Influences of time delay and connection topology on a multi-delay inertial neural system

Abstract

Multiple delays and connection topology are the key parameters for the realistic modeling of networks. This paper discusses the influences of time delays and connection weight on multi-delay artificial neural models with inertial couplings. Firstly, sufficient conditions of some singularities involving static bifurcation, Hopf bifurcation, and pitchfork-Hopf bifurcation are presented by analyzing the transcendental characteristic equation. Secondly, taking self-connection weight and coupling delays as adjusting parameters and utilizing the parameter perturbation with the aid of the non-reduced order technique for the first time, rich dynamics near zero-Hopf interaction are obtained on the plane with self-connected weight and coupling delay as abscissa and ordinate. The multi-delay inertial neural system can exhibit coexisting attractors such as a pair of nontrivial equilibrium points and a periodic orbit with nontrivial equilibrium points. Self-connected weight can affect the number and dynamics of the system equilibrium points, while time delays can contribute to both trivial equilibrium and non-trivial equilibrium losing their stability and generating limit cycles. Simulation plots are displayed with computer software to support the established main results. Compared with the traditional reduced-order method, the used method here is simple and valid with less computation. The key research findings of this paper have significant theoretical guiding value in dominating and optimizing networks.

Introduction

Until now, there has been an increasing interest in various modeling of neuron systems to mimic some biological neuron dynamics as closely as possible (Xu et al. 2023, 2022; Ding et al. 2023). It is well known that time delay is one of the universal phenomena in natural systems. Since Marcus and Westervelt (Marcus and Westervelt 1989) pointed out that the time delay should not be ignored due to the finite speeds of information propagation for neurons, delayed neural systems have drawn much attention from researchers. Delayed models are magnificent compared to non-delayed models as time delay could instigate more complex dynamics and provide new insight into neural dynamics (Hou and Zhang 2023; Bi et al. 2021; Xing et al. 2022; Yang et al. 2022; Wang and Liu 2016, Wang et al. 2019; Ma and Tang 2017). Most of the existing bifurcation works for delayed networks have been carried out with a single delay or the sum of time delays from the center manifold analysis (Guckenheimer and Holmes 1984)

Realistic modeling of networks needs to vary the connection topology, which can undergo various behaviors. And topology structure has some important implications since coupled synaptic can change through learning. Hence it is very meaningful and necessary to investigate the dynamics of delayed neural systems concerning connection topologies and time delay as the adjusting parameters. In fact, if the joint effects of two independent parameters are discussed, then codimension-two bifurcations occur such as zero-Hopf bifurcations (Érika et al. 2021; Dong et al. 2022; Ge 2022; Dong et al. 2021; Dong and Liao 2013; Bélair et al. 1996) and Hopf-Hopf bifurcations (Shayer and Campbell 2000; Song et al. 2019).

To enhance the performance of optimizing networks, the pioneering work of inertial characteristics was introduced to the Hopfield neural networks (Hopfield 1982, 1984) by Babcock and Westervelt (1986, 1987) who discussed the complex dynamics of even simple electronic neural networks. The inclusion of inertial terms also has strong biological support (Ashmore and Attwell 1985; Angelaki and Correia 1991; Mauro et al. 1970; Koch 1984). Afterward, more and more researchers are interested in inertial neural networks and obtained significant achievements in stability, bifurcating periodic oscillations (Wheeler and Schieve 1997; Liu et al. 2009a, b; Ge and Xu 2018), and synchronization (Liao et al. 2022; Zhang and Cao 2019). In recent years, complex dynamic behaviors have been studied in inertial neural networks with a single delay when two parameters are varied simultaneously (Ge and Xu 2012, 2013; Yao et al. 2019; Song and Xu 2022). In Yao et al. (2019) and Song and Xu (2022), the authors studied an inertial neural system with one coupling delay which is given by

$$\left\{\begin{array}{c}\frac{d^{2}x}{d{t}^2}=-k_1\frac{\mathit{dx}}{\mathit{dt}}-x\left(t\right)+c_{1}f\left(y\left(t-\tau \right)\right),\\ \frac{d^{2}y}{d{t}^2}=-k_1\frac{\mathit{dy}}{\mathit{dt}}-y\left(t\right)+c_{2}f\left(x\left(t-\tau \right)\right),\\ \end{array}\right)$$

where $f\left(u\right)=uexp\left(-u^2\left./\phantom{-u^{2}2}\right)2\right)$. By making some variable substitutions, interesting and complex dynamics are analyzed on the corresponding first-order differential system. To be closer to reality to contribute to understanding the mechanism of complex dynamics, Song et al. (2016) incorporated multiple time delays into inertial systems

$$\left\{\begin{array}{c}\frac{d^{2}x}{d{t}^2}=-k_1\frac{\mathit{dx}}{\mathit{dt}}-x\left(t-{\tau }_1\right)+c_{1}tanh\left(y\left(t-{\tau }_2\right)\right),\\ \frac{d^{2}y}{d{t}^2}=-k_1\frac{\mathit{dy}}{\mathit{dt}}-y\left(t-{\tau }_1\right)+c_{2}tanh\left(x\left(t-{\tau }_2\right)\right),\\ \end{array}\right)$$

and meaningful results were obtained on codimension-two bifurcations by analyzing the transformed first-order differential equations. Compared with the case of a single delay, the investigation of multiple delays is more challenging and realistic. However, so far few works directly deal with the bifurcation singularity of the second-order delayed differential systems without converting to the first-order delayed systems.

With inspiration from the above discussions, in this paper, a general inertial Hopfield model is considered with multiple delays and nonlinear self-connection simultaneously

$$\left\{\begin{array}{c}\frac{d^2{x}_1}{d{t}^2}=-\frac{d{x}_1}{\mathit{dt}}-k_1{x}_1\left(t\right)+atanh\left(x_1\left(t-s\right)\right)\\ +J_{12}tanh\left(x_2\left(t-{\tau }_1\right)\right),\\ \frac{d^2{x}_2}{d{t}^2}=-\frac{d{x}_2}{\mathit{dt}}-k_2{x}_2\left(t\right)+atanh\left(x_2\left(t-s\right)\right)\\ +J_{21}tanh\left(x_1\left(t-{\tau }_2\right)\right),\\ \end{array}\right)$$ 1

Where $x_1\left(t\right)$ and $x_2\left(t\right)$ are the states of two neurons at time $t$, $k_i$ $\left(i=1,2\right)$ are non-negative and denote adjustable parameters of neurons, $a$ represents the feedback weight and may choose any values, $J_{12}$ and $J_{21}$ denote the cross-interaction weights between two neurons, $s\left(\ge 0\right)$ represents time-delay feedback from each neuron to itself, and ${\tau }_i\left(\ge 0\right)\left(i=1,2\right)$ are time-delay connections between the neurons. The topological architecture of network (1) is displayed in Fig. 1.

Fig. 1.

Topological connections of a pair of neurons (1)

This paper is interested in how connection topology and time delay might affect the bifurcation dynamics of the network (1) without converting to the first-order delayed system. Specifically, the main contributions to this paper are as follows:

1. A more general inertial neural model is proposed with multiple delays and nonlinear self-connection simultaneously. Compared with the case of a single delay, the incorporation of multiple delays is more realistic and challenging.

2. This paper mainly focuses on the combined effect of self-connection weight and time delay on the stability of neural networks. Connecting weight can affect the number and dynamics of network equilibrium, while time delay can contribute to both trivial equilibrium and non-trivial equilibrium losing their stability and generating periodic solutions. It is more meaningful to study their joint influences on network dynamics.

3. The search for explicit bifurcating limit cycles is converted to solve the calculation problem of three algebraic equations. Limit cycles have modeled the behaviors of many real-world oscillatory systems (Van der Pol 1926), which play an important role in the qualitative and quantitative theory of differential equations.

4. For the first time, zero-Hopf interactions are discussed by the parameter perturbation with the aid of a non-reduced order technique. In contrast with the traditional reduced order method, it is simple and valid with less computation.

5. The results obtained in this paper can be considered as an extension of the works for neural networks without inertia to the case with inertial coupling.

Finally, the correctness of the key findings is illustrated by comprehensive computer numerical simulations.

