Periodic Oscillatory Solutions for a Nonlinear Model with Multiple Delays

Abstract

For systems such as the van der Pol and van der Pol–Duffing oscillators, the study of their oscillation is currently a very active area of research. Many authors have used the bifurcation method to try to determine oscillatory behavior. But when the system involves n separate delays, the equations for bifurcation become quite complex and difficult to deal with. In this paper, the existence of periodic oscillatory behavior was studied for a system consisting of n coupled equations with multiple delays. The method begins by rewriting the second-order system of differential equations as a larger first-order system. Then, the nonlinear system of first-order equations is linearized by disregarding higher-degree terms that are locally small. The instability of the trivial solution to the linearized equations implies the instability of the nonlinear equations. Periodic behavior often occurs when the system is unstable and bounded, so this paper also studied the boundedness here. It follows from previous work on the subject that the conditions here did result in periodic oscillatory behavior, and this is illustrated in the graphs of computer simulations.

1. Introduction

Periodic oscillation is one of the key characteristics in a general class of nonlinear systems, and in coupled van der Pol-like systems in particular. Bifurcation can arise as a periodic solution. Therefore, many researchers have investigated various models of van der Pol or van der Pol-like oscillators by means of bifurcation or other methods. For example, Torres et al. [1] studied a triplet of mechanical oscillators equipped with a van der Pol term and having a Huygens coupling. These have a special coordinated motion. A transformation was used to reduce the system. Then, the system was analyzed using Poincare’s perturbation method. In [2], accurate analytical solutions were obtained for a system of three coupled van der Pol–Duffing oscillators. In [3], there was a system of four mixed-mode vibration types with a pitchfork bifurcation delay phenomenon. In [4], the oscillations of a hybrid van der Pol–Duffing–Rayleigh system were studied, both numerically and theoretically. In [5], bursting oscillations were analyzed in a complex coupled four-dimensional Mathieu–van der Pol oscillator. In [6], sufficient conditions were obtained for the exponential input-to-state stability of a coupled van der Pol system using graph theory and Kirchoff’s matrix-tree theorem. In [7], fold bifurcation and Hopf bifurcation were studied using the formal adjoin theory, the center manifold theorem, and the normal form method for a neutral differential–difference equation of the van der Pol type. In [8], Ghosh et al. studied zones of stability for a delayed Duffing–van der Pol system, which gave rise to a variety of synchronization channels. In [9], the original system was transformed into a three-dimensional system, and then all of the Darboux polynomials of a Mathieu–van der Pol–Duffing system were characterized and a complete system of rational first integrals was obtained. In [10], Sysoev proposed a method involving multivariate time series for the reconstruction of nonlinear coupling functions for all elements of ensembles of coupled van der Pol systems. In [11], Hoskoti et al. proposed a reduced-order method to study the oscillatory nature of wake dynamics caused by vortex shedding. This used a spring-mounted airfoil coupled with a van der Pol oscillator. In [12], a chaotic system of van der Pol oscillators coupled to linear oscillators was stabilized using adaptive control and synchronization. The authors of [13] investigated the stability and dynamic behavior of the simultaneous primary and super-harmonic resonance of a van der Pol oscillator. The authors of [14] investigated bifurcation structures in a van der Pol–Mathieu oscillator with external excitation. In [15], modulated motion and bifurcation were studied for two van der Pol oscillators that were nonlinearly coupled and parametrically excited. The authors of [16] studied the dynamic behavior of a van der Pol biorhythmic oscillator subjected to both colored noise and harmonic excitation. The authors of [17] conducted a systematic search of the parameter space for bi-rhythmic and tri-rhythmic families of van der Pol and Rayleigh oscillators and their higher-order variants using a numerical simulation. The authors of [18,19] studied the bifurcation characteristics of a van der Pol–Duffing oscillator subjected to white noise excitation and adaptive synchronization. They identified parameters for chaos to occur. In [20], the lateral oscillations of a pedestrian walking on a periodically moving floor were modeled by a modified van der Pol–Rayleigh oscillator. The authors of [21] studied voltage oscillations in a Bonhoeffer–van der Pol oscillator with a non-ideal capacitor. In 2011, Zhang et al. discussed the dynamic behavior of the following symmetric system [22]:

