Analysis of Hopf–Hopf Interactions Induced by Multiple Delays for Inertial Hopfield Neural Models

Abstract

The investigation of dynamic behaviors of inertial neural networks depicted by second-order delayed differential equations has received considerable attention. Substantial research has been performed on the transformed first-order differential equations using traditional variable substitution. However, there are few studies on bifurcation dynamics using direct analysis. In this paper, a multi-delay Hopfield neural system with inertial couplings is considered. The perturbation scheme and non-reduced order technique are firstly combined into studying multi-delay induced Hopf–Hopf singularity. This combination avoids tedious computation and overcomes the disadvantages of the traditional variable-substitution reduced-order method. In the neighbor of Hopf–Hopf interaction points, interesting dynamics are found on the plane of self-connected delay and coupled delay. Multiple delays can induce the switching of stable periodic oscillation and periodic coexistence. The explicit expressions of periodic solutions are obtained. The validity of theoretical results is shown through consistency with numerical simulations.

1. Introduction

It is well-known that time delays are unavoidable due to the finite transmission speed of signals, system reaction, and processing time between synapses. Therefore, it is meaningful and interesting to focus on how time delays influence the dynamic behavior of natural systems. In the past few decades, numerous excellent results on dynamic analysis have been obtained for a natural system of first-order delayed differential equations such as stability [1,2,3,4], synchronization [5,6,7,8,9], bifurcation [10,11,12,13,14], and chaotic activity [15]. Artificial neural networks are very complex nonlinear dynamical systems. Hopf bifurcation is studied in fractional-order delayed neural models [16]. Recently, inertial neural models depicted using second-order delayed differential equations have received considerable attention because of their theoretical and practical significance [17,18,19]. The authors considered complex codimension-two bifurcations and coexistence dynamics for delayed neural networks with inertia terms [20,21,22,23,24,25,26]. The criteria for global synchronization and dissipativity are established on inertial neural networks with discrete-time or time-varying delays [27,28,29,30].

It is worth mentioning that some weaknesses need to be addressed in the existing research results. Most discussions of single delay-induced bifurcation dynamics employ central manifold analysis [31]. Despite being multiple-delay systems, authors usually convert them into single-delay systems under the limitation of delays for analysis convenience, which cannot well reveal the real and complex dynamic characteristics of a nonlinear neural system. To the best of our knowledge, for a system of second-order differential equations, studies are also always made on the rewritten first-order equations by introducing proper variable transformations. Some authors [32] pointed out that this substitution doubles the dimension of the addressed systems and greatly increases the complexity of theoretical analysis and the difficulty of studied results. Therefore, there is an urgent need for a new technique to avoid and overcome those questions arising from the traditional variable-substitution reduced-order method. To be more concise, a natural idea is to perform a dynamic analysis directly on the second-order differential systems. Very recently, authors began to develop non-reduced order strategies to reveal the global exponential stability or dissipativity, or state estimation for inertial network models with different kinds of delays [33,34,35,36,37]. Limit cycles have been investigated for second-order differential models with a single delay [38]. However, there are few studies on bifurcation dynamics based on the non-reduced method at present.

Motivated by the above discussion, and due to the high precision of artificial neural networks, we incorporate inertial couplings into a multi-delay neural system in this paper. We attempt to study the joint influences of multiple time delays on Hopf–Hopf interactions with the aid of the non-reduced order technique. The main works of this paper are as follows:

(1)

Diverse time delays are two major parameters of our concern. Compared with the previous related works, investigating the joint influences of diverse delays on inertial neural systems is more realistic and meaningful.

(2)

The perturbation scheme and non-reduced order technique are first combined into the study of Hopf–Hopf interactions. In contrast with the traditional reduced-order method, it is simple and valid with less computation.

(3)

The search for analytical bifurcating solutions is converted to the problem of solving four algebraic equations.

(4)

A developed model with inertial couplings and multiple delays is investigated and theoretical results demonstrate its validity.

This paper is arranged as follows. In Section 2, a perturbation scheme involving the non-reduced order technique is demonstrated to investigate Hopf–Hopf singularity induced by multiple delays. In Section 3, the Hopf–Hopf bifurcation of a multi-delay Hopfield inertial neural network is discussed using the method proposed in Section 2 and the numerical results are displayed to verify the accuracy of the theoretical analysis. Finally, a brief conclusion is given in Section 4.

2. Methodology Formulation

In this section, a perturbation scheme with the aid of the non-reduced order technique is presented to investigate Hopf–Hopf singularities near the bifurcation point when considering two delays as control parameters for a general second-order nonlinear differential system.

2.1. Hopf–Hopf Bifurcation Point

Consider a general delayed system of $n$-dimension second-order nonlinear equations with two discrete delays as follows

$$\ddot{Z}\left(t\right)=L\dot{Z}\left(t\right)+E_{0}Z\left(t\right)+E_{1}Z\left(t-{\tau }_1\right)+E_{2}Z\left(t-{\tau }_2\right)+\epsilon F\left(Z\left(t\right),Z\left(t-{\tau }_1\right),Z\left(t-{\tau }_2\right)\right),$$

(1)

where $Z\left(t\right)\in R^n$, $\ddot{Z}\left(t\right)=d^{2}Z\left(t\right)/d{t}^2$, $\dot{Z}\left(t\right)=dZ\left(t\right)/dt$, $L$, $E_0$, $E_1$, and $E_2$ are real square matrices; $\epsilon$ is a nonlinear coupling strength; ${\tau }_1$ and ${\tau }_2$ are discrete delays; and a nonlinear smooth function $F\left(\cdot \right)$ satisfies $F\left(0,0,\dots ,0\right)=0$.