The remainder of this paper is organized as follows. In Sect. Linear stability analysis of the trivial equilibrium, the linear stability of the trivial equilibrium point including a pitchfork, Hopf bifurcation, and zero-Hopf bifurcation is analyzed. Section Methodology Formulation introduces the perturbation combined with the non-reduced order technique to illustrate complex dynamics close to zero-Hopf bifurcations. Numerical results are given to support the main research findings in Sect. Numerical Simulations. Section Conclusions draws some conclusions.

Linear stability analysis of the trivial equilibrium

According to the Routh-Hurwitz stability criterion (Parks 1962), to study the singularity of the system (1), it is needed to consider the real part of a certain eigenvalue changes from negative to zero or positive. So the characteristic equation of the linearized one of the system (1) will be first analyzed.

Existence of pitchfork bifurcation

It is easy to see that $\left(0,0\right)$ is an equilibrium point of Eq. (1). The linear part of the system (1) at $\left(0,0\right)$ is

$$\left\{\begin{array}{c}\frac{d^2{x}_1}{d{t}^2}=-\frac{d{x}_1}{\mathit{dt}}-k_1{x}_1\left(t\right)+a{x}_1\left(t-s\right)+J_{12}{x}_2\left(t-{\tau }_1\right),\\ \frac{d^2{x}_2}{d{t}^2}=-\frac{d{x}_2}{\mathit{dt}}-k_2{x}_2\left(t\right)+a{x}_2\left(t-s\right)+J_{21}{x}_1\left(t-{\tau }_2\right).\\ \end{array}\right)$$ 2

The stability of the trivial equilibrium point is found by evaluating the characteristic equation.

$$\left|\begin{array}{cc}{\lambda }^2+\lambda +k_1-a{e}^{-\lambda s}& -J_{12}{e}^{-\lambda {\tau }_1}\\ -J_{21}{e}^{-\lambda {\tau }_2}& {\lambda }^2+\lambda +k_2-a{e}^{-\lambda s}\end{array}\right|=0,$$

which results in

$$\begin{array}{c}P\left(\lambda \right)=a^2{\text{e}}^{-2s\lambda }+2{\lambda }^3+{\lambda }^4-{\text{e}}^{-2\lambda \tau }{J}_{12}{J}_{21}-a{\text{e}}^{-s\lambda }{k}_1\\ -a{\text{e}}^{-s\lambda }{k}_2+k_1{k}_2+\lambda \left(-2a{\text{e}}^{-s\lambda }+k_1+k_2\right)\\ +{\lambda }^2\left(1-2a{\text{e}}^{-s\lambda }+k_1+k_2\right),\\ \end{array}$$ 3

with ${\tau }_1+{\tau }_2=2\tau$.

From Eq. (3), one has

$$P\left(0\right)=\left(a-k_1\right)\left(a-k_2\right)-J_{12}{J}_{21},$$

and

$$P^{\prime }\left(0\right)=2\tau J_{12}{J}_{21}-\left(1+as\right)\left(2a-k_1-k_2\right).$$

If $P\left(0\right)=0$ and $P^{\prime }\left(0\right)\ne 0$ hold, then the following statements on the zero root of Eq. (3) are correct.

Lemma 1

Based on $\left(a-k_1\right)\left(a-k_2\right)=J_{12}{J}_{21}$, Eq. (3) has only a single zero eigenvalue if and only if one of the following conditions holds.

• (I)$0<a<k_i\left(i=1,2\right)$ for all $\tau \ge 0$ and $s\ge 0$.
• (II)$s\le -\frac{1}{a}$ for all $\tau \ge 0$.
• (III)
  $min\{k_1,k_2\}<a<min\left\{max\{k_1,k_2\},\frac{k_1+k_2}{2}\right\}$, and

  $$\tau \ne \frac{\left(1+as\right)\left(2a-k_1-k_2\right)}{2\left(a-k_1\right)\left(a-k_2\right)}.$$
• (IV) $a>max\left\{k_1,k_2\right\}$, and $\tau \ne \frac{\left(1+as\right)\left(2a-k_1-k_2\right)}{2\left(a-k_1\right)\left(a-k_2\right)}$.

From Lemma 1, one is not difficult to get following transversality condition on static bifurcation.

${\lambda }^{\prime }\left(a\right)\left|_{\lambda =0}^{}\right)=\frac{-2a+k_1+k_2}{2\tau \left(a-k_1\right)\left(a-k_2\right)-\left(1+as\right)\left(2a-k_1-k_2\right)}\ne 0$.

Theorem 1

If Eq. (3) has only a single zero eigenvalue, then pitchfork bifurcation occurs around the trivial equilibrium point for the system (1). System (1) has a stable unique zero equilibrium at $\left(a-k_1\right)\left(a-k_2\right)>J_{12}{J}_{21}$ while the unstable trivial equilibrium and adding two stable nontrivial equilibria at $\left(a-k_1\right)\left(a-k_2\right)<J_{12}{J}_{21}$.

The null clines and fixed points indicate the intrinsic dynamics of the system. The intersection of the two null clines of the system is the equilibrium point on the state $\left(x_1,x_2\right)$ space, where $d{x}_1\left./\phantom{d{x}_1\mathit{dt}}\right)\mathit{dt}=0$ in the dashed line and $d{x}_2\left./\phantom{d{x}_2\mathit{dt}}\right)\mathit{dt}=0$ in the solid line are displayed in Fig. 2 where the parameter $\left(a,J_{21}\right)$ plane are divided into two distinct regions with a unique intersection point and three intersection points by the curve $\left(a-k_1\right)\left(a-k_2\right)=J_{12}{J}_{21}$. Specifically, a unique trivial attractor exists on the $\left(x_1,x_2\right)$ plane when $\left(a-k_1\right)\left(a-k_2\right)>J_{12}{J}_{21}$, and two nontrivial attractors emerge when $\left(a-k_1\right)\left(a-k_2\right)<J_{12}{J}_{21}$. These show that pitchfork bifurcation undergoes around the trivial equilibrium point for the system (1).

Fig. 2.

Static bifurcation curves divided the parameter $a-J_{21}$ plane into two distinct regions with the unique fixed point and three fixed points (top) $J_{12}=0.5$ and (bottom) $J_{12}=2$ with $k_1=k_2=1$

Next, in order to illustrate the stability of equilibrium points in the system (1), bifurcation diagrams, distribution of eigenvalues, and time histories are plotted when self-connection weight is viewed as an adjusting parameter.

• (I)For fixed $k_1=k_2=1$, when coupling weights are taken as $J_{12}=J_{21}=0.5$ which satisfy condition (I) of Lemma 1 for $0<a<1$. In terms of $\left(a-k_1\right)\left(a-k_2\right)=J_{12}{J}_{21}$, the critical value $a=a_0$$=0.5$ is displayed on the $\left(a,x_1\right)$ plane in Fig. 3 (top). If coupling weights $J_{12}=J_{21}=2$ satisfy condition (II) of Lemma 1 for $a<0$, then the critical value is solved as $a=a_0=-1$. Accordingly, pitchfork bifurcation diagram is displayed on the $\left(a,x_1\right)$ plane in Fig. 3 (bottom). From Fig. 3, it can be seen that two nontrivial equilibrium points are added when $a>a_0$ and the unique trivial equilibrium point exists when $a<a_0$. This proves Theorem 1.
• (II)When $J_{12}=1.5$, $J_{21}=-0.5$,$k_1=1,$ and $k_2=3$, the critical value of the adjusting parameter is computed from $\left(a-k_1\right)\left(a-k_2\right)=J_{12}{J}_{21}$ as $a=a_0$ $=1.5$ satisfying the condition (III) in Lemma 1. Supercritical pitchfork is displayed and the critical point $(1.5,0)$ is denoted in black dot in Fig. 4. The dashed line expresses the unstable solution while the solid line denotes the stable solution. This agrees with the statement in Theorem 1.
• (III)Now the stability of equilibrium points in Fig. 3(top) is only verified for $J_{12}=J_{21}=0.5$, and $k_1=k_2=1$. By taking advantage of the Matlab software, the stability of the equilibrium point is obtained from the real part of the rightmost root of the distribution of eigenvalues where the real part in green star is less than zero while the real part in red is greater than zero. When $a=0.4<a_0$, one can see that all eigenvalues have negative real parts at the trivial equilibrium point from Fig. 5 (top) and all solutions with different initial values asymptotically tend to the unique trivial equilibrium point as shown by time histories in Fig. 5 (bottom). With the increasing self-connection weight $a=0.6>a_0$, the trivial equilibrium become unstable and solutions converge to two stable nontrivial equilibria $(0.553235,0.553235)$ and $(-0.553235,-0.553235)$ are displayed in Fig. 6. The stability of three equilibrium points is verified from the real parts of the rightmost eigenvalues of the distribution of eigenvalues and time histories of $x_1$ as shown in Fig. 6 simultaneously. These are in good agreement with the statements of Theorem 1.