$$\left\{\begin{array}{c}{x}_1^{″}\left(t\right)+x_1\left(t\right)-\epsilon \left(1-x_1^2\left(t\right)\right)x_1^{\prime }\left(t\right)=k\left(x_2^{\prime }\left(t-\tau \right)-x_1^{\prime }\left(t-\tau \right)\right)+k\left(x_3^{\prime }\left(t-\tau \right)-x_1^{\prime }\left(t-\tau \right)\right),\\ x_2^{″}\left(t\right)+x_2\left(t\right)-\epsilon \left(1-x_2^2\left(t\right)\right)x_2^{\prime }\left(t\right)=k\left(x_1^{\prime }\left(t-\tau \right)-x_2^{\prime }\left(t-\tau \right)\right)+k\left(x_3^{\prime }\left(t-\tau \right)-x_2^{\prime }\left(t-\tau \right)\right),\\ x_3^{″}\left(t\right)+x_3\left(t\right)-\epsilon \left(1-x_3^2\left(t\right)\right)x_3^{\prime }\left(t\right)=k\left(x_1^{\prime }\left(t-\tau \right)-x_3^{\prime }\left(t-\tau \right)\right)+k\left(x_2^{\prime }\left(t-\tau \right)-x_3^{\prime }\left(t-\tau \right)\right).\end{array}\right.$$

(1)

The dynamics of System (1) were carried out using a symmetric Hopf bifurcation result. Some important results about the spontaneous bifurcations of multiple branches of periodic solutions and their spatiotemporal patterns were obtained. Recently, Liu and Zhang extended Model (1) to four coupled van der Pol oscillators:

$$\left\{\begin{array}{c}{x}_1^{″}\left(t\right)=\alpha \left(p^2-x_1^2\left(t\right)\right)x_1^{\prime }\left(t\right)-x_1\left(t\right)+a{x}_1^{\prime }\left(t-\tau \right)+b{x}_2^{\prime }\left(t-\tau \right)+c{x}_3^{\prime }\left(t-\tau \right)+b{x}_4^{\prime }\left(t-\tau \right),\\ x_2^{″}\left(t\right)=\alpha \left(p^2-x_2^2\left(t\right)\right)x_2^{\prime }\left(t\right)-x_2\left(t\right)+a{x}_2^{\prime }\left(t-\tau \right)+b{x}_3^{\prime }\left(t-\tau \right)+c{x}_4^{\prime }\left(t-\tau \right)+b{x}_1^{\prime }\left(t-\tau \right),\\ \begin{array}{c}{x}_3^{″}\left(t\right)=\alpha \left(p^2-x_3^2\left(t\right)\right)x_3^{\prime }\left(t\right)-x_3\left(t\right)+a{x}_3^{\prime }\left(t-\tau \right)+b{x}_4^{\prime }\left(t-\tau \right)+c{x}_1^{\prime }\left(t-\tau \right)+b{x}_2^{\prime }\left(t-\tau \right),\\ x_4^{″}\left(t\right)=\alpha \left(p^2-x_4^2\left(t\right)\right)x_4^{\prime }\left(t\right)-x_4\left(t\right)+a{x}_4^{\prime }\left(t-\tau \right)+b{x}_1^{\prime }\left(t-\tau \right)+c{x}_2^{\prime }\left(t-\tau \right)+b{x}_3^{\prime }\left(t-\tau \right).\end{array}\end{array}\right.$$

(2)

where $a$, $b$, $c$, and $\tau >0$ are constants. By means of the symmetric Hopf bifurcation theory, the normal form of the system on the central manifold, the multiple periodic solutions of spatiotemporal patterns of the system were obtained and the bifurcating periodic solutions were also derived [23]. On the other hand, Nganso et al. considered the van der Pol oscillator with extended nonlinearity described by the following system:

$$\begin{gathered} x_i^{\prime \prime }\left(t\right)=\mu \left(1-x_i^2\left(t\right)+\alpha x_i^4\left(t\right)-\beta x_i^6\left(t\right) \right)x_i^{\prime }\left(t\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{k}{2p}}{\displaystyle \sum _{k=i-p}^{i+p}}\left[\alpha \right(x_k\left(t\right)-x_i\left(t\right))+\beta \left({x^{\prime }}_k\left(t\right)-{x^{\prime }}_i\left(t\right)\right)] \end{gathered}$$