The equilibrium point of Equation (1) is the zero solution $\left(0,0,\dots ,0\right)$. Linearization for Equation (1) near $\left(0,0,\dots ,0\right)$ produces the following linear system

$$\ddot{Z}\left(t\right)=L\dot{Z}\left(t\right)+E_{0}Z\left(t\right)+E_{1}Z\left(t-{\tau }_1\right)+E_{2}Z\left(t-{\tau }_2\right).$$

The corresponding characteristic equation is

$$\det \left({\lambda }^{2}I-\lambda L-E_0-e^{-\lambda {\tau }_1}{E}_1-e^{-\lambda {\tau }_2}{E}_2\right)=0.$$

(2)

Our main goals concern the joint effects of two diverse delays on Hopf–Hopf interactions in Equation (1). Therefore, to obtain our main results, the following assumptions first need to be made.

(a1)

Diverse delays ${\tau }_1$ and ${\tau }_2$ are chosen as two control parameters.

(a2)

Increasing two control delays, Equation (1) undergoes weak resonant Hopf–Hopf bifurcations at the critical values ${\tau }_1={\tau }_1{}_c$ and ${\tau }_2={\tau }_2{}_c$. That is, the other roots of Equation (2) have negative real parts except for two distinct pairs of roots, $\pm i{k}_1\omega$ and $\pm i{k}_2\omega$ $\left(\omega >0\right)$ for ${\tau }_1={\tau }_1{}_c$ and ${\tau }_2={\tau }_2{}_c$ where $k_1$ and $k_2$ are positive real constants.

The critical values ${\tau }_1{}_c$ and ${\tau }_2{}_c$ of Hopf–Hopf interactions can be obtained by studying the distribution of the roots of the associated characteristic Equation (2).

2.2. Bifurcation Sets and Periodic Solutions

In this section, the non-reduced order technique combined with a perturbation scheme is constructed to derive the explicit expressions of stable and unstable periodic solutions and sets of dynamical classifications in the neighborhood of the critical point$\left({\tau }_1{}_c,{\tau }_2{}_c\right)$.

For control parameters ${\tau }_1$ and ${\tau }_2$, there always exists a small perturbation such as

$${\tau }_1={\tau }_1{}_c+{\delta }_1\epsilon ,{\tau }_2={\tau }_2{}_c+{\delta }_2\epsilon .$$

Accordingly, Equation (1) is equivalent to being transformed as

$$\ddot{Z}=L\dot{Z}+E_{0}Z+E_1{Z}_{{\tau }_{1c}}+E_2{Z}_{{\tau }_{2c}}+\widehat{F}\left(\cdot \right),$$

(3)

where

$$Z=Z\left(t\right),Z_{{\tau }_{1c}}=Z\left(t-{\tau }_{1c}\right),Z_{{\tau }_{2c}}=Z\left(t-{\tau }_{2c}\right),$$

$$\widehat{F}\left(\cdot \right)=E_1\left[Z_{{\tau }_1}-Z_{{\tau }_{1c}}\right]+E_2\left[Z_{{\tau }_2}-Z_{{\tau }_{2c}}\right]+\epsilon F\left(Z,Z_{{\tau }_1},Z_{{\tau }_2},\epsilon \right).$$

For $\epsilon =0$ in Equation (3), periodic solutions of the linearized equations are expressed by

$$Z_0\left(t\right)={\displaystyle \sum _{m=1}^2\left[\left(\begin{array}{c}{a}_{m1}\\ \begin{array}{l}{a}_{m2}\\ ⋮\\ a_{mn}\end{array}\end{array}\right)\cos \left(k_m\omega t\right)+\left(\begin{array}{c}{b}_{m1}\\ \begin{array}{l}{b}_{m2}\\ ⋮\\ b_{mn}\end{array}\end{array}\right)\sin \left(k_m\omega t\right)\right]},$$

(4)

where an integer $n$ is the dimension of Equation (1).

For $\epsilon =0$, taking Equation (4) to (3), we can solve the values of $a_{mj}$ and $b_{mj}$$\left(m=1,2;j=1,2,\dots ,n\right)$ in Equation (4) by satisfying the following equations

$$M_m\left(\begin{array}{c}{b}_{m1}\\ \begin{array}{l}{b}_{m2}\\ ⋮\\ b_{mn}\end{array}\end{array}\right)=N_m\left(\begin{array}{c}{a}_{m1}\\ \begin{array}{l}{a}_{m2}\\ ⋮\\ a_{mn}\end{array}\end{array}\right),-M_m\left(\begin{array}{c}{a}_{m1}\\ \begin{array}{l}{a}_{m2}\\ ⋮\\ a_{mn}\end{array}\end{array}\right)=N_m\left(\begin{array}{c}{b}_{m1}\\ \begin{array}{l}{b}_{m2}\\ ⋮\\ b_{mn}\end{array}\end{array}\right),$$

(5)

where

$$\begin{array}{l}{M}_m=k_m\omega L-E_1\sin \left(k_m\omega {\tau }_{1c}\right)-E_2\sin \left(k_m\omega {\tau }_{2c}\right),\\ N_m=-k_m{}^2{\omega }^{2}I-E_0-E_1\cos \left(k_m\omega {\tau }_{1c}\right)-E_2\cos \left(k_m\omega {\tau }_{2c}\right).\end{array}$$

