ \hfill & h_8=2g_2{g}_6+2g_3{g}_5+g_4^2+2g_{12}{g}_{16}+2g_{13}{g}_{15}+g_{14}^2-2g_7{g}_{10}-g_8^2,\hfill \\ \hfill & h_9=2g_3{g}_6+2g_4{g}_5+2g_{13}{g}_{16}+2g_{14}{g}_{15}-2g_8{g}_9,\hfill \\ \hfill & h_{10}=2g_4{g}_6+g_5^2+2g_{14}{g}_{16}+g_{15}^2-2g_8{g}_9-g_9^2,\hfill \\ \hfill & h_{11}=2g_5{g}_6+2g_{15}{g}_{16}-2g_9{g}_{10},\hfill \\ \hfill & h_{12}=g_6^2+g_{16}^2-g_{10}^2.\hfill \end{array}$$ 41

Here, we put forward the assumption

($H_6$): Eq. (40) has at most twelve positive roots.

According to Eq. (38), we get

$$\begin{array}{c}\hfill \begin{array}{cc}& T_{2k}^n=\frac{1}{2{\omega }_k}arccos\frac{g_{11}{\omega }_k^6+g_{12}{\omega }_k^4+g_{13}{\omega }_k^3+g_{14}{\omega }_k^2+g_{15}{\omega }_k+g_{16}}{g_7{\omega }_k^4+g_8{\omega }_k^2+g_9{\omega }_k+g_{10}}+\frac{n\pi }{{\omega }_k}\hfill \\ \hfill & (k=1,...,N;n=0,1,...).\hfill \end{array}\end{array}$$ 42

Let $T_2^0=\underset{k=1,...,N}{\mathit{min}}\{T_{2k}^n\}$ be the minimum critical value of $T_2$ corresponding to pure imaginary roots $\pm i{\omega }_0^{(2)}$.

Next, we prove the transversal condition about time delay $T_2$. Differentiating the two sides of Eq. (33) about time delay $T_2$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}& {(\frac{\text{d}\lambda }{\text{d}{T}_2})}^{-1}=\frac{P_3}{Q_3},\hfill \end{array}\end{array}$$ 43

where

$$\begin{array}{cc}& P_3=2{(\lambda +\frac{1}{\tau })}^3-(\lambda +\frac{1}{\tau })(a_{12}{a}_{21}{e}^{-2\lambda T_1}+a_{34}{a}_{43}{e}^{-2\lambda T_2}+a_{13}{a}_{31}{e}^{-2\lambda T_3})\hfill \\ \hfill & +{(\lambda +\frac{1}{\tau })}^2(a_{12}{a}_{21}{T}_1{e}^{-2\lambda T_1}+a_{34}{a}_{43}{T}_2{e}^{-2\lambda T_2}+a_{13}{a}_{31}{T}_3{e}^{-2\lambda T_3})\hfill \\ \hfill & -a_{12}{a}_{21}{a}_{34}{a}_{43}(T_1+T_2)e^{-2\lambda (T_1+T_2)},\hfill \\ \hfill & Q_3=a_{12}{a}_{21}{a}_{34}{a}_{43}\lambda e^{-2\lambda (T_1+T_2)}-a_{34}{a}_{43}\lambda {(\lambda +\frac{1}{\tau })}^2{e}^{-2\lambda T_2}.\hfill \end{array}$$ 44

Therefore, $Re\{{(\frac{\text{d}\lambda }{\text{d}{T}_2})}_{\lambda =i{\omega }_0^{(2)}}^{-1}\}=\frac{P_{3R}{Q}_{3R}+P_{3I}{Q}_{3I}}{Q_{3R}^2+Q_{3I}^2}$, where $P_{3R}(Q_{3R})$ and $P_{3I}(Q_{3I})$ represent the real and imaginary parts of $P_3(Q_3)$, respectively.

where

$$\begin{array}{cc}\hfill P_{3R}& =2(\frac{1}{{\tau }^3}-\frac{3{({\omega }_0^{(2)})}^2}{\tau })-\frac{1}{\tau }(a_{12}{a}_{21}cos(2{\omega }_0^{(2)}{T}_1)\hfill \\ \hfill & +a_{34}{a}_{43}cos(2{\omega }_0^{(2)}{T}_2)+a_{13}{a}_{31}cos(2{\omega }_0^{(2)}{T}_3))\hfill \\ \hfill & -{\omega }_0^{(2)}(a_{12}{a}_{21}sin(2{\omega }_0^{(2)}{T}_1)+a_{34}{a}_{43}sin(2{\omega }_0^{(2)}{T}_2)\hfill \\ \hfill & +a_{13}{a}_{31}sin(2{\omega }_0^{(2)}{T}_3))\hfill \\ \hfill & +(\frac{1}{{\tau }^2}-{({\omega }_0^{(2)})}^2)(a_{12}{a}_{21}{T}_{1}cos(2{\omega }_0^{(2)}{T}_1)\hfill \\ \hfill & +a_{34}{a}_{43}{T}_{2}cos(2{\omega }_0^{(2)}{T}_2)\hfill \\ \hfill & +a_{13}{a}_{31}{T}_{3}cos(2{\omega }_0^{(2)}{T}_3))\hfill \\ \hfill & +\frac{2{\omega }_0^{(2)}}{\tau }(a_{12}{a}_{21}{T}_{1}sin(2{\omega }_0^{(2)}{T}_1)\hfill \\ \hfill & +a_{34}{a}_{43}{T}_{2}sin(2{\omega }_0^{(2)}{T}_2)\hfill \\ \hfill & +a_{13}{a}_{31}{T}_{3}sin(2{\omega }_0^{(2)}{T}_3))\hfill \\ \hfill & -a_{12}{a}_{21}{a}_{34}{a}_{43}(T_1+T_2)cos(2{\omega }_0^{(2)}(T_1+T_2)),\hfill \\ \hfill P_{3I}& =2(\frac{3{\omega }_0^{(2)}}{{\tau }^2}-{({\omega }_0^{(2)})}^3)+\frac{1}{\tau }(a_{12}{a}_{21}sin(2{\omega }_0^{(2)}{T}_1)\hfill \\ \hfill & +a_{34}{a}_{43}sin(2{\omega }_0^{(2)}{T}_2)+a_{13}{a}_{31}sin(2{\omega }_0^{(2)}{T}_3))\hfill \\ \hfill & -{\omega }_0^{(2)}(a_{12}{a}_{21}cos(2{\omega }_0^{(2)}{T}_1)+a_{34}{a}_{43}cos(2{\omega }_0^{(2)}{T}_2)\hfill \\ \hfill & +a_{13}{a}_{31}cos(2{\omega }_0^{(2)}{T}_3))\hfill \\ \hfill & +\frac{2{\omega }_0^{(2)}}{\tau }(a_{12}{a}_{21}{T}_{1}cos(2{\omega }_0^{(2)}{T}_1)\hfill \\ \hfill & +a_{34}{a}_{43}{T}_{2}cos(2{\omega }_0^{(2)}{T}_2)+a_{13}{a}_{31}{T}_{3}cos(2{\omega }_0^{(2)}{T}_3))\hfill \\ \hfill & -(\frac{1}{{\tau }^2}-{({\omega }_0^{(2)})}^2)(a_{12}{a}_{21}{T}_{1}sin(2{\omega }_0^{(2)}{T}_1)+a_{34}{a}_{43}{T}_{2}sin(2{\omega }_0^{(2)}{T}_2)+a_{13}{a}_{31}{T}_{3}sin(2{\omega }_0^{(2)}{T}_3))\hfill \\ \hfill & +a_{12}{a}_{21}{a}_{34}{a}_{43}(T_1+T_2)sin(2{\omega }_0^{(2)}(T_1+T_2)),\hfill \\ \hfill Q_{3R}& =a_{12}{a}_{21}{a}_{34}{a}_{43}{\omega }_0^{(2)}sin(2{\omega }_0^{(2)}(T_1+T_2))\hfill \\ \hfill & +a_{34}{a}_{43}{\omega }_0^{(2)}[\frac{2{\omega }_0^{(2)}}{\tau }cos(2{\omega }_0^{(2)}{T}_2)\hfill \\ \hfill & -(\frac{1}{{\tau }^2}-{({\omega }_0^{(2)})}^2)sin(2{\omega }_0^{(2)}{T}_2)],\hfill \\ \hfill Q_{3I}& =a_{12}{a}_{21}{a}_{34}{a}_{43}{\omega }_0^{(2)}cos(2{\omega }_0^{(2)}(T_1+T_2))\hfill \\ \hfill & -a_{34}{a}_{43}{\omega }_0^{(2)}[(\frac{1}{{\tau }^2}-{({\omega }_0^{(2)})}^2)cos(2{\omega }_0^{(2)}{T}_2)\hfill \\ \hfill & +\frac{2{\omega }_0^{(2)}}{\tau }sin(2{\omega }_0^{(2)}{T}_2)].\hfill \end{array}$$ 45

We make the assumption ($H_7$): $P_{3R}{Q}_{3R}+P_{3I}{Q}_{3I}\ne 0$.