Fig. 3.

Pitchfork bifurcation on the $a-x_1$ plane for fixed $k_1=k_2=1$ (top) $J_{12}=J_{21}=0.5$(bottom) $J_{12}=J_{21}=2$. Bifurcation critical points are marked with coordinates in black dots. The dashed line expresses the unstable solution while solid line denotes the stable solution

Fig. 4.

Pitchfork bifurcations on the $a-x_1$ and $a-x_2$ planes for some parameters $k_1=1$, $k_2=3$, $J_{12}=1.5$, and $J_{21}=-0.5$. The black dot point is the bifurcation critical point $\left(1.5,0\right)$. The dashed line expresses the unstable solution while the solid line denotes the stable solution

Fig. 5.

(top) The distribution of partial eigenvalues of the characteristic equation at the trivial equilibrium point and (bottom) Time histories of $x_1$ with three different initial values where $a=0.4<a_0=0.5$, $J_{12}=J_{21}=0.5$, $k_1=k_2=1$, and $s={\tau }_1={\tau }_2=1$

Fig. 6.

Time histories of $x_1$ with three initial values and the distribution of partial eigenvalues around the trivial equilibrium point $\left(0,0\right)$ and two nontrivial equilibrium points $\left(\pm 0.553235,\pm 0.553235\right)$ respectively where $a=0.6>a_0$,$J_{12}=J_{21}=0.5$, $k_1=k_2=1$, $s={\tau }_1={\tau }_2=1$. The trivial equilibrium point $\left(0,0\right)$ is unstable and $\left(\pm 0.553235,\pm 0.553235\right)$ are stable

Remark 1

For inertial neural networks considered in this paper, a pitchfork bifurcation undergoes when coupling weights satisfy $J_{12}{J}_{21}\ne 0$. However, pitchfork bifurcation occurs only at $J_{12}{J}_{21}>0$ in Ge (2022). The results obtained in this paper can be considered as an extension of the works for neural networks without inertia to the case with inertial couplings.

Existence of pitchfork-Hopf bifurcation

Firstly, the stability of the initial state of system (2) is considered. For ${\tau }_1={\tau }_2=s=0$, the corresponding characteristic equation from (3) is simplified as

$$\begin{array}{c}{a}^2+2{\lambda }^3+{\lambda }^4-J_{12}{J}_{21}-a\left(k_1+k_2\right)+k_1{k}_2\\ +\lambda \left(-2a+k_1+k_2\right)+{\lambda }^2\left(1-2a+k_1+k_2\right)=0.\\ \end{array}$$ 4

According to stability criterion (Parks 1962), all roots of Eq. (4) have negative real parts if and only if connection weights satisfy the following conditions

$$\left\{\begin{array}{c}\left(-2+a\right)a+2J_{12}{J}_{21}-2\left(-1+a\right)k_1+k_1^2+k_2\left(2-2a+k_2\right)>0,\hfill \\ \left(a-k_1\right)\left(a-k_2\right)-J_{12}{J}_{21}>0,\hfill \\ 2-2a+k_1+k_2>0.\hfill \end{array}\right)$$ 5

Lemma 2

If Eq. (5) is satisfied, then the trivial equilibrium point is stable for the network (1) without any delay. That is, system orbits converge to the trivial equilibrium point asymptotically, and the neural network is finally resting.

Next, the effect of time delay on Hopf bifurcation is considered for the network (1). Fix $s$ and choose $\tau$ as a control parameter. $\lambda =i\omega$ $\left(\omega >0\right)$ is a purely imaginary root of Eq. (3) if and only

$$\begin{array}{c}{a}^{2}cos\left[2s\omega \right]-2a\omega sin\left[s\omega \right]-cos\left[2\tau \omega \right]J_{12}{J}_{21}\\ +{\omega }^2\left(-1+{\omega }^2-k_2\right)+acos\left[s\omega \right]\left(2{\omega }^2-k_2\right)\\ -k_1\left({\omega }^2+acos\left[s\omega \right]-k_2\right)=0,\\ \end{array}$$

which gets

$$\begin{array}{c}sin\left[2\tau \omega \right]J_{12}{J}_{21}=\left(\omega +asin\left[s\omega \right]\right)\left(2{\omega }^2-k_1-k_2\right)\\ +2acos\left[s\omega \right]\left(\omega +asin\left[s\omega \right]\right),\\ \end{array}$$

$$\begin{array}{c}cos\left(2\omega \tau \right)=\frac{{\omega }^4-{\omega }^2+2a{\omega }^{2}cos\left[s\omega \right]-{\omega }^2{k}_1}{J_{12}{J}_{21}}\\ +\frac{a^{2}cos\left[2s\omega \right]-2a\omega sin\left[s\omega \right]}{J_{12}{J}_{21}}\\ +\frac{k_1{k}_2-acos\left[s\omega \right]k_1}{J_{12}{J}_{21}}-\frac{{\omega }^2{k}_2+acos\left[s\omega \right]k_2}{J_{12}{J}_{21}},\\ sin\left(2\omega \tau \right)=\frac{\left(\omega +asin\left[s\omega \right]\right)\left(2{\omega }^2-k_1-k_2\right)}{J_{12}{J}_{21}}\\ +\frac{2acos\left[s\omega \right]\left(\omega +asin\left[s\omega \right]\right)}{J_{12}{J}_{21}}.\\ \end{array}$$ 6

In terms of ${cos}^2\left(2\omega \tau \right)+{sin}^2\left(2\omega \tau \right)=1$ in Eq. (6), one has the following equation on $\omega$

$$\begin{array}{c}G\left(\omega \right)=b_0+\omega b_1+{\omega }^2{b}_2+{\omega }^3{b}_3+{\omega }^4{b}_4\\ +{\omega }^5{b}_5+{\omega }^6{b}_6+{\omega }^8=0,\\ \end{array}$$ 7

where

$$\begin{array}{c}{b}_0=a^4+4a{\omega }^{5}sin\left[s\omega \right]-J_{12}^2{J}_{21}^2-2a^{3}cos\left[s\omega \right]cos\left[2s\omega \right]k_1\\ -4a^{3}cos\left[s\omega \right]sin{\left[s\omega \right]}^2\left(k_1+k_2\right)-2a^{3}cos\left[s\omega \right]cos\left[2s\omega \right]k_2\\ +2a^{2}cos\left[2s\omega \right]k_1{k}_2-2acos\left[s\omega \right]k_1^2{k}_2\left(k_1+k_2\right)\\ +k_1^2{k}_2^2+2a^2{k}_1{k}_2+a^2{k}_2^2+a^2{k}_1^2,\\ \end{array}$$

$$\begin{array}{c}{b}_1=8a^{3}cos{\left[s\omega \right]}^{2}sin\left[s\omega \right]-4a^{3}cos\left[2s\omega \right]sin\left[s\omega \right]\\ +2asin\left[s\omega \right]k_1^2-4a^{2}cos\left[s\omega \right]sin\left[s\omega \right]k_2\\ +2asin\left[s\omega \right]k_2^2-4a^{2}cos\left[s\omega \right]sin\left[s\omega \right]k_1,\\ \end{array}$$