(3)

where $i=1, 2, \cdots , n, k$ represents the strength of the coupling and $\alpha$ and $\beta$ are the interaction parameters. A phenomenon of coupling-induced multi-stability was found in Model (3). A large limit cycle and a smaller quasi-periodic-like attractor were discussed [24]. Ji and Zhang discussed a van der Pol–Duffing oscillator:

$$\begin{array}{c}{x}^{″}\left(t\right)-\mu x\left(t\right)+{\omega }^{2}x\left(t\right)-\mu x\left(t-\tau \right)-\epsilon x^{\prime }\left(t-\tau \right)+\beta x^2\left(t\right)x^{\prime }\left(t\right)+\alpha x^3\left(t\right)\\ -k_1{x}^3\left(t-\tau \right)-k_2({x\prime )}^3\left(t-\tau \right){-k}_3{x^{\prime }\left(t-\tau \right)x}^2\left(t-\tau \right)-{k_{4}x}^{\prime }\left(t-\tau \right)x\left(t-\tau \right)=0\end{array}$$

(4)

Hopf bifurcation and a quasi-periodic solution to Model (4) were investigated [25]. Motivated by the above models, in this paper, we studied the following n-coupled nonlinear differential system:

$$\left\{\begin{array}{c}{x}_1^{″}\left(t\right)={\mu }_1\left(1-x_1^2\left(t\right)+{\alpha }_1 x_1^4\left(t\right)-{\beta }_1 x_1^6\left(t\right)\right)x_1^{\prime }\left(t\right)-{\gamma }_1{x}_1\left(t\right)+k_1 x_1^3\left(t\right)\\ +a_{11}{x}_1^{\prime }\left(t-{\tau }_1\right)+\cdots +a_{1n}{x}_n^{\prime }\left(t-{\tau }_n\right)+{\displaystyle \sum _{k=2}^n}{p}_{k1}\left[x_k\left(t-{\tau }_k\right)-x_1\left(t-{\tau }_1\right)\right],\\ \begin{array}{c}{x}_2^{″}\left(t\right)={\mu }_2\left(1-x_2^2\left(t\right)+{\alpha }_2 x_2^4\left(t\right)-{\beta }_2 x_2^6\left(t\right)\right)x_2^{\prime }\left(t\right)-{\gamma }_2{x}_2\left(t\right)+k_2 x_2^3\left(t\right)\\ +a_{22}{x}_2^{\prime }\left(t-{\tau }_2\right)+\cdots +a_{2n}{x}_1^{\prime }\left(t-{\tau }_1\right)+a_{21}{x}_1^{\prime }\left(t-{\tau }_1\right)\\ \begin{array}{c}+{\displaystyle \sum _{k=1, k\ne 2}^n}{p}_{k2}\left[x_k\left(t-{\tau }_k\right)-x_2\left(t-{\tau }_2\right)\right],\\ x_3^{″}\left(t\right)={\mu }_3\left(1-x_3^2\left(t\right)+{\alpha }_3 x_3^4\left(t\right)-{\beta }_3 x_3^6\left(t\right)\right)x_3^{\prime }\left(t\right)-{\gamma }_3{x}_3\left(t\right)+k_3 x_3^3\left(t\right)\\ \begin{array}{c}+a_{33}{x}_3^{\prime }\left(t-{\tau }_3\right)+\cdots +a_{3n}{x}_1^{\prime }\left(t-{\tau }_1\right)+a_{31}{x}_1^{\prime }\left(t-{\tau }_1\right)+a_{32}{x}_2^{\prime }\left(t-{\tau }_2\right)\\ +{\displaystyle \sum _{k=1, i\ne 3}^n}{p}_{k3}\left[x_k\left(t-{\tau }_k\right)-x_3\left(t-{\tau }_3\right)\right],\\ \begin{array}{c} \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \begin{array}{c}{x}_n^{″}\left(t\right)={\mu }_n\left(1-x_n^2\left(t\right)+{\alpha }_n x_n^4\left(t\right)-{\beta }_n x_n^6\left(t\right)\right)x_n^{\prime }\left(t\right)-{\gamma }_n{x}_n\left(t\right)+k_n x_n^3\left(t\right)\\ +a_{nn}{x}_n^{\prime }\left(t-{\tau }_n\right)+a_{n1}{x}_1^{\prime }\left(t-{\tau }_1\right)+\cdots +a_{n,n-1}{x}_{n-1}^{\prime }\left(t-{\tau }_{n-1}\right)\\ +{\displaystyle \sum _{k=1}^{n-1}}{p}_{kn}\left[x_k\left(t-{\tau }_k\right)-x_n\left(t-{\tau }_n\right)\right],\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right.$$