If the condition ($H_7$) holds, $Re\{{(\frac{\text{d}\lambda }{\text{d}{T}_2})}_{\lambda =i{\omega }_0^{(2)}}^{-1}\}\ne 0$ is true, the transversal condition of Hopf bifurcation is satisfied.

From the above analysis, we get the following Theorem.

Theorem 4

For $T_2^0$ as the minimum value that makes the characteristic equation (33) has pure imaginary roots, if the conditions $H_1,H_6-H_7$ are true, combining with Lemma 1, we can draw the following conclusions,

1. For $T_1\in (0,T_1^0),T_3\in (0,T_3^0)$, all the roots of Eq. (33) have negative real parts for $T_2\in (0,T_2^0)$, the system (3) is asymptotically stable at the equilibrium $u_0$ for $T_2\in (0,T_2^0)$ and is unstable for $T_2>T_2^0$ .

2. The system (3) experiences Hopf bifurcation at the equilibrium $u_0$ when $T_2$ crosses the critical value $T_2^0$ for $T_1\in (0,T_1^0),T_3\in (0,T_3^0)$.

In summary, the conditions under which the system (3) undergoes Hopf bifurcation are given in Theorems 1 - 4 for four case (1) $T_1=T_2=T_3=0$ (2) $T_1=T_2=0$, $T_3>0$ (3) $T_2=0$, $T_1>0$, $T_3>0$ (4) $T_1>0$, $T_3>0$,$T_2>0$, respectively.

The direction and stability of Hopf bifurcation

In this section, for $T_1\in (0,T_1^0),T_3\in (0,T_3^0)$, we use central manifold theory and normal theory to judge the direction and stability of Hopf bifurcation at the critical value $T_2^0$.

First of all, let $v(t)=u(Tt)$, $T=T_2^0+\mu$, $\mu \in R$, and then the system (4) becomes the following form,

$$\begin{array}{c}\hfill \dot{v}(t)=L_{\mu }(v_t)+F(\mu ,v_t),\end{array}$$ 46

where $v_t(\theta )=v(t+\theta )$.

The $L_{\mu }:C\to R^4$,$F:R\times C\to R^4$ are given by

$$\begin{array}{cc}\hfill L_{\mu }(\mathrm{\Phi })=& (T_2^0+\mu )C_1\mathrm{\Phi }(0)+(T_2^0+\mu )C_2\mathrm{\Phi }(-\frac{T_1^{\ast }}{T_2^0})\hfill \\ \hfill & +(T_2^0+\mu )C_4\mathrm{\Phi }(-\frac{T_3^{\ast }}{T_2^0})+(T_2^0+\mu )C_3\mathrm{\Phi }(-1),\hfill \end{array}$$ 47

where

$$\begin{array}{c}\hfill \mathrm{\Phi }(t)={({\mathrm{\Phi }}_1(t),{\mathrm{\Phi }}_2(t),{\mathrm{\Phi }}_3(t),{\mathrm{\Phi }}_4(t))}^T,\end{array}$$ 48

and

$$\begin{array}{c}\hfill F(\mu ,v_t)=(T_2^0+\mu )\left(\begin{array}{c}{F}_{2S}+F_{3S}\\ F_{2G}+F_{3G}\\ F_{2E}+F_{3E}\\ F_{2I}+F_{3I}\end{array}\right),\end{array}$$ 49

with

$$\begin{array}{c}\hfill \begin{array}{cc}& \left(\begin{array}{c}{F}_{2S}\\ F_{2G}\\ F_{2E}\\ F_{2I}\end{array}\right)=\left(\begin{array}{c}{r}_S[{\omega }_{\mathit{GS}}^2{\mathrm{\Phi }}_2^2(t-\frac{T_1^{\ast }}{T_2^0})+{\omega }_{\mathit{CS}}^2{\mathrm{\Phi }}_3^2(t-\frac{T_3^{\ast }}{T_2^0})-{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}{\mathrm{\Phi }}_2(t-\frac{T_1^{\ast }}{T_2^0}){\mathrm{\Phi }}_3(t-\frac{T_3^{\ast }}{T_2^0})]\\ r_G[{\omega }_{\mathit{SG}}^2{\mathrm{\Phi }}_1^2(t-\frac{T_1^{\ast }}{T_2^0})]\\ r_E[{\omega }_{\mathit{SC}}^2{\mathrm{\Phi }}_1^2(t-\frac{T_3^{\ast }}{T_2^0})+{\omega }_{\mathit{CC}}^2{\mathrm{\Phi }}_4^2(t-1)+{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}{\mathrm{\Phi }}_1(t-\frac{T_3^{\ast }}{T_2^0}){\mathrm{\Phi }}_4(t-1)]\\ r_I[{\omega }_{\mathit{CC}}^2{\mathrm{\Phi }}_3^2(t-1)]\end{array}\right),\hfill \\ \hfill & \left(\begin{array}{c}{F}_{3S}\\ F_{3G}\\ F_{3E}\\ F_{3I}\end{array}\right)=\left(\begin{array}{c}{l}_1\\ l_2\\ l_3\\ l_4\end{array}\right),\hfill \end{array}\end{array}$$ 50

where

$$\begin{array}{cc}& l_1=e_S[{\omega }_{\mathit{CS}}^3{\mathrm{\Phi }}_3^3(t-\frac{T_3^{\ast }}{T_2^0})-{\omega }_{\mathit{GS}}^3{\mathrm{\Phi }}_2^3(t-\frac{T_1^{\ast }}{T_2^0})\hfill \\ \hfill & +{\omega }_{\mathit{GS}}^2{\omega }_{\mathit{CS}}{\mathrm{\Phi }}_2^2(t-\frac{T_1^{\ast }}{T_2^0}){\mathrm{\Phi }}_3(t-\frac{T_3^{\ast }}{T_2^0})\hfill \\ \hfill & -{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}^2{\mathrm{\Phi }}_2(t-\frac{T_1^{\ast }}{T_2^0}){\mathrm{\Phi }}_3^2(t-\frac{T_3^{\ast }}{T_2^0})],\hfill \\ \hfill & l_2=e_G[{\omega }_{\mathit{SG}}^3{\mathrm{\Phi }}_1^3(t-\frac{T_1^{\ast }}{T_2^0})],\hfill \\ \hfill & l_3=e_E[-{\omega }_{\mathit{SC}}^3{\mathrm{\Phi }}_1^3(t-\frac{T_3^{\ast }}{T_2^0})-{\omega }_{\mathit{CC}}^3{\mathrm{\Phi }}_4^3(t-1)\hfill \\ \hfill & -{\omega }_{\mathit{SC}}^2{\omega }_{\mathit{CC}}{\mathrm{\Phi }}_1^2(t-\frac{T_3^{\ast }}{T_2^0}){\mathrm{\Phi }}_4(t-1)\hfill \\ \hfill & -{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}^2{\mathrm{\Phi }}_1(t-\frac{T_3^{\ast }}{T_2^0}){\mathrm{\Phi }}_4^2(t-1)],\hfill \\ \hfill & l_4=e_I[{\omega }_{\mathit{CC}}^3{\mathrm{\Phi }}_3^3(t-1)],\hfill \end{array}$$ 51

and

$$\begin{array}{c}\hfill \begin{array}{cc}& r_X=\frac{-8X^{\ast }(M_X-X^{\ast })(2X^{\ast }-M_X)}{\tau M_X^4},\hfill \\ \hfill & e_X=\frac{32X^{\ast }(M_X-X^{\ast })(M_X^2+6{X^{\ast }}^2-6M_X{X}^{\ast })}{3\tau M_X^6}\hfill \\ \hfill & (X=S,G,E,I).\hfill \end{array}\end{array}$$ 52