$$\begin{array}{c}{b}_2=4a^2-2a^{2}cos\left[2s\omega \right]+4a^{3}cos\left[s\omega \right]cos\left[2s\omega \right]\\ -2acos\left[s\omega \right]k_1-4a^{2}cos{\left[s\omega \right]}^2{k}_1-2a^{2}cos\left[2s\omega \right]k_1\\ -2acos\left[s\omega \right]k_2-4a^{2}cos{\left[s\omega \right]}^2{k}_2-2a^{2}cos\left[2s\omega \right]k_2\\ -2k_1^2{k}_2-2k_1{k}_2^2+k_2^2+k_1^2+8a^{3}cos\left[s\omega \right]sin{\left[s\omega \right]}^2\\ +8acos\left[s\omega \right]k_1{k}_2+2acos\left[s\omega \right]k_1^2+2acos\left[s\omega \right]k_2^2\\ -4a^{2}sin{\left[s\omega \right]}^2{k}_1-4a^{2}sin{\left[s\omega \right]}^2{k}_2,\\ \end{array}$$

$$\begin{array}{c}{b}_3=4asin\left[s\omega \right]+8a^{2}cos\left[s\omega \right]sin\left[s\omega \right]\\ -4asin\left[s\omega \right]k_1-4asin\left[s\omega \right]k_2,\\ \end{array}$$

$$\begin{array}{c}{b}_4=4acos\left[s\omega \right]+2a^{2}cos\left[2s\omega \right]-6acos\left[s\omega \right]\left(k_1+k_2\right)\\ +4a^2+4k_1{k}_2+k_1^2+k_2^2-2k_1-2k_2+1,\\ \end{array}$$

$$b_5=4asin\left[s\omega \right],$$

$$b_6=2+4acos\left[s\omega \right]-2k_1-2k_2.$$

If Eq. (7) has positive and simple roots ${\omega }_i$$\left(i=1,2,\cdots \right)$, then the critical delay values are computed from Eq. (6)

$${\tau }_i^j=\frac{{\phi }_i+2j\pi }{2{\omega }_i},i=1,2,\cdots ,j=0,1,2,\cdots ,$$

where ${\phi }_i\left(\in \left[0,2\pi \right)\right)$ are satisfied with

$$\begin{array}{c}cos{\phi }_i=\frac{{\omega }_i^4-{\omega }_i^2+2a{\omega }_i^{2}cos\left[s{\omega }_i\right]-{\omega }_i^2{k}_1}{J_{12}{J}_{21}}\\ +\frac{a^{2}cos\left[2s{\omega }_i\right]-2a{\omega }_{i}sin\left[s{\omega }_i\right]}{J_{12}{J}_{21}}\\ +\frac{k_1{k}_2-acos\left[s{\omega }_i\right]k_1-{\omega }_i^2{k}_2-acos\left[s{\omega }_i\right]k_2}{J_{12}{J}_{21}},\\ \end{array}$$

$$sin{\phi }_i=\frac{\left({\omega }_i+asin\left[s{\omega }_i\right]\right)\left(2{\omega }_i^2+2acos\left[s{\omega }_i\right]-k_1-k_2\right)}{J_{12}{J}_{21}}.$$

If the first bifurcation point is only focused on, then the bifurcation critical delay is defined as

$${\tau }_0=min\left\{{\tau }_i^j\right\},i=1,2,\cdots ,j=0,1,2,\cdots .$$ 8

To make Hopf bifurcation occur, transversality condition is useful and necessary. The necessary condition is that the velocity of the critical eigenvalue through the imaginary axis is nonzero. So differentiating $\lambda$ concerning $\tau$ in Eq. (3), one gets

$${\left[{\lambda }^{\prime }\left(\tau \right)\right]}^{-1}=-\frac{E}{\lambda J_{12}{J}_{21}}-\frac{\tau }{\lambda },$$

where

$$E={\text{e}}^{2\lambda \left(-s+\tau \right)}\left(as+{\text{e}}^{s\lambda }\left(1+2\lambda \right)\right)\left(-2a+{\text{e}}^{s\lambda }\left(2\lambda \left(1+\lambda \right)+k_1+k_2\right)\right).$$

In terms of Eq. (6), one has

$$\begin{array}{c}{\left[\text{Re}\left({\lambda }^{\prime }\left(\tau \right)\right)\right]}^{-1}\left|_{\lambda =i\omega ,\tau ={\tau }_0}^{}\right)=\frac{E_1}{J_{12}^2{J}_{21}^2}+\frac{E_0+\omega E_1}{\omega J_{12}^2{J}_{21}^2}\\ +\frac{{\omega }^2{E}_2+{\omega }^3{E}_3+{\omega }^4{E}_4+{\omega }^5{E}_5+{\omega }^6{E}_6+4{\omega }^7}{\omega J_{12}^2{J}_{21}^2},\\ \end{array}$$ 9

where

$$\begin{array}{c}{E}_0=2a^{3}sin\left[s\omega \right]+a^{3}ssin\left[s\omega \right]k_1-a^{2}sin\left[2s\omega \right]k_1\\ +asin\left[s\omega \right]k_1^2+a^{3}ssin\left[s\omega \right]k_2-a^{2}sin\left[2s\omega \right]k_2\\ -4a^{2}scos\left[s\omega \right]sin\left[s\omega \right]k_1{k}_2+assin\left[s\omega \right]k_1^2{k}_2\\ +asin\left[s\omega \right]k_2^2+assin\left[s\omega \right]k_1{k}_2^2,\\ \end{array}$$

$$\begin{array}{cc}\hfill E_1& =2a^2+2a^3\left(2+s\right)cos\left[s\omega \right]+4a^{2}sin{\left[s\omega \right]}^2-4a^2{k}_1\hfill \\ \hfill & -2acos\left[s\omega \right]k_1-2a^2\left(1+s\right)cos\left[2s\omega \right]k_1\hfill \\ \hfill & +a\left(2+s\right)cos\left[s\omega \right]k_1^2+8acos\left[s\omega \right]k_1{k}_2\hfill \\ \hfill & -4a^2{k}_2-2acos\left[s\omega \right]k_2-2a^2\left(1+s\right)cos\left[2s\omega \right]k_2\hfill \\ \hfill & +k_2^2+a\left(2+s\right)cos\left[s\omega \right]k_2^2-2k_1{k}_2^2+k_1^2-2k_1^2{k}_2,\hfill \\ \hfill \end{array}$$

$$\begin{array}{c}{E}_2=6asin\left[s\omega \right]-2a^{3}ssin\left[s\omega \right]+4a^{2}cos\left[s\omega \right]sin\left[s\omega \right]\\ +4a^2\left(2+s\right)cos\left[s\omega \right]sin\left[s\omega \right]+a\left(-6+s\right)sin\left[s\omega \right]k_1\\ +a\left(-6+s\right)sin\left[s\omega \right]k_2+2a^{2}ssin\left[2s\omega \right]k_1-assin\left[s\omega \right]k_1^2\\ +2a^{2}ssin\left[2s\omega \right]k_2-4assin\left[s\omega \right]k_1{k}_2-assin\left[s\omega \right]k_2^2,\\ \end{array}$$

$$\begin{array}{c}{E}_3=2+4a^2+4acos\left[s\omega \right]+2a\left(2+s\right)cos\left[s\omega \right]-4k_1-4k_2\\ +4a^2\left(2+s\right)cos{\left[s\omega \right]}^2-4a^{2}ssin{\left[s\omega \right]}^2+2k_1^2+2k_2^2\\ -2a\left(6+s\right)cos\left[s\omega \right]k_1-2a\left(6+s\right)cos\left[s\omega \right]k_2+8k_1{k}_2,\\ \end{array}$$

$$\begin{array}{c}{E}_4=10asin\left[s\omega \right]-2assin\left[s\omega \right]-4a^{2}scos\left[s\omega \right]sin\left[s\omega \right]\\ +3assin\left[s\omega \right]k_1+3assin\left[s\omega \right]k_2,\\ \end{array}$$

$$E_5=6+8acos\left[s\omega \right]+2a\left(2+s\right)cos\left[s\omega \right]-6k_1-6k_2,E_6=-2assin\left[s\omega \right].$$

Noticing that

$$\text{sign}\left\{\text{Re}\left({\lambda }^{\prime }\left(\tau \right)\right)\left|_{\lambda =i\omega ,\tau ={\tau }_0}^{}\right)\right\}=\text{sign}\left\{{\left[\text{Re}\left({\lambda }^{\prime }\left(\tau \right)\right)\right]}^{-1}\left|_{\lambda =i\omega ,\tau ={\tau }_0}^{}\right)\right\}.$$

Theorem 2 Supposing $\text{Re}\left({\lambda }^{\prime }\left(\tau \right)\left|_{\lambda =i\omega ,\tau ={\tau }_0}^{}\right)\right)\ne 0$ and Eq. (5) are satisfied. The following results hold for network (1):

• (I)Network (1) undergoes a Hopf bifurcation at the trivial equilibrium when $\tau ={\tau }_0$. If $\text{Re}\left({\lambda }^{\prime }\left(\tau \right)\left|_{\lambda =i\omega ,\tau ={\tau }_0}^{}\right)\right)>0$, then the stable trivial equilibrium point becomes unstable and a branch of stable periodic solutions emerges from the trivial equilibrium point near $\tau ={\tau }_0$. Hopf bifurcations mean that the neurons in the model (1) can realize the transition from the resting state to periodic spiking near $\tau ={\tau }_0$.
• (II)If $\left(a-k_1\right)\left(a-k_2\right)=J_{12}{J}_{21}$, and $s<-\frac{1}{a}$ hold, then system (1) undergoes a pitchfork-Hopf interaction near the zero solution when $\tau ={\tau }_0$.