(6)

For a small value $\epsilon \left(>0\right)$, the bifurcation solutions of the addressed Equation (3) can be seen as a small perturbation of periodic solutions in Equation (4), which are expressed in a polar coordinate by the following

$$Z\left(t\right)=\left(\begin{array}{c}{\displaystyle \sum _{m=1}^2{r}_m\left(\epsilon \right)\cos \left(\left(k_m\omega +{\sigma }_m\left(\epsilon \right)\right)t+{\theta }_m\right)}\\ z_2\left(t\right)\\ ⋮\\ z_n\left(t\right)\end{array}\right),$$

(7)

where $r_m\left(0\right)=r_m,$ and ${\theta }_m$ are determined by initial values.

The general expressions of $z_2\left(t\right),$ $\dots ,$ and $z_n\left(t\right)$ in the above, Equation (7), are too complicated and omitted here. For a specific differential system, the expressions of solutions are easy to simplify. Furthermore, it is not difficult to derive the values of the coefficients $r_m\left(\epsilon \right)$ and perturbations ${\sigma }_m\left(\epsilon \right)$ $\left(m=1,2\right)$ based on the adjoint operator. That is, firstly, according to the definition of the adjoint operator, one can obtain the adjoint system (8) in Lemma 1 of the original linear system (see Appendix A). Secondly, based on Lemma 1, $r_m\left(\epsilon \right)$ and ${\sigma }_m\left(\epsilon \right)$ $\left(m=1,2\right)$ can be solved from Theorem 1.

Lemma 1.

If$Y\left(t\right)$is the periodic solution of the following linear system

$$\ddot{Y}=-L^T\dot{Y}+E_0{}^{T}Y+E_1{}^{T}Y\left(t+{\tau }_{1c}\right)+E_2{}^{T}Y\left(t+{\tau }_{2c}\right),$$

(8)

then the analytical expression$Y\left(t\right)$is given by

$$Y\left(t\right)=\left(\begin{array}{c}{y}_1\left(t\right)\\ y_2\left(t\right)\\ ⋮\\ y_n\left(t\right)\end{array}\right)={\displaystyle \sum _{m=1}^2\left[\left(\begin{array}{c}{p}_{m1}\\ \begin{array}{l}{p}_{m2}\\ ⋮\\ p_{mn}\end{array}\end{array}\right)\cos \left(k_i\omega t\right)+\left(\begin{array}{c}{q}_{m1}\\ \begin{array}{l}{q}_{m2}\\ ⋮\\ q_{mn}\end{array}\end{array}\right)\sin \left(k_i\omega t\right)\right]},$$

(9)

with$Y\left(t\right)=Y\left(t+\frac{2\pi }{\omega }\right)$, where the unknown coefficients$p_{mj}$and$q_{mj}$$\left(m=1,2;j=1,2,\dots ,n\right)$are computed by the following equations

$$M_m{}^T\left(\begin{array}{c}{p}_{m1}\\ \begin{array}{l}{p}_{m2}\\ ⋮\\ p_{mn}\end{array}\end{array}\right)=N_m{}^T\left(\begin{array}{c}{q}_{m1}\\ \begin{array}{l}{q}_{m2}\\ ⋮\\ q_{mn}\end{array}\end{array}\right),-M_m{}^T\left(\begin{array}{c}{q}_{m1}\\ \begin{array}{l}{q}_{m2}\\ ⋮\\ q_{mn}\end{array}\end{array}\right)=N_m{}^T\left(\begin{array}{c}{p}_{m1}\\ \begin{array}{l}{p}_{m2}\\ ⋮\\ p_{mn}\end{array}\end{array}\right),$$

(10)

and $M_m$ and $N_m$ are expressed by Equation (6).

Theorem 1.

Based on Lemma 1, the values of$r_m\left(\epsilon \right)$and${\sigma }_m\left(\epsilon \right)$$\left(m=1,2\right)$ in perturbation solutions in Equation (7) are derived by

$$\begin{array}{l}{Y}^T\left(0\right)\left[\dot{Z}\left(2\pi /\omega \right)-\dot{Z}\left(0\right)\right]-{\dot{Y}}^T\left(0\right)\left[Z\left(2\pi /\omega \right)-Z\left(0\right)\right]\\ -Y^T\left(0\right)L\left[Z\left(2\pi /\omega \right)-Z\left(0\right)\right]-{\displaystyle {\int }_{-{\tau }_{1c}}^0{Y}^T\left(t+{\tau }_{1c}\right)E_1\left[Z\left(t\right)-Z\left(t+2\pi /\omega \right)\right]dt}\\ -{\displaystyle {\int }_{-{\tau }_{2c}}^0{Y}^T\left(t+{\tau }_{2c}\right)E_2\left[Z\left(t\right)-Z\left(t+2\pi /\omega \right)\right]dt}-{\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t\right)\widehat{F}\left(\cdot \right)dt}=0.\end{array}$$

(11)

Proof.