(5)

where all the parameters are real numbers, $0<{\mu }_i\ll 1,0<{\alpha }_i,{\beta }_i,{\gamma }_i(i=1,2,\cdots ,n)$, and the time delays are $0<{\tau }_i\left(i=1, 2, \cdots , n\right)$. We were concerned with the existence of periodic oscillatory solutions to Model (5). It is known that periodic solutions can arise from bifurcation. However, if time delays involve several different numbers, the bifurcation method is hard to carry out in Model (5) due to the complexity of the bifurcation equations. In the present paper, the method of mathematical analysis was used to deal with the question of oscillatory solutions to this model.

2. Preliminaries

System (5) can be expressed in the following equivalent form:

$$\left\{\begin{array}{c}\begin{array}{c}{x}_1^{\prime }=x_2,\\ x_2^{\prime }={\mu }_1\left(1-x_1^2+{\alpha }_1 x_1^4-{\beta }_1 x_1^6\right)x_2-{\gamma }_1{x}_1+k_1 x_1^3+a_{11}{x}_2\left(t-{\tau }_1\right)\\ +a_{12}{x}_4\left(t-{\tau }_2\right)+\cdots +a_{1n-1}{x}_{2n-2}\left(t-{\tau }_{n-1}\right)+a_{1n}{x}_{2n}\left(t-{\tau }_n\right)\end{array}\\ +{\displaystyle \sum _{k=2}^n}{p}_{2k-\mathrm{1,1}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_1\left(t-{\tau }_1\right)\right],\\ \begin{array}{c}{x}_3^{\prime }=x_4,\\ x_4^{\prime }={\mu }_2\left(1-x_3^2+{\alpha }_2 x_3^4-{\beta }_2 x_3^6\right)x_4-{\gamma }_2{x}_3+k_2 x_3^3+a_{22}{x}_4\left(t-{\tau }_2\right)\\ \begin{array}{c}+a_{23}{x}_6\left(t-{\tau }_3\right)+\cdots +a_{2n}{x}_{2n}\left(t-{\tau }_n\right)+a_{21}{x}_2\left(t-{\tau }_1\right)\\ +{\displaystyle \sum _{k=1, k\ne 2}^n}{p}_{2k-\mathrm{1,3}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_3\left(t-{\tau }_2\right)\right],\\ \begin{array}{c}{x}_5^{\prime }=x_6,\\ x_6^{\prime }={\mu }_3\left(1-x_5^2+{\alpha }_3 x_5^4-{\beta }_3 x_5^6\right)x_6-{\gamma }_3{x}_5+k_3 x_5^3+a_{33}{x}_6\left(t-{\tau }_3\right)\\ \begin{array}{c}+a_{34}{x}_8\left(t-{\tau }_4\right)+\cdots +a_{3n}{x}_{2n}\left(t-{\tau }_n\right)+a_{31}{x}_2\left(t-{\tau }_1\right)+a_{32}{x}_4\left(t-{\tau }_2\right)\\ +{\displaystyle \sum _{k=1, k\ne 3}^n}{p}_{2k-\mathrm{1,5}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_5\left(t-{\tau }_3\right)\right],\\ \begin{array}{c}\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ x_{2n-1}^{\prime }=x_{2n},\\ \begin{array}{c}{x}_{2n}^{\prime }={\mu }_n\left(1-x_{2n-1}^2+{\alpha }_n x_{2n-1}^4-{\beta }_n x_{2n-1}^6\right)x_{2n}-{\gamma }_n{x}_{2n-1}+k_n x_{2n-1}^3\\ +a_{nn}{x}_{2n}\left(t-{\tau }_n\right)+a_{n1}{x}_2\left(t-{\tau }_1\right)+a_{n2}{x}_4\left(t-{\tau }_2\right)+\cdots +a_{n,n-1}{x}_{2n-2}\left(t-{\tau }_{n-1}\right)\\ +{\displaystyle \sum _{k=1}^{n-1}}{p}_{2k-\mathrm{1,2}n-1}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_{2n-1}\left(t-{\tau }_n\right)\right],\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right.$$