Numerical simulations are given to demonstrate the correction of Theorem 2 as follows.

(1) Some parameters are fixed as $a=-1.1,$ $k_1=k_2=1,$ $J_{12}=J_{21}=2$, and $s=0.01$ in system (1). Hopf bifurcation around the trivial equilibrium point emerges at $\tau =$ ${\tau }_0$ $=0.59936$ with ${\left[\text{Re}\left({\lambda }^{\prime }\left(\tau \right)\right)\right]}^{-1}\left|_{\lambda =1.75757i,\tau ={\tau }_0}^{}\right)$$=0.739164$$>0$. The trivial equilibrium point is asymptotically stable by time histories and phase portraits for $\tau =0.5\in \left[0,{\tau }_0\right)$ as shown in Fig. 7. With the increasing of coupling delays $\tau =0.7\in \left({\tau }_0,\infty \right)$, the trivial equilibrium point loses its stability and the stable periodic solution is derived by time histories and phase portraits as shown in Fig. 8. This proves conclusion (1) of Theorem 2.

Fig. 7.

Stable trivial equilibrium point (top) Time histories of $x_1$ (bottom) Phase portrait on the state $x_1-x_2$ plane where $a=-1.1$, $k_1=k_2=1$, $J_{12}=J_{21}=2$, $s=0.01$, and $\tau =0.5<{\tau }_0=0.59936$

Fig. 8.

Stable periodic oscillation with three different initial values where $a=-1.1$, $k_1=k_2=1$ , $J_{12}=J_{21}=2$, $s=0.01$, and $\tau =0.7>{\tau }_0$

(II) Some parameters are fixed as $a=-1$,$k_1=k_2=1$, $J_{12}=J_{21}=2$, and $s=0.01$. Equation (3) has only one simple zero root $\lambda =0$ and a pair of simple purely imaginary roots $\lambda =\pm 1.73776i$ exhibited in Fig. 9. One can obtain the critical time delay $\tau ={\tau }_0=0.595965$ from Eq. (8) and the condition ${\left[\text{Re}\left({\lambda }^{\prime }\left(\tau \right)\right)\right]}^{-1}\left|_{\lambda =1.7377i,\tau ={\tau }_0}^{}\right)=0.755025$$\ne 0$. Zero-Hopf bifurcation produces at $\left(\alpha ,\tau \right)=\left(-1,0.595965\right)$ for the network (1). It is good agreement with conclusion (II) of Theorem 2.

Fig. 9.

(top) The numbers of the positive real root $\omega$ with the increasing of self-connection delay $s$(bottom)$G\left(\omega \right)$ has only a simple zero root $\omega =0$ and a real root $\omega =1.73776$ with $s=0.01$ where $a=-1$, $k_1=k_2=1$, $J_{12}=J_{21}=2$

Next, the rich dynamic behaviors near the pitchfork-Hopf singularity in Fig. 9 are focused on discussing by using the perturbation combined non-reduced order method based on the method (Xu et al. 2007). The methodology formulation firstly will be presented in detail.

Methodology formulation

Consider the second-order delayed differential system with multiple delays

$$\begin{array}{cc}\hfill \frac{d^{2}X}{d{t}^2}& =K\frac{\mathit{dX}}{\mathit{dt}}+BX+D_{1}X\left(t-s\right)+D_{2}X\left(t-\tau \right)\hfill \\ \hfill & +\epsilon F\left(X,X\left(t-s\right),X\left(t-\tau \right)\right),\hfill \\ \hfill \end{array}$$ 10

where $X={\left(x_1,x_2,...,x_n\right)}^T\in R^n$, $n\times n$ matrices $K$, $B$, $D_1$ and $D_2$ are real constants, nonlinear function $F\left(·\right)$ is smooth with $F\left(0,0,...,0\right)=0$, $\epsilon$ is a parameter representing the strength of nonlinear coupling, $s$ and $\tau$ are discrete delays.

A zero solution $\left(0,0,...,0\right)$ is the trivial equilibrium point of system (10). Assume that system (10) undergoes zero-Hopf bifurcation at the zero solution $\left(0,0,...,0\right)$ for the critical values $D_1=D_{10}$ and $\tau ={\tau }_0$. That is to say, the solutions of the characteristic equation at $\left(0,0,\cdots ,0\right)$ are of the form $\lambda =0$ and $\lambda =\pm i\omega$$\left(\omega >0\right)$ at $D_1=D_{10}$ and $\tau ={\tau }_0$.

In the next, the perturbation scheme will be presented with the aid of the non-reduced order technique to obtain bifurcation sets and periodic solutions near $\left(D_{10},{\tau }_0\right)$.

There exists a small perturbation such as $D_1=D_{10}+\epsilon D_{\epsilon }$, and $\tau ={\tau }_0+\epsilon {\tau }_{\epsilon }$. System (10) is rewritten as the form

$$\frac{d^{2}X}{d{t}^2}=K\frac{\mathit{dX}}{\mathit{dt}}+BX\left(t\right)+D_{10}X\left(t-s\right)+D_{2}X\left(t-{\tau }_0\right)+\overline{F}\left(·\right),$$ 11

where

$$\begin{array}{c}\overline{F}\left(·\right)=D_2\left[X\left(t-\tau \right)-X\left(t-{\tau }_0\right)\right]+\epsilon D_{\epsilon }X\left(t-s\right)\\ +\epsilon F\left(X\left(t\right),X\left(t-s\right),X\left(t-\tau \right)\right).\\ \end{array}$$

In Eq. (11) for $\epsilon =0$, the periodic solution is expressed as

$$X(t)\left|_{\epsilon =0}^{}\right)=A_{1}cos\left(\omega t\right)+A_{2}sin\left(\omega t\right)+z,$$ 12

where

$$\begin{array}{c}{A}_1={\left(m_1,m_2,...,m_n\right)}^T,A_2={\left(l_1,l_2,...,l_n\right)}^T,\\ z={\left(z_1,z_2,...,z_n\right)}^T.\\ \end{array}$$

Bringing the solution (12) into Eq. (11) for $\epsilon =0$, using the harmonic balance, it is easy to get the unknown coefficients $A_1$, $A_2$ and $z$ in Eq. (12) satisfying

$$F_1\left(\begin{array}{c}{m}_1\\ m_2\\ ...\\ m_n\end{array}\right)=F_2\left(\begin{array}{c}{l}_1\\ l_2\\ ...\\ l_n\end{array}\right),-F_1\left(\begin{array}{c}{l}_1\\ l_2\\ ...\\ l_n\end{array}\right)=F_2\left(\begin{array}{c}{m}_1\\ m_2\\ ...\\ m_n\end{array}\right),F_3\left(\begin{array}{c}{z}_1\\ z_2\\ ...\\ z_n\end{array}\right)=0,$$ 13

where

$$F_1=-\omega K+D_{10}sin\left(\omega s\right)+D_{2}sin\left(\omega {\tau }_0\right),$$

$$F_2=-{\omega }^{2}I-B-D_{10}cos\left(\omega s\right)-D_{2}cos\left(\omega {\tau }_0\right),F_3=B+D_{10}+D_2.$$

Here $F_1$ and $F_2$ are also the imaginary and real parts of characteristic matrix $\left({\lambda }^{2}I-\lambda K-B-D_{10}{e}^{-\lambda s}-D_2{e}^{-\lambda {\tau }_0}\right)\left|_{\lambda =i\omega }^{}\right)$.. So there are $2n-2$ independent equations to solve $m_i$ and $l_i$ in (13). If $m_1$ and $l_1$ are chosen to be independent, then $m_i$ and $l_i$ $\left(i=2,3,...,n\right)$ are solved from (13) based on $m_1$ and $l_1$. Similarly, $z_i$ $\left(i=2,3,...,n\right)$ can be derived from (13) in terms of $z_1$.