Based on the Riesz representation theorem, there is a $4\times 4$ matrix function $\eta (\theta ,\mu )$ in $\theta \in [-1,0]$ such that

$$\begin{array}{c}\hfill L_{\mu }(\mathrm{\Phi })={\int }_{-1}^0\text{d}\eta (\theta ,\mu )\mathrm{\Phi }(\theta ),\mathrm{\Phi }\in C([-1,0],R^4).\end{array}$$ 53

Now we choose

$$\begin{array}{c}\hfill \eta (\theta ,\mu )=\left\{\begin{array}{cc}& (T_2^0+\mu )(C_1+C_2+C_4+C_3)\theta =0,\hfill \\ \hfill & (T_2^0+\mu )(C_2+C_4+C_3)\theta \in [-\frac{T_1^{\ast }}{T_2^0},0),\hfill \\ \hfill & (T_2^0+\mu )(C_4+C_3)\theta \in [-\frac{T_3^{\ast }}{T_2^0},-\frac{T_1^{\ast }}{T_2^0}),\hfill \\ \hfill & (T_2^0+\mu )C_3\theta \in (-1,-\frac{T_3^{\ast }}{T_2^0}),\hfill \\ \hfill & 0\theta =-1.\hfill \end{array}\right)\end{array}$$ 54

For $\mathrm{\Phi }\in C^1([-1,0],R^4)$, we define

$$\begin{array}{c}\hfill A(\mu )\mathrm{\Phi }=\left\{\begin{array}{cc}& \frac{\text{d}\mathrm{\Phi }(\theta )}{\text{d}\theta }\theta \in [-1,0),\hfill \\ \hfill & {\int }_{-1}^0{\text{d}}_{\xi }\eta (\xi ,\mu )\mathrm{\Phi }(\xi )\theta =0.\hfill \\ \hfill \end{array}\right)\end{array}$$ 55

and

$$\begin{array}{c}\hfill R(\mu )\mathrm{\Phi }=\left\{\begin{array}{cc}& 0\theta \in [-1,0),\hfill \\ \hfill & F(\mu ,\mathrm{\Phi })\theta =0.\hfill \\ \hfill \end{array}\right)\end{array}$$ 56

Then, Eq. (46) can be converted into the following abstract ODE form,

$$\begin{array}{c}\hfill \dot{v}(t)=A(\mu )v(t)+R(\mu )v(t).\end{array}$$ 57

For $\mathrm{\Psi }\in C^1([0,1],{R^4}^{\ast })$, ${R^4}^{\ast }$ is the 4 dimensional complex row vector space. The adjoint operator $A^{\ast }$ of A can be represented as follows,

$$\begin{array}{c}\hfill A^{\ast }\mathrm{\Psi }(s)=\left\{\begin{array}{cc}& \frac{-\text{d}\mathrm{\Psi }(s)}{\text{d}s}s\in (0,1],\hfill \\ \hfill & {\int }_{-1}^0{\text{d}}_s{\eta }^T(s,\mu )\mathrm{\Psi }(-s)s=0.\hfill \\ \hfill \end{array}\right)\end{array}$$ 58

In which, $\mathrm{\Phi }\in C^1([-1,0],R^4)$ and $\mathrm{\Psi }\in C^1([0,1],{R^4}^{\ast })$ are suited to double linear inner product of complex vector,

$$\begin{array}{c}\hfill <\mathrm{\Psi },\mathrm{\Phi }>=\overline{\mathrm{\Psi }}(0)\mathrm{\Phi }(0)-{\int }_{-1}^0{\int }_{\xi =0}^{\theta }\overline{\mathrm{\Psi }}(\xi -\theta )\text{d}\eta (\theta )\mathrm{\Phi }(\xi )\text{d}\xi ,\end{array}$$ 59

which satisfies

$$\begin{array}{c}\hfill <\mathrm{\Psi },A\mathrm{\Phi }>=<A^{\ast }\mathrm{\Psi },\mathrm{\Phi }>.\end{array}$$ 60

According to the above analysis, we know that $\pm i{\omega }_0^{(2)}{T}_2^0$ are the eigenvalues of A(0) and $A^{\ast }(0)$, let $q(\theta )={(1,\alpha ,\beta ,\gamma )}^{-1}{e}^{i{\omega }_0^{(2)}{T}_2^0\theta }$ and $q^{\ast }(s)=D{(1,{\alpha }^{\ast },{\beta }^{\ast },{\gamma }^{\ast })}^{-1}{e}^{i{\omega }_0^{(2)}{T}_2^{0}s}$ are the eigenvectors of A(0) and $A^{\ast }(0)$ corresponding to $i{\omega }_0^{(2)}{T}_2^0$ and $-i{\omega }_0^{(2)}{T}_2^0$, respectively. With a simple calculation, we can obtain

$$\begin{array}{cc}& \alpha =\frac{a_{21}{e}^{-i{\omega }_0^{(2)}{T}_1^{\ast }}}{i{\omega }_0^{(2)}+\frac{1}{\tau }},\hfill \\ \hfill & \beta =\frac{{(i{\omega }_0^{(2)}+\frac{1}{\tau })}^2-a_{12}{a}_{21}{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}}{(i{\omega }_0^{(2)}+\frac{1}{\tau })a_{13}{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }}},\hfill \\ \hfill & \gamma =\frac{{(i{\omega }_0^{(2)}+\frac{1}{\tau })}^2-a_{12}{a}_{21}{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}}{a_{13}{a}_{34}{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }}{e}^{-i{\omega }_0^{(2)}{T}_2^0}}-a_{13}{a}_{34}{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }}{e}^{-i{\omega }_0^{(2)}{T}_2^0},\hfill \\ \hfill & {\alpha }^{\ast }=\frac{a_{12}{e}^{-i{\omega }_0^{(2)}{T}_1^{\ast }}}{\frac{1}{\tau }-i{\omega }_0^{(2)}},\hfill \\ \hfill & {\beta }^{\ast }=\frac{{(\frac{1}{\tau }-i{\omega }_0^{(2)})}^2-a_{12}{a}_{21}{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}}{(\frac{1}{\tau }-i{\omega }_0^{(2)})a_{31}{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }}},\hfill \\ \hfill & {\gamma }^{\ast }=\frac{a_{34}{e}^{-i{\omega }_0^{(2)}{T}_2^0}}{\frac{1}{\tau }-i{\omega }_0^{(2)}}{\beta }^{\ast }.\hfill \end{array}$$ 61

By bilinear inner product, we know

$$\begin{array}{c}\hfill \begin{array}{cc}& <q^{\ast }(s),q(\theta )>={{\overline{q}}^{\ast }}^T(0)q(0)-{\int }_{-1}^0{\int }_{\xi =0}^{\theta }\overline{D}(1,{\overline{\alpha }}^{\ast },{\overline{\beta }}^{\ast },{\overline{\gamma }}^{\ast })e^{-i(\xi -\theta ){\omega }_0^{(2)}{T}_2^0}\text{d}\eta (\theta ,0){(1,\alpha ,\beta ,\gamma )}^T{e}^{i\xi {\omega }_0^{(2)}{T}_2^0}\text{d}\xi \hfill \\ \hfill & ={{\overline{q}}^{\ast }}^T(0)q(0)-{{\overline{q}}^{\ast }}^T(0){\int }_{-1}^0\theta e^{i\theta {\omega }_0^{(2)}{T}_2^0}\text{d}\eta (\theta ,0)q(0)\hfill \\ \hfill & ={{\overline{q}}^{\ast }}^T(0)q(0)+{{\overline{q}}^{\ast }}^T(0)(T_1^{\ast }{C}_2{e}^{-i{\omega }_0^{(2)}{T}_1^{\ast }}+T_2^0{C}_3{e}^{-i{\omega }_0^{(2)}{T}_2^0}+T_3^{\ast }{C}_4{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }})q(0)\hfill \\ \hfill & =\overline{D}[1+\alpha {\overline{\alpha }}^{\ast }+\beta {\overline{\beta }}^{\ast }+\gamma {\overline{\gamma }}^{\ast }\hfill \\ \hfill & +T_1^{\ast }{e}^{-i{\omega }_0^{(2)}{T}_1^{\ast }}(a_{21}{\overline{\alpha }}^{\ast }+a_{12}\alpha )+T_3^{\ast }{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }}(a_{31}{\overline{\beta }}^{\ast }+a_{13}\beta )\hfill \\ \hfill & +T_2^0{e}^{-i{\omega }_0^{(2)}{T}_2^0}(a_{43}\beta {\overline{\gamma }}^{\ast }{a}_{34}\gamma {\overline{\beta }}^{\ast })]\hfill \\ \hfill & =1.\hfill \end{array}\end{array}$$ 62