(6)

where $x_i^{\prime }=x_i^{\prime }\left(t\right),x_i=x_i\left(t\right)(i=1,2,\cdots ,2n)$. System (6) can be expressed as the following:

$$\left\{\begin{array}{c}\begin{array}{c}{x}_1^{\prime }=x_2,\\ x_2^{\prime }=-{\gamma }_1{x}_1+{{\mu }_{1}x}_2+a_{11}{x}_2\left(t-{\tau }_1\right)+a_{12}{x}_4\left(t-{\tau }_2\right)+\cdots +a_{1n-1}{x}_{2n-2}\left(t-{\tau }_{n-1}\right)\\ +a_{1n}{x}_{2n}\left(t-{\tau }_n\right)+{\displaystyle \sum _{k=2}^n}{p}_{2k-\mathrm{1,1}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_1\left(t-{\tau }_1\right)\right]\end{array}\\ +k_1 x_1^3{+\mu }_1\left(-x_1^2+{\alpha }_1 x_1^4-{\beta }_1 x_1^6\right)x_2\\ \begin{array}{c}{x}_3^{\prime }=x_4,\\ x_4^{\prime }=-{\gamma }_2{x}_3+{{\mu }_{2}x}_4+a_{22}{x}_4\left(t-{\tau }_2\right)+a_{23}{x}_6\left(t-{\tau }_3\right)+\cdots +a_{2n}{x}_{2n}\left(t-{\tau }_n\right)\\ \begin{array}{c}+a_{21}{x}_2\left(t-{\tau }_1\right)+{\displaystyle \sum _{k=1, k\ne 2}^n}{p}_{2k-\mathrm{1,3}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_3\left(t-{\tau }_2\right)\right]\\ {+k_2 x_3^3+\mu }_2\left(-x_3^2+{\alpha }_2 x_3^4-{\beta }_2 x_3^6\right)x_4\\ \begin{array}{c}{x}_5^{\prime }=x_6,\\ x_6^{\prime }=-{\gamma }_3{x}_5+{\mu }_3{x}_6+a_{33}{x}_6\left(t-{\tau }_3\right)+a_{34}{x}_8\left(t-{\tau }_4\right)+\cdots +a_{3n}{x}_{2n}\left(t-{\tau }_n\right)\\ \begin{array}{c}+a_{31}{x}_2\left(t-{\tau }_1\right)+a_{32}{x}_4\left(t-{\tau }_2\right)+{\displaystyle \sum _{k=1, k\ne 3}^n}{p}_{2k-\mathrm{1,5}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_5\left(t-{\tau }_3\right)\right]\\ {+k_3 x_5^3+\mu }_3\left(1-x_5^2+{\alpha }_3 x_5^4-{\beta }_3 x_5^6\right)x_6\\ \begin{array}{c}\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ x_{2n-1}^{\prime }=x_{2n},\\ \begin{array}{c}{x}_{2n}^{\prime }=-{\gamma }_n{x}_{2n-1}+{\mu }_n{x}_{2n}+a_{nn}{x}_{2n}\left(t-{\tau }_n\right)+a_{n1}{x}_2\left(t-{\tau }_1\right)+a_{n2}{x}_4\left(t-{\tau }_2\right)\\ +\cdots +a_{n,n-1}{x}_{2n-2}\left(t-{\tau }_{n-1}\right)+{\displaystyle \sum _{k=1}^{n-1}}{p}_{2k-\mathrm{1,2}n-1}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_{2n-1}\left(t-{\tau }_n\right)\right]\\ +k_n x_{2n-1}^3+{\mu }_n\left(-x_{2n-1}^2+{\alpha }_n x_{2n-1}^4-{\beta }_n x_{2n-1}^6\right)x_{2n}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right.$$