The periodic solution (12) is rewritten in a polar coordinate as

$$X(t)\left|_{\epsilon =0}^{}\right)=\left(\begin{array}{c}{r}_1\\ r_2\\ \dots \\ r_n\end{array}\right)cos\left(\omega t+{\theta }_i\right)+\left(\begin{array}{c}{z}_1\\ z_2\\ \dots \\ z_n\end{array}\right),$$ 14

where $r_i$ is function of $r_1$, $z_i$ are functions of $z_1$, ${\theta }_i$ is function of ${\theta }_1$$\left(i=2,...,n\right)$.

For a small $\epsilon$, the $\mathit{ith}$ component of periodic solution of system (11) can be thought as a perturbation to the solution (14),

$$x_i\left(t\right)=r_i\left(\epsilon \right)cos\left(\phi \right)+z_i\left(\epsilon \right),$$ 15

where

$$r_i\left(\epsilon \right)=r_i\left(r_1\left(\epsilon \right)\right),z_i\left(\epsilon \right)=z_i\left(z_1\left(\epsilon \right)\right),\frac{d\phi }{\mathit{dt}}=\omega +\sigma \left(\epsilon \right),$$

$$\sigma \left(0\right)=0,r_i\left(0\right)=r_i,z_i\left(0\right)=z_i\left(i=2,...,n\right).$$

Remark 2

It is very hard to derive directly the expression of periodic solution (15). However, with the aid of the adjoint operator, solutions can be easily obtained. So the definition of the adjoint operator is firstly given for reader’s convenience as follows.

$L_{}^{\ast }$ is the adjoint operator of $L$ if $〈L\left(X\right),Y〉=〈X,L_{}^{\ast }\left(Y\right)〉$, and $〈X,Y〉={\int }_0^m{Y}^{T}Xdt$ for any vectors $X$ and $Y$$\left(\in C\left[0,m\right]\right)$, where $T$ denotes a transpose symbol, and $m$ is a real constant.

In system (11) for $\epsilon =0$, let

$$\begin{array}{c}L\left(X\right)=\frac{d^{2}X}{d{t}^2}-K\frac{\mathit{dX}}{\mathit{dt}}-BX-D_{10}X\left(t-s\right)\\ -D_{2}X\left(t-{\tau }_0\right)=0.\\ \end{array}$$ 16

According to the definition of the adjoint operator, one can obtain

$$\begin{array}{c}L_{}^{\ast }\left(Y\right)=\frac{d^{2}Y}{d{t}^2}+K^T\frac{\mathit{dY}}{\mathit{dt}}-B^{T}Y-D_{10}^{T}Y\left(t+s\right)\\ -D_2^{T}Y\left(t+{\tau }_0\right)=0,\\ \end{array}$$

$$\frac{d^{2}Y}{d{t}^2}=-K^T\frac{\mathit{dY}}{\mathit{dt}}+B^{T}Y+D_{10}^{T}Y\left(t+s\right)+D_2^{T}Y\left(t+{\tau }_0\right).$$ 17

Lemma 3

If $Y\left(t\right)$ is the periodic solution of Eq. (17), then it can be expressed as

$$Y\left(t\right)=\left(\begin{array}{c}{p}_1\\ p_2\\ ...\\ p_n\end{array}\right)cos\left(\omega t\right)+\left(\begin{array}{c}{q}_1\\ q_2\\ ...\\ q_n\end{array}\right)sin\left(\omega t\right)+\left(\begin{array}{c}{s}_1\\ s_2\\ ...\\ s_n\end{array}\right),$$ 18

where the unknown coefficients $p_i$, $q_i$ and $s_i$ are determined by the following equations

$$-F_1^T\left(\begin{array}{c}{p}_1\\ p_2\\ \dots \\ p_n\end{array}\right)=F_2^T\left(\begin{array}{c}{q}_1\\ q_2\\ \dots \\ q_n\end{array}\right),F_1^T\left(\begin{array}{c}{q}_1\\ q_2\\ \dots \\ q_n\end{array}\right)=F_2^T\left(\begin{array}{c}{p}_1\\ p_2\\ \dots \\ p_n\end{array}\right),F_3^T\left(\begin{array}{c}{s}_1\\ s_2\\ \dots \\ s_n\end{array}\right)=0,$$

which are obtained to see Appendix A.

If $p_1$ and $q_1$ are chosen to be independent, then $p_i$ and $q_i$ $\left(i=2,3,...,n\right)$ are solved in terms of $p_1$ and $q_1$. Similarly, $s_i$ $\left(i=2,3,...,n\right)$ are also derived in terms of $s_1$.

Combining with Eq. (17), the perturbation solution (15) can be obtained by the following theorem.

Theorem 3

If $Y$ is the periodic solution of Eq. (17), and $X$ is periodic solution of system (10) for a small $\epsilon \left(>0\right)$, then $X$ satisfies the following equations

$$\begin{array}{c}\frac{d{Y}^T\left(0\right)}{\mathit{dt}}\left[X\left(\frac{2\pi }{\omega }\right)-X\left(0\right)\right]-Y^T\left(0\right)\left[\frac{dX\left(\frac{2\pi }{\omega }\right)}{\mathit{dt}}-\frac{dX\left(0\right)}{\mathit{dt}}\right]\\ +Y^T\left(0\right)K\left[X\left(\frac{2\pi }{\omega }\right)-X\left(0\right)\right]\\ +{\int }_{-s}^0{Y}^T\left(t+s\right)D_{10}\left[X-X\left(t+\frac{2\pi }{\omega }\right)\right]dt\\ +{\int }_{-{\tau }_0}^0{Y}^T\left(t+{\tau }_0\right)D_2\left[X-X\left(t+\frac{2\pi }{\omega }\right)\right]dt\\ +{\int }_0^{\frac{2\pi }{\omega }}{Y}^T\overline{F}\left(·\right)dt=0.\\ \end{array}$$ 19

Proof

Due to $Y\left(t\right)=Y\left(t+\frac{2\pi }{\omega }\right)$, taking advantage of partial integral method, one gets the following equations

$$\begin{array}{c}{\int }_0^{\frac{2\pi }{\omega }}{Y}^{T}K\frac{\mathit{dX}}{\mathit{dt}}dt=Y^T\left(0\right)K\left[X\left(\frac{2\pi }{\omega }\right)-X\left(0\right)\right]\\ -{\int }_0^{\frac{2\pi }{\omega }}\frac{d{Y}^T}{\mathit{dt}}KXdt,\\ \end{array}$$ 20

$$\begin{array}{c}{\int }_0^{\frac{2\pi }{\omega }}{Y}^T{D}_{10}X\left(t-s\right)dt={\int }_0^{\frac{2\pi }{\omega }}{Y}^T\left(t+s\right)D_{10}Xdt\\ +{\int }_{-s}^0{Y}^T\left(t+s\right)D_{10}\left[X-X\left(t+\frac{2\pi }{\omega }\right)\right]dt,\\ \end{array}$$ 21

$$\begin{array}{c}{\int }_0^{\frac{2\pi }{\omega }}{Y}^T\frac{d^{2}X}{d{t}^2}dt=Y^T\left(0\right)\left[\frac{dX\left(\frac{2\pi }{\omega }\right)}{\mathit{dt}}-\frac{dX\left(0\right)}{\mathit{dt}}\right]\\ -\frac{d{Y}^T\left(0\right)}{\mathit{dt}}\left[X\left(\frac{2\pi }{\omega }\right)-X\left(0\right)\right]-{\int }_0^{\frac{2\pi }{\omega }}\frac{d{Y}^T}{\mathit{dt}}KXdt\\ +{\int }_0^{\frac{2\pi }{\omega }}{Y}^{T}BXdt+{\int }_0^{\frac{2\pi }{\omega }}{Y}^T\left(t+s\right)D_{10}Xdt\\ +{\int }_0^{\frac{2\pi }{\omega }}{Y}^T\left(t+{\tau }_0\right)D_{2}Xdt,\\ \end{array}$$ 22

$$\begin{array}{c}{\int }_0^{\frac{2\pi }{\omega }}{Y}^T{D}_{2}X\left(t-{\tau }_0\right)dt={\int }_0^{\frac{2\pi }{\omega }}{Y}^T\left(t+{\tau }_0\right)D_{2}Xdt\\ +{\int }_{-{\tau }_0}^0{Y}^T\left(t+{\tau }_0\right)D_2\left[X-X\left(t+\frac{2\pi }{\omega }\right)\right]dt.\\ \end{array}$$ 23