Therefore, we have,

$$\begin{array}{c}\hfill \overline{D}=\frac{1}{m},\end{array}$$ 63

and

$$\begin{array}{c}\hfill D=\frac{1}{n},\end{array}$$ 64

where

$$\begin{array}{cc}& m=1+\alpha {\overline{\alpha }}^{\ast }+\beta {\overline{\beta }}^{\ast }+\gamma {\overline{\gamma }}^{\ast }+T_1^{\ast }{e}^{-i{\omega }_0^{(2)}{T}_1^{\ast }}(a_{21}{\overline{\alpha }}^{\ast }+a_{12}\alpha )\hfill \\ \hfill & +T_3^{\ast }{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }}(a_{31}{\overline{\beta }}^{\ast }+a_{13}\beta )+T_2^0{e}^{-i{\omega }_0^{(2)}{T}_2^0}(a_{43}\beta {\overline{\gamma }}^{\ast }+a_{34}\gamma {\overline{\beta }}^{\ast }),\hfill \\ \hfill & n=1+\overline{\alpha }{\alpha }^{\ast }+\overline{\beta }{\beta }^{\ast }+\overline{\gamma }{\gamma }^{\ast }+T_1^{\ast }{e}^{i{\omega }_0^{(2)}{T}_1^{\ast }}(a_{21}{\alpha }^{\ast }+a_{12}\overline{\alpha })\hfill \\ \hfill & +T_3^{\ast }{e}^{i{\omega }_0^{(2)}{T}_3^{\ast }}(a_{31}{\beta }^{\ast }+a_{13}\overline{\beta })+T_2^0{e}^{i{\omega }_0^{(2)}{T}_2^0}(a_{43}\overline{\beta }{\gamma }^{\ast }+a_{34}\overline{\gamma }{\beta }^{\ast }).\hfill \end{array}$$ 65

We compute the coordinates to describe the center manifold $C_0$ at $\mu =0$. Let $v_t$ be the solution of Eq. (46) at $\mu =0$.

Define

$$\begin{array}{c}\hfill z(t)=<q^{\ast }(s),v_t>,\end{array}$$ 66

and

$$\begin{array}{c}\hfill \begin{array}{cc}& W(t,\theta )=v_t(\theta )-z(t)q(\theta )-\overline{z}(t)\overline{q}(\theta )\hfill \\ \hfill & =v_t(\theta )-2Re\{z(t)q(\theta )\}.\hfill \end{array}\end{array}$$ 67

On the center manifold $C_0$, $W(t,\theta )=W(z,\overline{z},\theta )$, in which $W(z,\overline{z},\theta )$ can be expressed as,

$$\begin{array}{c}\hfill W(z,\overline{z},\theta )=W_{20}(\theta )\frac{z^2}{2}+W_{11}(\theta )z\overline{z}+W_{02}(\theta )\frac{{\overline{z}}^2}{2}+...,\end{array}$$ 68

where z and $\overline{z}$ are local coordinates for the $C_0$ in the direction of $q^{\ast }$ and $\overline{q}\ast$ , respectively. The flow of Eq. (46) on the central manifold is determined by the following equation,

$$\begin{array}{c}\hfill \dot{z}(t)=i{\omega }_0^{(2)}{T}_2^{0}z(t)+{\overline{q}}^{\ast }(0)F_0(z,\overline{z}),\end{array}$$ 69

where

$$\begin{array}{c}\hfill F_0(z,\overline{z})=F(0,W(z,\overline{z},\theta )+z(t)q(0)+\overline{z}(t)\overline{q}(0)).\end{array}$$ 70

Let

$$\begin{array}{c}\hfill F_0(z,\overline{z})=F_{z^2}\frac{z^2}{2}+F_{z\overline{z}}z\overline{z}+F_{{\overline{z}}^2}\frac{{\overline{z}}^2}{2}+F_{z^2\overline{z}}\frac{z^2\overline{z}}{2}+...\end{array}$$ 71

Eq. (69) can be rewritten as,

$$\begin{array}{c}\hfill \dot{z}(t)=i{\omega }_0^{(2)}{T}_2^{0}z(t)+g(z,\overline{z})(t),\end{array}$$ 72

where

$$\begin{array}{c}\hfill g(z,\overline{z})(t)=g_{20}\frac{z^2}{2}+g_{11}z\overline{z}+g_{02}\frac{{\overline{z}}^2}{2}+g_{21}\frac{z^2\overline{z}}{2}+...\end{array}$$ 73

and

$$\begin{array}{c}\hfill \begin{array}{cc}& g_{20}={\overline{q}}^{\ast }(0)F_{z^2},g_{11}={\overline{q}}^{\ast }(0)F_{z\overline{z}},g_{02}={\overline{q}}^{\ast }(0)F_{{\overline{z}}^2},g_{21}={\overline{q}}^{\ast }(0)F_{z^2\overline{z}}.\hfill \end{array}\end{array}$$ 74

According to Eqs. (66) - (68), we have

$$\begin{array}{c}\hfill \begin{array}{cc}& v_t=W(t,\theta )+2Re\{z(t)q(\theta )\}\hfill \\ \hfill & =W_{20}(\theta )\frac{z^2}{2}+W_{11}(\theta )z\overline{z}+W_{02}(\theta )\frac{{\overline{z}}^2}{2}+z(t)q(\theta )+\overline{z}(t)\overline{q}(\theta ).\hfill \end{array}\end{array}$$ 75

From Eq.(49), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}& g(z,\overline{z})(t)={\overline{q}}^{\ast }(0)F_0(z,\overline{z})\hfill \\ \hfill & ={\overline{q}}^{\ast }(0)F(0,v_t).\hfill \end{array}\end{array}$$ 76

Inserting Eq. (75) into Eq.(76) and comparing the coefficients with Eq. (73), we get,

$$\begin{array}{cc}& g_{20}=\frac{{\partial }^{2}g(0,0)}{\partial z^2}=T_2^0\overline{D}(\frac{{\partial }^2{F}_{2S}}{\partial z^2}+{\overline{\alpha }}^{\ast }\frac{{\partial }^2{F}_{2G}}{\partial z^2}+{\overline{\beta }}^{\ast }\frac{{\partial }^2{F}_{2E}}{\partial z^2}+{\overline{\gamma }}^{\ast }\frac{{\partial }^2{F}_{2I}}{\partial z^2}),\hfill \\ \hfill & g_{02}=\frac{{\partial }^{2}g(0,0)}{\partial {\overline{z}}^2}=T_2^0\overline{D}(\frac{{\partial }^2{F}_{2S}}{\partial {\overline{z}}^2}+{\overline{\alpha }}^{\ast }\frac{{\partial }^2{F}_{2G}}{\partial {\overline{z}}^2}+{\overline{\beta }}^{\ast }\frac{{\partial }^2{F}_{2E}}{\partial {\overline{z}}^2}+{\overline{\gamma }}^{\ast }\frac{{\partial }^2{F}_{2I}}{\partial {\overline{z}}^2}),\hfill \\ \hfill & g_{11}=\frac{{\partial }^{2}g(0,0)}{\partial z\partial \overline{z}}=T_2^0\overline{D}(\frac{{\partial }^2{F}_{2S}}{\partial z\partial \overline{z}}+{\overline{\alpha }}^{\ast }\frac{{\partial }^2{F}_{2G}}{\partial z\partial \overline{z}}+{\overline{\beta }}^{\ast }\frac{{\partial }^2{F}_{2E}}{\partial z\partial \overline{z}}+{\overline{\gamma }}^{\ast }\frac{{\partial }^2{F}_{2I}}{\partial z\partial \overline{z}}).\hfill \end{array}$$ 77