(7)

The matrix form of System (7) is as follows:

$$x^{\prime }\left(t\right)=Ax\left(t\right)+B\left(t-\tau \right)+f(x\left(t\right),$$

(8)

where $x\left(t\right)={[x_1\left(t\right),x_2\left(t\right),\cdots ,x_{2n}\left(t\right)]}^T$, $x\left(t-\tau \right)={[x_1\left(t-{\tau }_1\right),x_2\left(t-{\tau }_1\right),x_3\left(t-{\tau }_2,\right),\cdots ,x_{2n-1}\left(t-{\tau }_n\right),x_{2n}\left(t-{\tau }_n\right)]}^T$, $A$ and $B$ are both $2n\times 2n$ matrices, and $f\left(x\left(t\right)\right)$ is a $2n$-by-1 vector:

$$A={\left(a_{ij}\right)}_{2n\times 2n}=\left(\begin{array}{ccccccc}0& 1& 0& 0& \cdots & 0& 0\\ -{\gamma }_1& {\mu }_1& 0& 0& \cdots & 0& 0\\ 0& 0& 0& 1& \cdots & 0& 0\\ 0& 0& -{\gamma }_2& {\mu }_2& \cdots & 0& 0\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0& 0& 0& 0& \cdots & 0& 1\\ 0& 0& 0& 0& \cdots & -{\gamma }_{\mathrm{n}}& {\mu }_{\mathrm{n}}\end{array}\right),$$

$$B={\left(b_{ij}\right)}_{2n\times 2n}=\left(\begin{array}{ccccccc}0& 0& 0& 0& \cdots & 0& 0\\ b_{21}& b_{22}& b_{23}& b_{24}& \cdots & \cdots & b_{\mathrm{2,2}n}\\ 0& 0& 0& 0& \cdots & 0& 0\\ b_{41}& b_{42}& b_{43}& b_{44}& \cdots & \cdots & b_{\mathrm{4,2}n}\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ 0& 0& 0& 0& \cdots & 0& 0\\ b_{2n,1}& b_{2n,2}& b_{2n,3}& b_{2n,4}& \cdots & \cdots & b_{2n,2n}\end{array}\right),$$

where $b_{21}=$ ${\displaystyle {\sum }_{k=2}^n}{p}_{2k-\mathrm{1,1}},b_{22}=a_{11},{b_{22}=p}_{31},b_{24}=a_{12},\cdots ,b_{\mathrm{2,2}n-1}=p_{2n-\mathrm{1,1}},b_{22n}=a_{1n},{b_{41}=p}_{13},{b_{42}=a}_{21},$ $b_{43}={\displaystyle {\sum }_{k=1,k\ne 2}^n}{p}_{2k-\mathrm{1,3}},b_{44}=a_{22},\cdots ,b_{\mathrm{4,2}n-1}=p_{2n-\mathrm{1,3}},b_{\mathrm{4,2}n}=,\cdots ,$ ${b_{2n,1}=p}_{\mathrm{1,2}n-1},{b_{2n,2}=a}_{n1},{b_{2n,3}=p}_{\mathrm{3,2}n-1},{b_{2n,4}=a}_{n2},$ $b_{2n,2n-1}={\displaystyle {\sum }_{k=1}^{n-1}}{p}_{2k-\mathrm{1,2}n-1},b_{2n2n}=a_{nn},\mathrm{a}\mathrm{n}\mathrm{d} f\left(x\left(t\right)\right)={\left[0, k_1 x_1^3{+\mu }_1\left(-x_1^2+{\alpha }_1 x_1^4-{\beta }_1 x_1^6\right)x_2, \cdots 0, k_n x_{2n-1}^3+{\mu }_n\left(-x_{2n-1}^2+{\alpha }_n x_{2n-1}^4-{\beta }_n x_{2n-1}^6\right)x_{2n}\right]}^T.$ The linearized System (8) is as follows:

$$x^{\prime }\left(t\right)=Ax\left(t\right)+B\left(t-\tau \right).$$

(9)

Lemma 1. 

If matrix$S=A+B$is a nonsingular matrix for selected parameters, then there exists a unique equilibrium point for System (7) (or (8)).