Multiplying both sides of Equation (3) by $Y^T\left(t\right)$ and integrating on $t$, we have

$${\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t\right)\ddot{Z}dt}={\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t\right)\left(L\dot{Z}+E_{0}Z+E_1{Z}_{{\tau }_{1c}}+E_2{Z}_{{\tau }_{2c}}+\widehat{F}\right)dt.}$$

(12)

Due to $Y\left(t\right)=Y\left(t+2\pi /\omega \right)$, we have the following equations through combination with the partial integral method

$$\begin{array}{l}{\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t\right)\ddot{Z}\left(t\right)dt=Y^T\left(0\right)\left[\dot{Z}\left(2\pi /\omega \right)-\dot{Z}\left(0\right)\right]-{\dot{Y}}^T\left(0\right)\left[Z\left(2\pi /\omega \right)-Z\left(0\right)\right]-{\displaystyle {\int }_0^{2\pi /\omega }{\dot{Y}}^T\left(t\right)LZdt}}\\ {\displaystyle +{\int }_0^{2\pi /\omega }{Y}^T\left(t\right)E_{0}Zdt+{\int }_0^{2\pi /\omega }{Y}^T\left(t+{\tau }_{1c}\right)E_{1}Zdt}+{\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t+{\tau }_{2c}\right)E_{2}Zdt},\end{array}$$

(13)

$${\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t\right)L\dot{Z}dt}=Y^T\left(0\right)L\left[Z\left(2\pi /\omega \right)-Z\left(0\right)\right]-{\displaystyle {\int }_0^{2\pi /\omega }{\dot{Y}}^T\left(t\right)LZdt},$$

(14)

$${\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t\right)E_1{Z}_{{\tau }_{1c}}dt}={\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t+{\tau }_{1c}\right)E_{1}Zdt}+{\displaystyle {\int }_{-{\tau }_{1c}}^0{Y}^T\left(t+{\tau }_{1c}\right)E_1\left[Z-Z\left(t+2\pi /\omega \right)\right]dt},$$

(15)

$${\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t\right)E_2{Z}_{{\tau }_{2c}}dt}={\displaystyle {\int }_0^{2\pi /\omega }{Y}^T\left(t+{\tau }_{2c}\right)E_{2}Zdt}+{\displaystyle {\int }_{-{\tau }_{2c}}^0{Y}^T\left(t+{\tau }_{2c}\right)E_2\left[Z-Z\left(t+2\pi /\omega \right)\right]dt}.$$

(16)

Equation (11) holds from Equations (13)–(16). □

Remark 1.

Equation (8) is the adjoint system of linearized system for the addressed Equation (1), which is obtained in Appendix A.

Remark 2.

The perturbation scheme and non-reduced order technique are first combined to discuss bifurcation dynamics. The incorporation of the non-reduced order technique can be considered as a complement of bifurcation investigation on the second-order differential model with the traditional reduced-order method [14,20].

Remark 3.

In contrast to the perturbation scheme with the traditional reduced order technique in [14], the main difference is the increase in two items in the key Equation (11) in Theorem 1, i.e.,$Y^T\left(0\right)\left[\dot{Z}\left(2\pi /\omega \right)-\dot{Z}\left(0\right)\right]$ and ${\dot{Y}}^T\left(0\right)\left[Z\left(2\pi /\omega \right)-Z\left(0\right)\right]$. However, the two terms are very simple to calculate because the expressions of $Z\left(\cdot \right)$,$\dot{Z}\left(\cdot \right)$ and $Y\left(\cdot \right)$,$\dot{Y}\left(\cdot \right)$ are obtained easily by the perturbation solution (7) and periodic solution (9), respectively. The non-reduced order technique introduced in this paper avoids and overcomes the above-mentioned problem of twice raising the dimension arising from the traditional reduced order technique. Furthermore, it has a very clear procedure. Therefore, it can be easier to program the computation of the periodic solution, using a direct analysis, than the reduced order technique in [14].

Remark 4.

Equation (11) can yield four algebraic equations for$r_1\left(\epsilon \right)$, $r_2\left(\epsilon \right)$, ${\sigma }_1\left(\epsilon \right)$, and ${\sigma }_2\left(\epsilon \right)$. When bifurcation parameters are in the neighborhood of Hopf–Hopf bifurcation points (i.e.,${\delta }_1\epsilon$and${\delta }_2\epsilon$are very small), it is not difficult to analytically derive periodic solutions in Equation (7) from them.

3. Multiple-Delay Neural Networks with Inertial Couplings

In this section, we employ the methodology in Section 2 to investigate multiple delay-induced Hopf–Hopf bifurcations for a pair of neurons’ multi-delay systems with inertial couplings.

3.1. Existence of Hopf–Hopf Bifurcation Point

Due to the high-precision description of artificial neural networks, we incorporate inertial couplings into the dynamical model [26]. The developed model with inertial couplings and multiple delays can be taken into account

$$\left\{\begin{array}{l}{M}_1\ddot{x}\left(t\right)=-m_1\dot{x}\left(t\right)-K_{1}x\left(t-{\tau }_1\right)+c_1\tanh \left(y\left(t-{\tau }_2\right)\right),\\ M_2\ddot{y}\left(t\right)=-m_2\dot{y}\left(t\right)-K_{2}y\left(t-{\tau }_1\right)+c_2\tanh \left(x\left(t-{\tau }_2\right)\right),\end{array}\right.$$

(17)

where $M_i$ and $m_i$ $\left(i=1,2\right)$ are positive real constants; $K_i$$\left(i=1,2\right)$ denotes self-connection weights from one neuron to itself; $c_i$$\left(i=1,2\right)$ are the cross-interaction weights through two neurons; and discrete delays ${\tau }_1$ and ${\tau }_2$ are non-negative real constants due to some physical meanings.