where

$$\begin{array}{cc}& \frac{{\partial }^2{F}_{2S}}{\partial z^2}=2r_S[{\omega }_{\mathit{GS}}^2{\alpha }^2{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}+{\omega }_{\mathit{CS}}^2{\beta }^2{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}\hfill \\ \hfill & -2{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}\alpha \beta e^{-i{\omega }_0^{(2)}(T_1^{\ast }+T_3^{\ast })}],\hfill \\ \hfill & \frac{{\partial }^2{F}_{2G}}{\partial z^2}=2r_G[{\omega }_{\mathit{SG}}^2{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}],\hfill \\ \hfill & \frac{{\partial }^2{F}_{2E}}{\partial z^2}=2r_E[{\omega }_{\mathit{SC}}^2{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}+{\omega }_{\mathit{CC}}^2{\gamma }^2{e}^{-2i{\omega }_0^{(2)}{T}_2^0}+2{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}{e}^{-i{\omega }_0^{(2)}(T_3^{\ast }+T_2^0)}],\hfill \\ \hfill & \frac{{\partial }^2{F}_{2I}}{\partial z^2}=2r_I[{\omega }_{\mathit{CC}}^2{\beta }^2{e}^{-2i{\omega }_0^{(2)}{T}_2^0}],\hfill \\ \hfill & \frac{{\partial }^2{F}_{2X}}{\partial {\overline{z}}^2}=conj(\frac{{\partial }^2{F}_{2X}}{\partial z^2}),(X=S,G,E,I),\hfill \\ \hfill & \frac{{\partial }^2{F}_{2S}}{\partial z\partial \overline{z}}=2r_S[{\omega }_{\mathit{GS}}^2\alpha \overline{\alpha }+{\omega }_{\mathit{CS}}^2\beta \overline{\beta }-{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}(\alpha \overline{\beta }{e}^{-i{\omega }_0^{(2)}(T_1^{\ast }-T_3^{\ast })}+\overline{\alpha }\beta e^{i{\omega }_0^{(2)}(T_1^{\ast }-T_3^{\ast })})],\hfill \\ \hfill & \frac{{\partial }^2{F}_{2G}}{\partial z\partial \overline{z}}=2r_G{\omega }_{\mathit{SG}}^2,\hfill \\ \hfill & \frac{{\partial }^2{F}_{2E}}{\partial z\partial \overline{z}}=2r_E[{\omega }_{\mathit{SC}}^2+{\omega }_{\mathit{CC}}^2\gamma \overline{\gamma }+{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}(\overline{\gamma }{e}^{-i{\omega }_0^{(2)}(T_3^{\ast }-T_2^0)}+\gamma e^{i{\omega }_0^{(2)}(T_3^{\ast }-T_2^0)})],\hfill \\ \hfill & \frac{{\partial }^2{F}_{2I}}{\partial z\partial \overline{z}}=2r_I[{\omega }_{\mathit{CC}}^2\beta \overline{\beta }].\hfill \end{array}$$ 78

About $g_{21}$, we have,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill g_{21}=\frac{{\partial }^{3}g(0,0)}{\partial z^2\partial \overline{z}}=T_2^0\overline{D}(\frac{{\partial }^3{F}_{2S}}{\partial z^2\partial \overline{z}}+{\overline{\alpha }}^{\ast }\frac{{\partial }^3{F}_{2G}}{\partial z^2\partial \overline{z}}+{\overline{\beta }}^{\ast }\frac{{\partial }^3{F}_{2E}}{\partial z^2\partial \overline{z}}+{\overline{\gamma }}^{\ast }\frac{{\partial }^3{F}_{2I}}{\partial z^2\partial \overline{z}})\\ \hfill +T_2^0\overline{D}(\frac{{\partial }^3{F}_{3S}}{\partial z^2\partial \overline{z}}+{\overline{\alpha }}^{\ast }\frac{{\partial }^3{F}_{3G}}{\partial z^2\partial \overline{z}}+{\overline{\beta }}^{\ast }\frac{{\partial }^3{F}_{3E}}{\partial z^2\partial \overline{z}}+{\overline{\gamma }}^{\ast }\frac{{\partial }^3{F}_{3I}}{\partial z^2\partial \overline{z}}),\end{array}\end{array}$$ 79

in which

$$\begin{array}{cc}& \frac{{\partial }^3{F}_{2S}}{\partial z^2\partial \overline{z}}=2r_S[{\omega }_{\mathit{GS}}^2(2\alpha W_{11}^{(2)}(-\frac{T_1^{\ast }}{T_2^0})e^{-i{\omega }_0^{(2)}{T}_1^{\ast }}+\overline{\alpha }{W}_{20}^{(2)}\hfill \\ \hfill & (-\frac{T_1^{\ast }}{T_2^0})e^{i{\omega }_0^{(2)}{T}_1^{\ast }})+{\omega }_{\mathit{CS}}^2(2\beta W_{11}^{(3)}(-\frac{T_3^{\ast }}{T_2^0})e^{-i{\omega }_0^{(2)}{T}_3^{\ast }}\hfill \\ \hfill & +\overline{\beta }{W}_{20}^{(3)}(-\frac{T_3^{\ast }}{T_2^0})e^{i{\omega }_0^{(2)}{T}_3^{\ast }})-2{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}(\beta W_{11}^{(2)}\hfill \\ \hfill & (-\frac{T_1^{\ast }}{T_2^0})e^{-i{\omega }_0^{(2)}{T}_3^{\ast }}+\alpha W_{11}^{(3)}(-\frac{T_3^{\ast }}{T_2^0})e^{-i{\omega }_0^{(2)}{T}_1^{\ast }})\hfill \\ \hfill & -{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}(\overline{\beta }{W}_{20}^{(2)}(-\frac{T_1^{\ast }}{T_2^0})e^{i{\omega }_0^{(2)}{T}_3^{\ast }}\hfill \\ \hfill & +\overline{\alpha }{W}_{20}^{(3)}(-\frac{T_3^{\ast }}{T_2^0})e^{i{\omega }_0^{(2)}{T}_1^{\ast }})],\hfill \\ \hfill & \frac{{\partial }^3{F}_{2G}}{\partial z^2\partial \overline{z}}=2r_G[{\omega }_{\mathit{SG}}^2(2W_{11}^{(1)}(-\frac{T_1^{\ast }}{T_2^0})e^{-i{\omega }_0^{(2)}{T}_1^{\ast }}+W_{20}^{(1)}\hfill \\ \hfill & (-\frac{T_1^{\ast }}{T_2^0})e^{i{\omega }_0^{(2)}{T}_1^{\ast }})],\hfill \\ \hfill & \frac{{\partial }^3{F}_{2E}}{\partial z^2\partial \overline{z}}=2r_E[{\omega }_{\mathit{SC}}^2(2W_{11}^{(1)}(-\frac{T_3^{\ast }}{T_2^0})e^{-i{\omega }_0^{(2)}{T}_3^{\ast }}+W_{20}^{(1)}\hfill \\ \hfill & (-\frac{T_3^{\ast }}{T_2^0})e^{i{\omega }_0^{(2)}{T}_3^{\ast }})+{\omega }_{\mathit{CC}}^2(2\gamma W_{11}^{(4)}(-1)e^{-i{\omega }_0^{(2)}{T}_2^0}\hfill \\ \hfill & +\overline{\gamma }{W}_{20}^{(4)}(-1)e^{i{\omega }_0^{(2)}{T}_2^0})+2{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}(\gamma W_{11}^{(1)}\hfill \\ \hfill & (-\frac{T_3^{\ast }}{T_2^0})e^{-i{\omega }_0^{(2)}{T}_2^0}+W_{11}^{(4)}(-1)e^{-i{\omega }_0^{(2)}{T}_3^{\ast }})\hfill \\ \hfill & +{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}(\overline{\gamma }{W}_{20}^{(1)}(-\frac{T_3^{\ast }}{T_2^0})e^{i{\omega }_0^{(2)}{T}_2^0}\hfill \\ \hfill & +W_{20}^{(4)}(-1)e^{i{\omega }_0^{(2)}{T}_3^{\ast }}))],\hfill \\ \hfill & \frac{{\partial }^3{F}_{2I}}{\partial z^2\partial \overline{z}}=2r_I[{\omega }_{\mathit{CC}}^2(2\beta W_{11}^{(3)}(-1)e^{-i{\omega }_0^{(2)}{T}_2^0}+\overline{\beta }{W}_{20}^{(3)}(-1)e^{i{\omega }_0^{(2)}{T}_2^0})],\hfill \\ \hfill & \frac{{\partial }^3{F}_{3S}}{\partial z^2\partial \overline{z}}=6e_S[{\omega }_{\mathit{CS}}^3{\beta }^2\overline{\beta }{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }}-{\omega }_{\mathit{GS}}^3{\alpha }^2\overline{\alpha }{e}^{-i{\omega }_0^{(2)}{T}_1^{\ast }}\hfill \\ \hfill & +{\omega }_{\mathit{GS}}^2{\omega }_{\mathit{CS}}({\alpha }^2\overline{\beta }{e}^{-i{\omega }_0^{(2)}(2T_1^{\ast }-T_3^{\ast })}+2\alpha \overline{\alpha }\beta e^{-i{\omega }_0^{(2)}{T}_3^{\ast }})\hfill \\ \hfill & -{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}^2({\beta }^2\overline{\alpha }{e}^{-i{\omega }_0^{(2)}(2T_3^{\ast }-T_1^{\ast })}+2\beta \overline{\beta }\alpha e^{-i{\omega }_0^{(2)}{T}_1^{\ast }})],\hfill \\ \hfill & \frac{{\partial }^3{F}_{3G}}{\partial z^2\partial \overline{z}}=6e_G[{\omega }_{\mathit{SG}}^3{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}],\hfill \\ \hfill & \frac{{\partial }^3{F}_{3E}}{\partial z^2\partial \overline{z}}=6e_E[-{\omega }_{\mathit{SC}}^3{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }}-{\omega }_{\mathit{CC}}^3{\gamma }^2\overline{\gamma }{e}^{-i{\omega }_0^{(2)}{T}_2^0}\hfill \\ \hfill & -{\omega }_{\mathit{SC}}^2{\omega }_{\mathit{CC}}(\overline{\gamma }{e}^{-i{\omega }_0^{(2)}(2T_3^{\ast }-T_2^0)}+2\gamma e^{-i{\omega }_0^{(2)}{T}_2^0})\hfill \\ \hfill & -{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}^2({\gamma }^2{e}^{-i{\omega }_0^{(2)}(2T_2^0-T_3^{\ast })}\hfill \\ \hfill & +2\gamma \overline{\gamma }{e}^{-i{\omega }_0^{(2)}{T}_3^{\ast }})],\hfill \\ \hfill & \frac{{\partial }^3{F}_{3I}}{\partial z^2\partial \overline{z}}=6e_I[{\omega }_{\mathit{CC}}^3{\beta }^2\overline{\beta }{e}^{-i{\omega }_0^{(2)}{T}_2^0}].\hfill \end{array}$$ 80