Multiplying $Y^T\left(t\right)$ to both sides of Eq. (11) and integrating concerning from 0 to $2\pi \left./\phantom{2\pi \omega }\right)\omega$, one obtains

$$\begin{array}{c}{\int }_0^{\frac{2\pi }{\omega }}{Y}^T\frac{d^{2}X}{d{t}^2}dt={\int }_0^{\frac{2\pi }{\omega }}{Y}^T\left(K\frac{\mathit{dX}}{\mathit{dt}}+BX+D_{10}X\left(t-s\right)\right)dt\\ +{\int }_0^{\frac{2\pi }{\omega }}{Y}^T\left(D_{2}Z\left(t-{\tau }_0\right)+\overline{F}\left(·\right)\right)dt.\\ \end{array}$$ 24

Substituting Eqs. (20) to (23) into Eq. (24), Eq. (19) is obtained. The theorem is finished.

Remark 3

Equation (19) is a transcendental equation in $r_1(\epsilon )$, $z_1\left(\epsilon \right)$, and $\sigma \left(\epsilon \right)$. In order to get the analytical expression of periodic solution (15), Eq. (19) need to be expanded into $\epsilon$ series, neglect high order terms in $\epsilon$, and yield three algebraic equations in $r_1(\epsilon )$,$z_1\left(\epsilon \right)$, and $\sigma (\epsilon )$. Hence, two control parameters are close to the zero-bifurcation point, the analytical periodic solution is no difficulty to obtain from these algebraic equations.

Zero-Hopf bifurcation and coexistence of attractors

As displayed in Fig. 9, taking self-connection weight and coupling time delay as two adjusting parameters, network (1) undergoes a pitchfork-Hopf bifurcation at the critical value $a_0=-1$ and ${\tau }_0=0.595965$ where $k_1=1,$$k_2=1$, $J_{12}=2,$ $J_{21}=2,$ and $s=0.01$. To study the dynamics in the neighbor of the critical point $\left(a_0,{\tau }_0\right)$, one needs to make the parameter perturbations as $a=a_0+{\epsilon }^2{\delta }_1,$ $\tau ={\tau }_0+{\epsilon }^2{\delta }_2$ and let $x_i\to \epsilon x_i$. Then system (1) can become the form (11) where

$$K=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right),B=\left(\begin{array}{cc}-k_1& 0\\ 0& -k_2\end{array}\right),$$

$$D_{10}=\left(\begin{array}{cc}{a}_0& 0\\ 0& a_0\end{array}\right),D_2=\left(\begin{array}{cc}0& J_{12}\\ J_{21}& 0\end{array}\right),$$

$F={\epsilon }^2\left(\begin{array}{c}{\delta }_1{x}_1\left(t-s\right)-\frac{a}{3}{x}_1^3\left(t-s\right)-\frac{J_{12}}{3}{x}_2^3\left(t-\tau \right)+h.o.t.\\ {\delta }_1{x}_2\left(t-s\right)-\frac{a}{3}{x}_2^3\left(t-s\right)-\frac{J_{12}}{3}{x}_1^3\left(t-\tau \right)+h.o.t.\end{array}\right)$.

Classification sets and periodic solutions

According to Eq. (13), the periodic solution of system (1) for $\epsilon =0$ is derived as

$$X(t)\left|_{\epsilon =0}^{}\right)=\left(\begin{array}{c}{m}_{1}cos\left(\omega t\right)+l_{1}sin\left(\omega t\right)+z_1\\ -m_{1}cos\left(\omega t\right)-l_{1}sin\left(\omega t\right)+z_1\end{array}\right),$$ 25

which is rewritten in polar coordinate with $m_1=r_{0}cos\left(\theta \right)$ and $l_1=-r_{0}sin\left(\theta \right)$

$$X(t)\left|_{\epsilon =0}^{}\right)=\left(\begin{array}{c}{z}_1+r_{0}cos\left(\omega t+\theta \right)\\ z_1-r_{0}cos\left(\omega t+\theta \right)\end{array}\right),$$

which leads to

$$X(t)\left|_{\epsilon >0}^{}\right)=\left(\begin{array}{c}{z}_1+r_{0}cos\left(\left(\omega +{\epsilon }^2\sigma \right)t+\theta \right)\\ z_1-r_{0}cos\left(\left(\omega +{\epsilon }^2\sigma \right)t+\theta \right)\end{array}\right).$$ 26

Based on Lemma 3, the periodic solution of the adjoint Eq. (17) is derived as

$$Y(t)=\left(\begin{array}{c}{p}_{1}cos\left(\omega t\right)+q_{1}sin\left(\omega t\right)+w\\ -p_{1}cos\left(\omega t\right)-q_{1}sin\left(\omega t\right)+w\end{array}\right).$$ 27

Substituting Eqs. (26) to (27) into Eq. (19), expanding into $\epsilon$ series, and omitting the higher order on $O\left({\epsilon }^2\right)$, three algebraic equations are derived as follows

$$\left\{\begin{array}{c}-7.30296{\epsilon }^2{z}_1^2{r}_0-1.82574{\epsilon }^2{r}_0^3-3.61514r_0{\epsilon }^2{\delta }_1\\ -10.8095r_0{\epsilon }^2{\delta }_2-16.2741r_0{\epsilon }^2\sigma =0,\\ 6.28319{\epsilon }^2{z}_1^2{r}_0+1.5708{\epsilon }^2{r}_0^3+0.0628287r_0{\epsilon }^2{\delta }_1\\ -6.40852r_0{\epsilon }^2{\delta }_2+1.38173r_0{\epsilon }^2\sigma =0,\\ -2.41046{\epsilon }^2{z}_1^3-3.61569z_1{\epsilon }^2{r}_0^2+7.23138z_1{\epsilon }^2{\delta }_1=0,\\ \end{array}\right)$$

which produces some solutions on $\left(r_0,z_1\right)$ except for $\left(0,0\right)$ as follows

$$\left(0,1\text{.73205}\sqrt{1+a}\right),\left(2.2748\sqrt{\tau -0.562645+0.0333197a},0\right),$$

$$\left(\begin{array}{c}1.01732\sqrt{2.88161-\tau +2.28565a},\\ 1.24596\sqrt{\tau -0.94914-0.353175a}\\ \end{array}\right),$$

and ${\epsilon }^2\sigma =-0.241484{\epsilon }^2{\delta }_1-1.24475{\epsilon }^2{\delta }_2$.

And four critical lines are derived according to the existences of the above solutions:

$$l_1:a=-1,$$

$$l_2:\tau =0.562645-0.0333197a,$$

$$l_3:\tau =2.88161+2.28565a,$$

$$l_4:\tau =0.94914+0.353175a.$$

The bifurcation diagram on the $\left(a,\tau \right)$ plane is given close to zero-Hopf bifurcation points in Fig. 10. It can be seen that there are six regions denoted by I, II, III, IV, V, and VI, which are divided by line $l_i$ $\left(i=1,2,3,4\right)$ and their dynamics depends on self-connected weight $a$ and coupling time delay $\tau$. The interesting phenomena are exhibited involving the stable equilibrium, the coexistence of nontrivial equilibrium points, and multi-stability of periodic solution and two nontrivial equilibrium points.

1. $\left(a,\tau \right)\in \text{region}\text{I}$, the trivial equilibrium of system (1) are asymptotically stable. And system (1) undergoes a Hopf bifurcation at line $l_2$.