The characteristic equation of the linearization system for (17) around the trivial equilibrium point is determined by

$$\begin{array}{l}{\lambda }^4+{\lambda }^3\left(\frac{m_1}{M_1}+\frac{m_2}{M_2}\right)+{\lambda }^2\left(\frac{K_1}{M_1}{e}^{-\lambda {\tau }_1}+\frac{K_2}{M_2}{e}^{-\lambda {\tau }_2}+\frac{m_1{m}_2}{M_1{M}_2}\right)\\ +\lambda \left(\frac{K_2{m}_1{e}^{-\lambda {\tau }_1}}{M_1{M}_2}+\frac{K_1{m}_2{e}^{-\lambda {\tau }_2}}{M_1{M}_2}\right)+\frac{K_1{K}_2{e}^{-2\lambda {\tau }_1}}{M_1{M}_2}-\frac{c_1{c}_2{e}^{-2\lambda {\tau }_2}}{M_1{M}_2}=0.\end{array}$$

(18)

For $K_1{K}_2\ne c_1{c}_2$ (i.e.,$\lambda \ne 0$), substituting $\lambda =\pm i\omega \left(\omega >0\right)$ into the above characteristic Equation (18) and separating real and imaginary parts yields the following equations

$$\left\{\begin{array}{cc}\cos \left(2\omega {\tau }_2\right)& =\frac{K_1{K}_2\cos \left(2\omega {\tau }_1\right)+\left(K_2{m}_1+K_1{m}_2\right)\omega \sin \left(\omega {\tau }_1\right)}{c_1{c}_2}+\\ & \frac{{\omega }^4{M}_1{M}_2-{\omega }^2{m}_1{m}_2-\left(K_2{M}_1+K_1{M}_2\right){\omega }^2\cos \left(\omega {\tau }_1\right)}{c_1{c}_2},\\ \sin \left(2\omega {\tau }_2\right)& =\frac{K_1{K}_2\sin \left(2\omega {\tau }_1\right)-\left(K_2{m}_1+K_1{m}_2\right)\omega \cos \left(\omega {\tau }_1\right)}{c_1{c}_2}+\\ & \frac{{\omega }^3\left(m_2{M}_1+m_1{M}_2\right)-\left(K_2{M}_1+K_1{M}_2\right){\omega }^2\sin \left(\omega {\tau }_1\right)}{c_1{c}_2}.\end{array}\right.$$

(19)

Eliminating ${\tau }_2$ from ${\cos }^2\left(2\omega {\tau }_2\right)+{\sin }^2\left(2\omega {\tau }_2\right)=1$ in (19), we obtain the following equation on $\omega$

$$\begin{array}{l}{k}_1{}^2{k}_2{}^2-c_1{}^2{c}_2{}^2+{\omega }^8{M}_1^2{M}_2^2-2\omega k_1{k}_2\left(m_1{k}_2+k_1{m}_2\right)\sin \left(\omega {\tau }_1\right)\\ +{\omega }^2\left(-2\cos \left[2\omega {\tau }_1\right]k_1{k}_2{m}_1{m}_2+{\left(k_2{m}_1+k_1{m}_2\right)}^2-2\cos \left[\omega {\tau }_1\right]k_1{k}_2\left(k_2{M}_1+k_1{M}_2\right)\right)\\ 2{\omega }^3\left(-\sin \left[\omega {\tau }_1\right]m_1{m}_2\left(k_2{m}_1+k_1{m}_2\right)+\sin \left[2\omega {\tau }_1\right]k_1{k}_2\left(m_2{M}_1+m_1{M}_2\right)\right)\\ +{\omega }^4\left(-2\cos \left[\omega {\tau }_1\right]k_1{m}_2^2{M}_1+2\cos \left[2\omega {\tau }_1\right]k_1{k}_2{M}_1{M}_2\right)\\ +{\omega }^4\left({\left(k_2{M}_1+k_1{M}_2\right)}^2+m_1^2\left(m_2^2-2\cos \left[\omega {\tau }_1\right]k_2{M}_2\right)\right)+\omega m_1^2{M}_2^2{\omega }^5\\ +{\omega }^5\left(-2k_1\sin \left[\omega {\tau }_1\right]m_1{M}_2^2-2k_1\omega \cos \left[\omega {\tau }_1\right]M_1{M}_2^2+\omega m_2^2{M}_1^2\right)\end{array}$$

$$-2k_2{M}_1^2\left(\sin \left[\omega {\tau }_1\right]m_2+\omega \cos \left[\omega {\tau }_1\right]M_2\right){\omega }^5=0.$$