Since $W_{11}(\theta )$ and $W_{20}(\theta )$ are unknown, we further calculate them in the following. From Eq. (67), we have $\dot{W}={\dot{v}}_t-\dot{z}q-\dot{\overline{z}}\overline{q}$, with Eqs. (57), (66) and (68), and catch

$$\begin{array}{c}\hfill \dot{W}=\left\{\begin{array}{cc}& AW-gq(\theta )-\overline{g}\overline{q}(\theta ),\theta \in [-1,0),\hfill \\ \hfill & AW-gq(0)-\overline{g}\overline{q}(0)+F_0\theta =0.\hfill \end{array}\right)\end{array}$$ 81

On the other hand, when $C_0$ is near to the origin, from Eq. (68)

$$\begin{array}{c}\hfill \begin{array}{cc}& \dot{W}=W_z\dot{z}+W_{\overline{z}}\dot{\overline{z}}\hfill \\ \hfill & =[W_{20}(\theta )z+W_{11}(\theta )\overline{z}]\dot{z}+[W_{11}(\theta )z+W_{02}(\theta )\overline{z}]\dot{\overline{z}}\hfill \\ \hfill & =[W_{20}(\theta )z+W_{11}(\theta )\overline{z}](i{\omega }_{0}z+g(z,\overline{z}))+[W_{11}(\theta )z+W_{02}(\theta )\overline{z}](-i{\omega }_0\overline{z}+\overline{g}(z,\overline{z}))+...\hfill \\ \hfill & =2i{\omega }_0{W}_{20}\frac{z^2}{2}-i{\omega }_0{W}_{02}{\overline{z}}^2+...\hfill \end{array}\end{array}$$ 82

Substitute Eqs. (72) and (73) into Eq. (81), comparing coefficient with Eq. (82) about $\frac{z^2}{2}$ and $z\overline{z}$, then

$$\begin{array}{c}\hfill (2i{\omega }_0^{(2)}{T}_2^{0}I-A)W_{20}(\theta )=\left\{\begin{array}{cc}& -g_{20}q(\theta )-{\overline{g}}_{02}\overline{q}(\theta )\theta \in [-1,0),\hfill \\ \hfill & -g_{20}q(0)-{\overline{g}}_{02}\overline{q}(0)+F_{z^2}\theta =0.\hfill \end{array}\right)\end{array}$$ 83

and

$$\begin{array}{c}\hfill -A{W}_{11}(\theta )=\left\{\begin{array}{cc}& -g_{11}q(\theta )-{\overline{g}}_{11}\overline{q}(\theta )\theta \in [-1,0),\hfill \\ \hfill & -g_{11}q(0)-{\overline{g}}_{11}\overline{q}(0)+F_{z\overline{z}}\theta =0.\hfill \end{array}\right)\end{array}$$ 84

From Eq. (55) and Eq. (83), when $\theta \in [-1,0)$

$$\begin{array}{c}\hfill W_{20}^{\prime }(\theta )=2i{\omega }_0^{(2)}{T}_2^0{W}_{20}(\theta )+g_{20}q(\theta )+{\overline{g}}_{02}\overline{q}(\theta ).\end{array}$$ 85

We get

$$\begin{array}{c}\hfill W_{20}(\theta )=\frac{-i{g}_{20}}{{\omega }_0^{(2)}{T}_2^0}q(0)e^{i{\omega }_0^{(2)}{T}_2^0\theta }+\frac{i{\overline{g}}_{02}}{{\omega }_0^{(2)}{T}_2^0}\overline{q}(0)e^{-i{\omega }_0^{(2)}{T}_2^0\theta }+E_1{e}^{2i{\omega }_0^{(2)}{T}_2^0\theta }.\end{array}$$ 86

For $\theta =0$

$$\begin{array}{c}\hfill {\int }_{-1}^0{\text{d}}_{\theta }\eta (0,\theta )W_{20}(\theta )=2i{\omega }_0^{(2)}{T}_2^0{W}_{20}(\theta )+g_{20}q(0)+{\overline{g}}_{02}\overline{q}(0)-F_{z^2}.\end{array}$$ 87

Substituting Eq. (86) into Eq. (87), and $(3i{\omega }_0^{(2)}{T}_2^{0}I-{\int }_{-1}^0{e}^{i{\omega }_0^{(2)}{T}_2^0\theta }{\text{d}}_{\theta }\eta (0,\theta ))q(0)=0$, we have

$$\begin{array}{c}\hfill E_1={[2i{\omega }_0^{(2)}{T}_2^{0}I-{\int }_{-1}^0{e}^{2i{\omega }_0^{(2)}{T}_2^0\theta }\text{d}\eta (0,\theta )]}^{-1}{F}_{z^2},\end{array}$$ 88

where

$$\begin{array}{c}\hfill \begin{array}{cc}& {\int }_{-1}^0{e}^{2i{\omega }_0^{(2)}{T}_2^0\theta }{\text{d}}_{\theta }\eta (0,\theta )=T_2^0{C}_1+T_2^0{C}_2{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}+T_2^0{C}_4{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}+T_2^0{C}_3{e}^{-2i{\omega }_0^{(2)}{T}_2^0}\hfill \\ \hfill & =T_2^0\left(\begin{array}{cccc}-\frac{1}{\tau }& a_{12}{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}& a_{13}{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}& 0\\ a_{21}{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}& -\frac{1}{\tau }& 0& 0\\ a_{31}{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}& 0& -\frac{1}{\tau }& a_{34}{e}^{-2i{\omega }_0^{(2)}{T}_2^0}\\ 0& 0& a_{43}{e}^{-2i{\omega }_0^{(2)}{T}_2^0}& -\frac{1}{\tau }\end{array}\right),\hfill \end{array}\end{array}$$ 89