2. $\left(a,\tau \right)\in \text{region}\text{II}$, the trivial equilibrium become unstable and periodic solution bifurcating from the trivial equilibrium point is stable. when $\left(a,\tau \right)$ crosses line $l_1$, two unstable nontrivial equilibrium points emerge.

3. $\left(a,\tau \right)\in \text{region}\text{III}$, when $\left(a,\tau \right)$ crosses line $l_3$, system (1) undergoes a secondary Hopf bifurcation. The new emerging periodic solutions and trivial equilibrium is unstable, while the coexistence of periodic oscillation and two nontrivial equilibrium points are found.

4. $\left(a,\tau \right)\in \text{region}\text{IV}$, when $\left(a,\tau \right)$ crosses line $l_4$, a pitchfork of limit cycles occurs where two unstable periodic solutions disappear and the third periodic solution becomes from stable to unstable.

5. $\left(a,\tau \right)\in \text{region}\text{V}$, when $\left(a,\tau \right)$ crosses line $l_2$, unstable periodic solutions disappear by Hopf bifurcation.

6. $\left(a,\tau \right)\in \text{region}\text{VI}$, there coexist attractors of two stable nontrivial equilibria and the trivial equilibrium is unstable.

Fig. 10.

(top) Bifurcation Classification near $\left(a_0,{\tau }_0\right)=\left(-1,0.595965\right)$ (bottom) Phase portraits corresponding to classification sets

Remark 4

Different bifurcations correspond to different firing characteristics of neurons in the neural network model. Pitchfork-Hopf bifurcations mean that the neurons in the model are not only in the resting or periodic spiking state but also the multi-stability coexistence of the resting state and periodic spiking. It is also found that the state of the neuron system can realize the transition from resting state to periodic spiking as well as from periodic spiking to resting state with the varying of the two bifurcation parameters.

Numerical simulations

In this section, some numerical results are provided to support the results of classification near the pitchfork-Hopf bifurcation point $\left(a_0,{\tau }_0\right)=\left(-1,0.595965\right)$ by using Matlab software.

Firstly, the distributions of eigenvalues are plotted to verify the stability of trivial equilibrium point in each region as shown in Fig. 11. In region I, all eigenvalues around the trivial equilibrium point have negative real parts and the trivial equilibrium point is asymptotically stable. For the other five regions, at least one eigenvalue has positive real part where is marked with red star and the trivial equilibrium point is unstable.

Fig. 11.

The real parts of maximum eigenvalues on the characteristic Eq. (3) for network (1) are exhibited respectively in six regions. The eigenvalues with positive real parts are marked by the red star while the green star represents the eigenvalues with negative real parts

Secondly, some phase portraits of $x_1-x_2$ and time histories of $x_1$ are further displayed near zero-Hopf bifurcation point with three initial values in each region shown in Fig. 12. One can see that neural system can display multi-stability. There coexist periodic oscillation and two nontrivial equilibrium points in region IV. The coexistences of nontrivial fixed points are located in regions V to VI.

Fig. 12.

Phase portraits of $x_1-x_2$ and time histories of $x_1$ near zero-Hopf bifurcation point with $J_{12}=2,$, $J_{21}=2,$, $s=0.01.$ Neural system can display multi-stability. There coexist periodic oscillation and two nontrivial equilibrium points in region IV. The coexistence of nontrivial fixed points is located in regions V to VI

All numerical simulations are in good agreement with the theoretical results. The coexistence phenomena are exhibited and very important dynamical behaviors in nonlinear dynamics.

Conclusions

It is well known that to be closer to reality to help to understand the mechanism of complex dynamics, multiple delays and inertial characteristics need to be incorporated into the neural system. Furthermore, realistic modeling of networks inevitably also needs to vary connection topology. So the investigation of multiple delays and connection weight is more realistic and challenging for inertial neural systems. The results presented in this paper can be considered as an extension of the works for neural networks without inertia to the case with inertial couplings.

This paper mainly focuses on the effects of connection topology and time delay on the stability of multi-delay inertial neural networks. Firstly, some sufficient conditions of pitchfork bifurcation include the results presented in Ge (2022). Secondly, for the first time, utilizing the perturbation scheme with the help of the non-reduced order technique, rich dynamics near zero-Hopf interactions are derived with the variation of the two adjusting parameters and coexisting multiple attractors’ behaviors are discussed.

Investigation shows that self-connected weight can affect the number and dynamics of system equilibrium, while time delay can contribute to equilibrium losing its stability and generating periodic solutions. Limit cycles play an important role in the qualitative and quantitative theory of differential equations and model the behaviors of many real-world oscillatory systems (Van der Pol 1926). The obtained main findings of bifurcation analysis have significant theoretical guiding value in dominating and optimizing networks.

By using the perturbation involving the non-reduced order method, the search for explicit bifurcating periodic solutions is converted to solve the calculation problem of three algebraic equations. In contrast with the traditional reduced-order method, the straightforward analysis is simple and valid with less computation. As an analytical tool, the advantage of the scheme also lies in its simplicity and ease of implementation. It has a very clear procedure and can easily be programmed to calculate the bifurcating solution. As a future direction, it will be extended to discuss mainly codimension-two and chaos in inertial networks with multiple delays.

Appendix A

Proof

For simplification, let

$$P=\left(\begin{array}{c}{p}_1\\ p_2\\ \dots \\ p_n\end{array}\right),Q=\left(\begin{array}{c}{q}_1\\ q_2\\ \dots \\ q_n\end{array}\right),A_0=\left(\begin{array}{c}{s}_1\\ s_2\\ \dots \\ s_n\end{array}\right).$$

The solution (18) is rewritten as

$$Y\left(t\right)=Pcos\left(\omega t\right)+Qsin\left(\omega t\right)+A_0.$$ A.1

One can obtain the following equations from (A.1)

$$\frac{\mathit{dY}}{\mathit{dt}}=-\omega Psin\left(\omega t\right)+\omega Qcos\left(\omega t\right).$$ A.2

$$\frac{d^{2}Y}{d{t}^2}=-{\omega }^{2}Pcos\left(\omega t\right)-{\omega }^{2}Qsin\left(\omega t\right).$$ A.3

$$Y\left(t+s\right)=Pcos\left(\omega t+\omega s\right)+Qsin\left(\omega t+\omega s\right)+A_0.$$ A.4

$$Y\left(t+{\tau }_0\right)=Pcos\left(\omega t+\omega {\tau }_0\right)+Qsin\left(\omega t+\omega {\tau }_0\right)+A_0.$$ A.5

Substituting Eqs. (A.1) to (A.5) into Eq. (17), the following three equations are derived as

$$\begin{array}{c}\left[\omega K^T-D_{10}^{T}sin\left(\omega s\right)-D_2^{T}sin\left(\omega {\tau }_0\right)\right]P=\\ \left[-{\omega }^{2}I-B^T-D_{10}^{T}cos\left(\omega s\right)-D_2^{T}cos\left(\omega {\tau }_0\right)\right]Q,\\ \end{array}$$ A.6

$$\begin{array}{c}\left[-\omega K^T+D_{10}^{T}sin\left(\omega s\right)+D_2^{T}sin\left(\omega {\tau }_0\right)\right]Q=\\ \left[-{\omega }^{2}I-B^T-D_{10}^{T}cos\left(\omega s\right)-D_2^{T}cos\left(\omega {\tau }_0\right)\right]P,\\ \end{array}$$ A.7

$$\left(B^T+D_{10}^T+D_2^T\right)A_0=0.$$ A.8

That is,

$$-F_1^T\left(\begin{array}{c}{p}_1\\ p_2\\ \dots \\ p_n\end{array}\right)=F_2^T\left(\begin{array}{c}{q}_1\\ q_2\\ \dots \\ q_n\end{array}\right),F_1^T\left(\begin{array}{c}{q}_1\\ q_2\\ \dots \\ q_n\end{array}\right)=F_2^T\left(\begin{array}{c}{p}_1\\ p_2\\ \dots \\ p_n\end{array}\right),F_3^T\left(\begin{array}{c}{s}_1\\ s_2\\ \dots \\ s_n\end{array}\right)=0,$$

where the expressions of $F_i\left(i=1,2,3\right)$ are consistent with those in Eq. (10).

This completes this proof of the lemma.