The transversality condition on Hopf bifurcations occurring should be satisfied. That is, $\mathrm{Re}\left({\lambda }^{\prime }\left({\tau }_2\right)\right)\left|{}_{\lambda =\pm i\omega }\right.\ne 0$, where

$${\lambda }^{\prime }\left({\tau }_2\right)=-\left(2c_1{c}_2\lambda e^{2{\tau }_1\lambda }\right)/G,$$

and

$$\begin{array}{l}G=2c_1{c}_2{\tau }_2{e}^{2{\tau }_1\lambda }+\left(K_2{m}_1+K_1{m}_2\right)e^{\left(2{\tau }_2+{\tau }_1\right)\lambda }-2K_1{K}_2{\tau }_1{e}^{2{\tau }_2\lambda }+2m_1{m}_2\lambda e^{2\left({\tau }_2+{\tau }_1\right)\lambda }\\ +2\left(K_2{m}_1+K_1{m}_2\right)\lambda e^{\left(2{\tau }_2+{\tau }_1\right)\lambda }-\left(K_2{m}_1+K_1{m}_2\right){\tau }_1\lambda e^{\left(2{\tau }_2+{\tau }_1\right)\lambda }+3\left(m_2{M}_1{\lambda }^2\right)e^{2\left({\tau }_2+{\tau }_1\right)\lambda }\\ -\left(K_2{m}_1+K_1{m}_2\right){\tau }_1{\lambda }^2{e}^{\left(2{\tau }_2+{\tau }_1\right)\lambda }+4M_1{M}_2{\lambda }^3{e}^{2\left({\tau }_2+{\tau }_1\right)\lambda }.\end{array}$$

If Equation (18) has two pairs of purely imaginary roots ${\lambda }_1=\pm i{\omega }_1$ and ${\lambda }_2=\pm i{\omega }_2$$\left({\omega }_1>{\omega }_2>0\right)$, then the critical delays are expressed by

$${\tau }_2^{1,j}=\frac{{\varphi }_1+2j\pi }{2{\omega }_1},j=0,1,2,\dots ,{\varphi }_1\in \left[0,2\pi \right)$$

where

$$\left\{\begin{array}{ll}\cos & \left({\varphi }_1\right)=\frac{K_1{K}_2\cos \left(2{\omega }_1{\tau }_1\right)+\left(K_2{m}_1+K_1{m}_2\right){\omega }_1\sin \left({\omega }_1{\tau }_1\right)}{c_1{c}_2}+\\ & \frac{{\omega }_1{}^4{M}_1{M}_2-\left(K_2{M}_1+K_1{M}_2\right){\omega }_1{}^2\cos \left({\omega }_1{\tau }_1\right)-{\omega }_1{}^2{m}_1{m}_2}{c_1{c}_2},\\ \sin & \left({\varphi }_1\right)=\frac{K_1{K}_2\sin \left(2{\omega }_1{\tau }_1\right)-\left(K_2{m}_1+K_1{m}_2\right){\omega }_1\cos \left({\omega }_1{\tau }_1\right)}{c_1{c}_2}+\\ & \frac{{\omega }_1{}^3\left(m_2{M}_1+m_1{M}_2\right)-\left(K_2{M}_1+K_1{M}_2\right){\omega }_1{}^2\sin \left({\omega }_1{\tau }_1\right)}{c_1{c}_2}.\end{array}\right.$$

and

$${\tau }_2^{2,j}=\frac{{\varphi }_2+2j\pi }{2{\omega }_2},j=0,1,2,\dots ,{\varphi }_2\in \left[0,2\pi \right)$$

where

$$\left\{\begin{array}{ll}\cos & \left({\varphi }_1\right)=\frac{K_1{K}_2\cos \left(2{\omega }_2{\tau }_1\right)+\left(K_2{m}_1+K_1{m}_2\right){\omega }_2\sin \left({\omega }_2{\tau }_1\right)}{c_1{c}_2}+\\ & \frac{{\omega }_2{}^4{M}_1{M}_2-\left(K_2{M}_1+K_1{M}_2\right){\omega }_2{}^2\cos \left({\omega }_2{\tau }_1\right)-{\omega }_2{}^2{m}_1{m}_2}{c_1{c}_2},\\ \sin & \left({\varphi }_1\right)=\frac{K_1{K}_2\sin \left(2{\omega }_2{\tau }_1\right)-\left(K_2{m}_1+K_1{m}_2\right){\omega }_2\cos \left({\omega }_2{\tau }_1\right)}{c_1{c}_2}+\\ & \frac{{\omega }_2{}^3\left(m_2{M}_1+m_1{M}_2\right)-\left(K_2{M}_1+K_1{M}_2\right){\omega }_2{}^2\sin \left({\omega }_2{\tau }_1\right)}{c_1{c}_2}.\end{array}\right.$$

It is well-known that double-Hopf bifurcation points are the intersection points of two Hopf bifurcation curves. Due to the existence of multiple delays in the neural system (17), double-Hopf bifurcation points cannot be derived analytically. However, for given parameter values, bifurcation points can be easily obtained.

For example, some values of parameters are chosen as follows

$$M_1=M_2=1.0,K_1=K_2=1.0,m_1=1.0,m_2=1.6,c_1=0.8,c_2=0.2.$$

Then, Equation (18) has only two pairs of simple purely imaginary roots $\pm i8\omega$ and $\pm i11\omega$, with $\omega =0.080575$ in red and all the other eigenvalues with negative real parts in green, in Figure 1. Therefore, we can obtain that Hopf–Hopf bifurcation occurs at ${\tau }_1={\tau }_{1c}=1.1383$ and ${\tau }_2={\tau }_{2c}=4.2847$.