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill F_{z^2}=T_2^0\left(\begin{array}{c}2r_S[{\omega }_{\mathit{GS}}^2{\alpha }^2{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}+{\omega }_{\mathit{CS}}^2{\beta }^2{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}-2{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}\alpha \beta e^{-i{\omega }_0^{(2)}(T_1^{\ast }+T_3^{\ast })}]\\ 2r_G[{\omega }_{\mathit{SG}}^2{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}]\\ 2r_E[{\omega }_{\mathit{SC}}^2{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}+{\omega }_{\mathit{CC}}^2{\gamma }^2{e}^{-2i{\omega }_0^{(2)}{T}_2^0}+2{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}{e}^{-i{\omega }_0^{(2)}(T_3^{\ast }+T_2^0)}]\\ 2r_I[{\omega }_{\mathit{CC}}^2{\beta }^2{e}^{-2i{\omega }_0^{(2)}{T}_2^0}]\end{array}\right).\end{array}\end{array}$$ 90

So, we can obtain $E_1$ as follows

$$\begin{array}{c}\hfill \begin{array}{cc}& E_1={\left(\begin{array}{cccc}2i{\omega }_0^{(2)}{T}_2^0+\frac{1}{\tau }& -a_{12}{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}& -a_{13}{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}& 0\\ -a_{21}{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}& 2i{\omega }_0^{(2)}{T}_2^0+\frac{1}{\tau }& 0& 0\\ -a_{31}{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}& 0& 2i{\omega }_0^{(2)}{T}_2^0+\frac{1}{\tau }& -a_{34}{e}^{-2i{\omega }_0^{(2)}{T}_2^0}\\ 0& 0& -a_{43}{e}^{-2i{\omega }_0^{(2)}{T}_2^0}& 2i{\omega }_0^{(2)}{T}_2^0+\frac{1}{\tau }\end{array}\right)}^{-1}\hfill \\ \hfill & \left(\begin{array}{c}2r_S[{\omega }_{\mathit{GS}}^2{\alpha }^2{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}+{\omega }_{\mathit{CS}}^2{\beta }^2{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}-2{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}\alpha \beta e^{-i{\omega }_0^{(2)}(T_1^{\ast }+T_3^{\ast })}]\\ 2r_G[{\omega }_{\mathit{SG}}^2{e}^{-2i{\omega }_0^{(2)}{T}_1^{\ast }}]\\ 2r_E[{\omega }_{\mathit{SC}}^2{e}^{-2i{\omega }_0^{(2)}{T}_3^{\ast }}+{\omega }_{\mathit{CC}}^2{\gamma }^2{e}^{-2i{\omega }_0^{(2)}{T}_2^0}+2{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}{e}^{-i{\omega }_0^{(2)}(T_3^{\ast }+T_2^0)}]\\ 2r_I[{\omega }_{\mathit{CC}}^2{\beta }^2{e}^{-2i{\omega }_0^{(2)}{T}_2^0}]\end{array}\right).\hfill \end{array}\end{array}$$ 91

Substituting $E_1$ into Eq. (86), we can calculate the value of $W_{20}(\theta )$. Similarly, by Eq. (55) and Eq. (84) for $\theta \in [-1,0)$ we will get

$$\begin{array}{c}\hfill W_{11}^{\prime }(\theta )=g_{11}q(\theta )+{\overline{g}}_{11}\overline{q}(\theta ).\end{array}$$ 92

We have,

$$\begin{array}{c}\hfill W_{11}(\theta )=\frac{-i{g}_{11}}{{\omega }_0^{(2)}{T}_2^0}q(0)e^{i{\omega }_0^{(2)}{T}_2^0\theta }+\frac{i{\overline{g}}_{11}}{{\omega }_0^{(2)}{T}_2^0}\overline{q}(0)e^{-i{\omega }_0^{(2)}{T}_2^0\theta }+E_2.\end{array}$$ 93

For $\theta =0$,

$$\begin{array}{c}\hfill {\int }_{-1}^0\text{d}\eta (0,\theta )W_{11}(\theta )=g_{11}q(0)+{\overline{g}}_{11}\overline{q}(0)-F_{z\overline{z}},\end{array}$$ 94

Substituting Eq. (93) into Eq. (94), one can obtain

$$\begin{array}{c}\hfill E_2=-{[{\int }_{-1}^0{\text{d}}_{\theta }\eta (0,\theta )]}^{-1}{F}_{z\overline{z}},\end{array}$$ 95

where

$$\begin{array}{c}\hfill \begin{array}{cc}& {\int }_{-1}^0{\text{d}}_{\theta }\eta (0,\theta )\hfill \\ \hfill & =T_2^0{C}_1+T_2^0{C}_2+T_2^0{C}_4+T_2^0{C}_3=T_2^0\left(\begin{array}{cccc}-\frac{1}{\tau }& a_{12}& a_{13}& 0\\ a_{21}& -\frac{1}{\tau }& 0& 0\\ a_{31}& 0& -\frac{1}{\tau }& a_{34}\\ 0& 0& a_{43}& -\frac{1}{\tau }\end{array}\right),\hfill \end{array}\end{array}$$ 96

and

$$\begin{array}{c}\hfill \begin{array}{c}\hfill F_{z\overline{z}}=T_2^0\left(\begin{array}{c}2r_S[{\omega }_{\mathit{GS}}^2\alpha \overline{\alpha }+{\omega }_{\mathit{CS}}^2\beta \overline{\beta }-{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}(\alpha \overline{\beta }{e}^{-i{\omega }_0^{(2)}(T_1^{\ast }-T_3^{\ast })}+\overline{\alpha }\beta e^{i{\omega }_0^{(2)}(T_1^{\ast }-T_3^{\ast })})]\\ 2r_G{\omega }_{\mathit{SG}}^2\\ 2r_E[{\omega }_{\mathit{SC}}^2+{\omega }_{\mathit{CC}}^2\gamma \overline{\gamma }+{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}(\overline{\gamma }{e}^{-i{\omega }_0^{(2)}(T_3^{\ast }-T_2^0)}+\gamma e^{i{\omega }_0^{(2)}(T_3^{\ast }-T_2^0)})]\\ 2r_I({\omega }_{\mathit{CC}}^2\beta \overline{\beta })\end{array}\right).\end{array}\end{array}$$ 97

Therefore, we can infer the following equation,

$$\begin{array}{c}\hfill \begin{array}{cc}& E_2=-{\left(\begin{array}{cccc}-\frac{1}{\tau }& a_{12}& a_{13}& 0\\ a_{21}& -\frac{1}{\tau }& 0& 0\\ a_{31}& 0& -\frac{1}{\tau }& a_{34}\\ 0& 0& a_{43}& -\frac{1}{\tau }\end{array}\right)}^{-1}\hfill \\ & \left(\begin{array}{c}2r_S[{\omega }_{\mathit{GS}}^2\alpha \overline{\alpha }+{\omega }_{\mathit{CS}}^2\beta \overline{\beta }-{\omega }_{\mathit{GS}}{\omega }_{\mathit{CS}}(\alpha \overline{\beta }{e}^{-i{\omega }_0^{(2)}(T_1^{\ast }-T_3^{\ast })}+\overline{\alpha }\beta e^{i{\omega }_0^{(2)}(T_1^{\ast }-T_3^{\ast })})]\\ 2r_G{\omega }_{\mathit{SG}}^2\\ 2r_E[{\omega }_{\mathit{SC}}^2+{\omega }_{\mathit{CC}}^2\gamma \overline{\gamma }+{\omega }_{\mathit{SC}}{\omega }_{\mathit{CC}}(\overline{\gamma }{e}^{-i{\omega }_0^{(2)}(T_3^{\ast }-T_2^0)}+\gamma e^{i{\omega }_0^{(2)}(T_3^{\ast }-T_2^0)})]\\ 2r_I({\omega }_{\mathit{CC}}^2\beta \overline{\beta })\end{array}\right).\hfill \end{array}\end{array}$$ 98

Finally, we should compute the following equation,

$$\begin{array}{cc}& C_1(0)=\frac{i}{2{\omega }_0^{(2)}}(g_{11}{g}_{20}-2|g_{11}{|}^2-\frac{|g_{02}{|}^2}{3})+\frac{g_{21}}{2},\hfill \\ \hfill & {\mu }_2=-\frac{Re{C}_1(0)}{Re({\lambda }^{\prime }(T_2^0))},\hfill \\ \hfill & {\beta }_2=2Re{C}_1(0),\hfill \\ \hfill & {\tau }_2=-\frac{Im{C}_1(0)+{\mu }_{2}Im({\lambda }^{\prime }(T_2^0))}{{\omega }_0^{(2)}}.\hfill \end{array}$$ 99

Therefore, we have the following results,

1. The sign of ${\mu }_2$ determines the direction of the Hopf bifurcation, if ${\mu }_2>0({\mu }_2<0)$, the Hopf bifurcation is supercritical (subcritical);

2. The sign of ${\tau }_2$ determines the period of the bifurcating periodic solutions: if ${\tau }_2>0({\tau }_2<0)$, the period increase (decrease);

3. The sign of ${\beta }_2$ determines the stability of the bifurcating periodic solutions: if ${\beta }_2<0({\beta }_2>0)$, the bifurcating periodic solutions are stable (unstable).

Hopf bifurcation induced by multiple time delay through numerical simulations

In this section, some numerical simulations are given to support the theoretical results in Sect. “Hopf bifurcation induced by multiple time  delay through theoretical analyses” through time series of firing rate of STN, GPE, EXI and INN in Fig. 2, in Figs. 3, 4, 5b–d, and one parameter bifurcation diagrams of firing rate of STN with respect to time delay in Fig. 3a, 4a, 5a. These figures are drawn by MATLAB software with all parameters in Table 1.

Fig. 2.

Time series of firing rate of STN, GPe, EXN and INN at $T_1=T_2=T_3=0$

Fig. 3.

(a) Bifurcation diagram of firing rate of STN with respect to $T_3$ at $T_1=T_2=0$. HB is Hopf bifurcation point with $T_3=0.00185$. (b) - (d) Time series of firing rate of STN at $T_3=0.0014$ (b) $T_3=0.00185$ (c) $T_3=0.0022$ (d)

Fig. 4.

(a) Bifurcation diagram of firing rate of STN with respect to $T_1$ at $T_2=0$ and $T_3=0.00136$ . HB is Hopf bifurcation point with $T_1=0.0021$. (b) - (d) Time series of firing rate of STN at $T_1=0.0015$ (b) $T_1=0.0021$ (c) $T_1=0.0026$ (d)

Fig. 5.

(a) Bifurcation diagram of firing rate of STN with respect to $T_2$ at $T_1=0.0011$ and $T_3=0.00136$. HB is Hopf bifurcation point with $T_2=0.00095$. (b) - (d) Time series of firing rate of STN at $T_2=0.0005$ (b) $T_2=0.00095$ (c) $T_2=0.0013$ (d)

First, for $T_1=0$, $T_2=0$ and $T_3=0$ in Case 1, the system (3) has a positive equilibrium $u=(17.71,77.98,38.54,22.39)$ corresponding to a stable steady state on the time series of firing rate of STN, GPe, EXN and INN in Fig. 2. Then the conditions in Theorem 1 are satisfied with $b_1=400>0$, $b_2=1.3562\times {10}^5>0$, $b_3=1.9125\times {10}^7>0$, $b_4=8.2277\times {10}^8>0$, $b_1{b}_2-b_3=3.5125\times {10}^7>0$, $b_1{b}_2{b}_3-b_1^2{b}_4-b_3^2=5.401\times {10}^{14}>0$ in Eq. (10).

Then, in Case 2 - Case 4, for the conditions in Theorem 2 - Theorem 4, the critical time delays are obtained and verified through one parameter bifurcation diagrams of the firing rate of STN with respect to time delay in Fig. 3(a), Fig. 4(a) and Fig. 5(a), where the black solid lines are stable steady state and green dots are the maximum and minimum of oscillation.

In Case 2, for $T_1=T_2=0$ and the conditions in Theorem 2, we get the critical value of $T_3$, $T_3^0=0.00185$ labeled by HB in Fig. 3(a). Furthermore, the firing rate of STN reaches a stable steady state for $T_3=0.0014<T_3^0$ in Fig. 3(b) and shows oscillation for $T_3=0.0022>T_3^0$ in Fig. 3(d), and undergoes damped oscillation for $T_3^0=0.00185$ in Fig. 3(c).

Similarly, for $T_2=0,T_3=0.00136\in (0,0.00184)$ in Case 3, the critical value of $T_1$, $T_1^0=0.0021$, is obtained for the conditions in Theorem 3. Fig. 4(a) shows the firing rate of STN undergoes Hopf bifurcation at $T_1^0=0.0021$. Also, the firing rate of STN switches from a stable steady state at $T_1=0.0015<T_1^0$ in Fig. 4(b) to oscillation at $T_1=0.0026>T_1^0$ in Fig. 4(d) through damped oscillation at $T_1^0=0.0021$ in Fig. 4(c).

Lastly, for $T_1=0.0011\in (0,T_1^0),T_3=0.00136\in (0,T_3^0)$ in Case 4, the critical value of $T_2^0=0.00095$ is consistent with Hopf bifurcation point HB in Fig. 5(a). The firing rate of STN show damped oscillation at $T_2^0=0.00095$ in Fig.5(c), and converges to a stable steady state at $T_2=0.0005<T_2^0$ in Fig.5(b) while show oscillation at $T_2=0.0013>T_2^0$ in Fig.5(d).

Furthermore, in order to gain a clearer and more comprehensive understanding of the impact of these time delays on beta - band oscillations in the cortex - basal ganglia loop, the critical surface between the stable steady state and the oscillating state are given in Fig. 6, where the surface is composed of the critical time delays, and SSS and OS denote the stable steady state and oscillating state. As shown in Fig. 6, the beta - band oscillation of the cortex - basal ganglia loop is sensitive to $T_3$, and the system (3) show oscillation for $T_3>0.00185$ regardless of the values of $T_1$ and $T_2$.

Fig. 6.

(a)The boundary plane of Hopf bifurcation with respect to $T_1,T_2,T_3$. SSS represents stable steady state and OS denotes oscillating state. (b) - (d) Time series of firing rate of STN at $T_3=0.0012$ (b) $T_3=0.00127$ (c) $T_3=0.00136$ (d) with $T_1=T_2=0.0015$

To sum up, these numerical simulations confirm to the theoretical results in Sect. “Hopf bifurcation induced by multiple time delay  through theoretical analyses”.

Conclusion

The synaptic transmission time delay plays an important role in inducing beta - band oscillation in the cortex - basal ganglia loop for PD. In this paper, based on a cortex - basal ganglia model with three excitatory - inhibitory loops, STN - GPe, EXN - INN and STN - EXN, we investigate the effect of three transmission time delays $T_1,T_2,T_3$ in STN - GPe, EXN - INN, STN - EXN loops on the critical condition for pathological beta - band oscillation through stability theory, manifold theorem and normal form analysis in four cases (1) $T_1=T_2=T_3=0$ (2) $T_1=T_2=0$, $T_3>0$ (3) $T_2=0$, $T_1>0$, $T_3>0$ (4) $T_1>0$, $T_3>0$,$T_2>0$. In each case, we provide the conditions for Hopf bifurcation induced by time delay in theorems 1 - 4, which are verified through time series of firing rate of STN, GPe, EXN, INN and bifurcation diagram of STN with respect to time delays. Our results show that suitable time delay $T_1$, $T_2$ and $T_3$ may induce Hopf bifurcation, which is sensitive to synaptic transmission time delay $T_3$ in STN - EXN loop. When $T_3<0.00097$, firing rate of STN reach a stable steady state for any $T_1$ and $T_2$ and show oscillating state for $T_3>0.00185$, regardless of $T_1$ and $T_2$. For $T_3$ between 0.00097 and 0.00185, the critical transition points from stable steady state to oscillating state decreases with the increase of $T_1$. Our investigation has demonstrated that reducing the synaptic transmission time delays between cortex and basal ganglia and between STN and GPe will suppress beta - band oscillation of Parkinson’s disease. These results may provide a clue to clinical physician in treating Parkinson’s disease.

In this study, we focus on beta - band oscillation in a cortex - basal ganglia model with three transmission time delays. Of course, a more complete network composed of cortex, striatum, basal ganglia and thalamus is certainly necessary to explore beta - band oscillation in response to different transmission time delays and connect weights between neuron populations (Yi et al. 2022; Yu et al. 2021; Xu et al. 2023). In addition, the effect of both environmental fluctuations and transmission time delay on beta - band oscillation in fractional-order neural network are worthy to be explored in future (Huang et al. 2023; Zhao et al. 2022; Bönsel et al. 2022).

Declarations

Conflicts of interest

The authors declare that they have no conflict of interest.