3.2. Bifurcation Sets and Numerical Simulations

To apply the mentioned non-reduced order method, let some perturbations on bifurcation parameters

$${\tau }_1={\tau }_1{}_c+{\delta }_1{\epsilon }^2,{\tau }_2={\tau }_2{}_c+{\delta }_2{\epsilon }^2,$$

and variable transformation

$$x\left(t\right)\to \epsilon x\left(t\right),y\left(t\right)\to \epsilon y\left(t\right).$$

System (17) becomes Equation (1)

$$L=\left(\begin{array}{cc}-m_1& 0\\ 0& -m_2\end{array}\right),E_0=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right),E_1=\left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right),$$

$$E_2=\left(\begin{array}{cc}0& c_1\\ c_2& 0\end{array}\right),F\left(\cdot \right)=\left[\begin{array}{l}-\frac{\epsilon }{3}{c}_1{y}^3\left(t-{\tau }_1\right)+h.o.t\\ -\frac{\epsilon }{3}{c}_2{x}^3\left(t-{\tau }_2\right)+h.o.t\end{array}\right].$$

Based on Equations (5) and (6), the perturbation solution of system (17) close to Hopf–Hopf bifurcation points can be expressed as

$$Z\left(t\right)=\left(\begin{array}{c}\epsilon r_1\cos \left(\left(8\omega +{\epsilon }^2{\sigma }_1\right)t+{\theta }_1\right)+\epsilon r_2\cos \left(\left(11\omega +{\epsilon }^2{\sigma }_2\right)t+{\theta }_2\right)\\ \begin{array}{l}-0.368219\epsilon r_1\cos \left(\left(8\omega +{\epsilon }^2{\sigma }_1\right)t+{\theta }_1\right)-0.180657\epsilon r_1\sin \left(\left(8\omega +{\epsilon }^2{\sigma }_1\right)t+{\theta }_1\right)+\\ 0.280884\epsilon r_2\cos \left(\left(11\omega +{\epsilon }^2{\sigma }_2\right)t+{\theta }_2\right)-0.152974\epsilon r_2\sin \left(\left(11\omega +{\epsilon }^2{\sigma }_2\right)t+{\theta }_2\right)\end{array}\end{array}\right).$$

(20)

Using Lemma 1, the adjoint periodic solutions in Equation (9) is written as

$$Y\left(t\right)=\left[\begin{array}{c}{p}_{11}\cos \left(8\omega t\right)+q_{11}\sin \left(8\omega t\right)+p_{21}\cos \left(11\omega t\right)+q_{21}\sin \left(11\omega t\right)\\ p_{12}\cos \left(8\omega t\right)+q_{12}\sin \left(8\omega t\right)+p_{22}\cos \left(11\omega t\right)+q_{22}\sin \left(11\omega t\right)\end{array}\right],$$

(21)

where

$$p_{11}=-0.547394p_{12}+0.268365q_{12},q_{11}=-0.268365p_{12}-0.547394q_{12},$$

$$p_{21}=0.686302p_{22}+0.373805q_{22},q_{21}=-0.373805p_{22}+0.686302q_{22}.$$

Putting Equations (20) and (21) into Equation (11), omitting the higher order on, four algebraic equations on ${\epsilon }^2{r}_1,$${\epsilon }^2{r}_2,$${\epsilon }^2{\sigma }_1,$ and ${\epsilon }^2{\sigma }_2$ are derived as

$$\left\{\begin{array}{l}{\epsilon }^2{r}_1\left(-0.843989r_1{}^2-1.59277r_2{}^2+18.5664{\delta }_1-9.33851{\delta }_2+69.1896{\sigma }_1\right)=0,\\ {\epsilon }^2{r}_1\left(2.11533r_1{}^2+3.99186r_2{}^2-13.773{\delta }_1-3.72481{\delta }_2+94.2979{\sigma }_1\right)=0,\\ {\epsilon }^2{r}_2\left(-3.60921r_1{}^2-1.70279r_2{}^2-32.3505{\delta }_1-8.43129{\delta }_2+144.122{\sigma }_2\right)=0,\\ {\epsilon }^2{r}_2\left(2.77835r_1{}^2+1.31079r_2{}^2-11.3497{\delta }_1-10.9523{\delta }_2+7.97716{\sigma }_2\right)=0,\end{array}\right.$$

(22)

where

$${\epsilon }^2{\sigma }_1=-0.122374{\delta }_1{\epsilon }^2+0.101342{\delta }_2{\epsilon }^2,{\epsilon }^2{\sigma }_2=0.304847{\delta }_1{\epsilon }^2+0.146674{\delta }_2{\epsilon }^2.$$

Substituting the perturbed expressions of time delays ${\tau }_1={\tau }_1{}_c+{\epsilon }^2{\delta }_1,{\tau }_2={\tau }_2{}_c+{\epsilon }^2{\delta }_2$ in Equation (22), except for the solution $\left(0,0\right)$, we can easily obtain the other three solutions $\left(r_1,r_2\right)$ depending on two parameters ${\tau }_1$ and ${\tau }_2$, as follows

$$\left(1.66036\sqrt{-0.65625+4.34064{\tau }_1-{\tau }_2},0\right)\mathrm{for}{\tau }_2<-0.65625+4.34064{\tau }_1,$$

$$\left(0,2.73183\sqrt{-5.32241+0.911631{\tau }_1+{\tau }_2}\right)\mathrm{for}{\tau }_2>5.32241-0.911631{\tau }_1,$$

$$\left(2.36928\sqrt{-4.34369+0.0518228{\tau }_1+{\tau }_2},2.10601\sqrt{2.69693+1.39486{\tau }_1-{\tau }_2}\right)$$

$$\mathrm{for}4.34369-0.0518228{\tau }_1<{\tau }_2<2.69693+1.39486{\tau }_1.$$

In addition, four critical bifurcation lines are given as follows:

$$l_1:{\tau }_2=4.34064{\tau }_1-0.65625,l_2:{\tau }_2=5.32241-0.911631{\tau }_1,$$

$$l_3:{\tau }_2=4.34369-0.0518228{\tau }_1,l_4:{\tau }_2=1.39486{\tau }_1+2.69693.$$

The bifurcation diagram on the plane $\left({\tau }_1,{\tau }_2\right)$ is given close to the Hopf–Hopf interaction points in Figure 2, where the bifurcation point (1.1383,4.2847) is indicated by the intersection black dotted point. The delayed parameter plane is divided into six regions denoted by I, II, III, IV, V, and VI, which are divided by the line $l_i$ $\left(i=1,2,3,4\right)$ in red and their dynamic classifications depend on two delays. The interesting phenomena are exhibited involving stable equilibrium, the switching of periodic motions, and the coexistence of periodic solutions.

(1)

$\left({\tau }_1,{\tau }_2\right)\in$ region (I), the trivial equilibrium of Equation (17) is asymptotically stable. In addition, Equation (17) undergoes a Hopf bifurcation at the line $l_1$. Region (I) is a stability zone.

(2)

$\left({\tau }_1,{\tau }_2\right)\in$ region (II), $\left(0,0\right)$ loses its stability and a stable periodic oscillation emerges.

(3)

$\left({\tau }_1,{\tau }_2\right)\in$ region (III), when $\left({\tau }_1,{\tau }_2\right)$ crosses the line $l_2$, there exist two periodic solutions and the trivial equilibrium point. A periodic solution has high frequency and the trivial equilibrium point is unstable and the other periodic solution with low frequency is stable.

(4)

$\left({\tau }_1,{\tau }_2\right)\in$ region (IV), when $\left({\tau }_1,{\tau }_2\right)$ crosses the line $l_3$, there exist three periodic solutions and the trivial equilibrium point. The new emerging periodic solution and the trivial equilibrium are unstable.

(5)

$\left({\tau }_1,{\tau }_2\right)\in$ region (V), when $\left({\tau }_1,{\tau }_2\right)$ crosses the line $l_4$, the unstable periodic solution disappears. The periodic solution with high frequency is stable and the other periodic solution and the trivial equilibrium point are unstable.

(6)

$\left({\tau }_1,{\tau }_2\right)\in$ region (VI), there exists a periodic solution with high frequency and a trivial equilibrium point. The periodic solution bifurcating from the trivial equilibrium point is stable.

Next, we plot some stable-state solutions using the software Matlab. Firstly, the comparison of analytical periodic solutions from the reduced-order method [14] and the non-reduced order method from this paper is displayed in Figure 3, using time-response trajectories (left) and phase portraits (right) with $\left(1.15,4.2\right)\in \mathrm{case}(\mathrm{II})$.

Secondly, for $\left({\tau }_1,{\tau }_2\right)=\left(1.0,4.3\right)\in \mathrm{region}(\mathrm{I})$, time histories and all the eigenvalues of the characteristic equation (roots with negative real parts denoted by green) are given to verify a stable trivial equilibrium point in Figure 4. Figure 5 shows the multi-stability coexistence on periodic solutions, where $\left({\tau }_1,{\tau }_2\right)=\left(1.25,4.31\right)\in \mathrm{region}(\mathrm{IV})$. Finally, Figure 6 exhibits the stability switching of period oscillation from low frequency to high frequency corresponding to four bifurcation sets, where $\left({\tau }_1,{\tau }_2\right)=\left(1.15,4.2\right)\in \mathrm{region}(\mathrm{II})$, $\left(1.25,4.25\right)\in \mathrm{region}(\mathrm{II}\mathrm{I})$, $\left(1.25,4.35\right)\in \mathrm{region}(\mathrm{V})$ and $\left(1.1,4.45\right)\in$$\mathrm{region}(\mathrm{VI})$, respectively.

It is clear to see that our analytical results given by the non-reduced order technique are in excellent agreement with those displayed using the reduced-order method [14]. All the numerical results also show that the perturbation scheme in combination with the non-reduced order method is effective and easier for discussing bifurcation analytical solutions and dynamical classification in general second-order differential systems.

4. Conclusions

In this article, in order to clearly reveal the real and complex dynamic characteristics of neural systems, a multi-delay neural model with inertial couplings is considered. Inertia and multiple delays can produce richer nonlinear dynamics, including the switching of periodic oscillation and periodic coexistence. The perturbation scheme and non-reduced order technique are first combined to analyze the joint influences of diverse delays on Hopf–Hopf interactions. Results from this combination are in good agreement with those from the traditional reduced-order method. Furthermore, it has a very clear procedure and can easily be programmed to derive dynamical sets and explicit bifurcation solutions in closed form in the neighborhood of the critical point.

The inclusion of the non-reduced order technique avoids and overcomes the disadvantages of the traditional variable substitution reduced order method. These advantages have been verified in comparison with the previous related studies. The key research findings of our work have significant theoretical guiding value in dominating and optimizing networks.

To the best of our knowledge, bifurcation dynamics involving the non-reduced order technique are discussed for the first time in this paper. Further research will consider absolute synchronization based on the non-reduced order technique for a system of second-order delayed differential equations.