Proof of Lemma 1. 

Assume that $x^{*}={[x_1^{*}, x_2^{*},\cdots ,x_{2n}^{*}]}^T$ is an equilibrium point of System (7); then, we have the following algebraic equations:

$$A{x}^{*}+B{x}^{*}+f\left(x^{*}\right)=\mathbf{0}$$

(10)

Since matrix $S=A+B$, Equation (10) can be written as

$$S{x}^{*}=-f\left(x^{*}\right)$$

(11)

Note that $f\left(\mathbf{0}\right)=\mathbf{0}$. So, there exists a trivial solution to (11). Since $\mathrm{S}=\mathrm{A}+\mathrm{B}$ is a nonsingular matrix, according to Cramer’s rule of linear algebra, Equation (11) has a unique solution, implying that System (7) has a unique trivial equilibrium. The proof is completed. □

Lemma 2. 

All solutions to System (6) (or (7)) are bounded, assuming that ${\beta }_i>0 \left(i=1, 2, \cdots , n\right).$

Proof of Lemma 2. 

To prove the boundedness of the solutions in System (6), we constructed a Lyapunov function $V\left(t\right)={\displaystyle {\sum }_{i=1}^{2n}}{\displaystyle \frac{1}{2}}{x}_i^2(t).$ By calculating the derivative of $V\left(t\right)$ through System (6), we have

$$\begin{gathered} V^{\prime }\left(t\right){|}_{\left(6\right)}={\displaystyle \sum _{i=1}^{2n}}{x}_i^{\prime }\left(t\right)x_i\left(t\right) \\ =-{\displaystyle \sum _{i=1}^n}{\beta }_i{x}_{2i-1}^6\left(t\right)x_{2i}^2\left(t\right)+{\displaystyle \sum _{i=1}^n}{\alpha }_i{x}_{2i-1}^4\left(t\right)x_{2i}^2\left(t\right) \\ -{\displaystyle \sum _{i=1}^n}{\mu }_i{x}_{2i-1}^2\left(t\right)x_{2i}^2\left(t\right)+{\displaystyle \sum _{i=1}^n}{k}_i{x}_{2i-1}^3\left(t\right)x_{2i}\left(t\right)+{\displaystyle \sum _{i=1}^n}{\mu }_i{x}_{2i}^2\left(t\right) \end{gathered}$$

(12)

Note that ${\beta }_i\left(i=1,2,\cdots ,n\right)$ are positive constants. Obviously, when $x_i\to +\mathrm{\infty }(1\le i\le 2n)$, $x_{2i-1}^6{x}_{2i}^2$ are higher-order infinity than $x_{2i-1}^4{x}_{2i}^2$, $x_{2i-1}^2{x}_{2i}^2\left(i=1,2,\cdots ,n\right),$ and so on. Therefore, there exists a suitably large $K>0$ such that $V\prime \left(t\right){|}_{\left(6\right)}<0$, as $\left|x_i\right|>K \left(i=1, 2, \cdots , 2n\right).$ This means that all solutions to System (6) (or (7)) are bounded. □

3. The Existence of Oscillatory Solutions

Theorem 1. 

Assume that System (7) (or (8)) has the unique trivial equilibrium point for selected parameter values. Let ${\alpha }_1, {\alpha }_2, \cdots , {\alpha }_{2n}$  and 0, ${\beta }_1, 0, {\beta }_2, { \cdots , 0, \beta }_n$ be characteristic values of matrix $A$and matrix $B$, respectively. Assume that Re(${\alpha }_{2i-1}$) > 0 ($i=1, 2, \cdots , n$), or there exists a characteristic value, say ${\alpha }_{2j}$, and Re(${\alpha }_{2j})>0$with Re(${\alpha }_{2j})>|Re\left({\beta }_j\right)|+|Im \left({\beta }_j\right)|$; then, the unique trivial solution to System (7) is unstable, implying that there exists an oscillatory solution in System (7).

Proof of Theorem 1.

Obviously, if the trivial solution to System (9) is unstable, then the trivial solution to System (7) (or (8)) is also unstable. Therefore, to discuss the instability of the trivial solution to System (7), we only need to deal with the instability of the trivial solution to System (9). The characteristic equation associated with System (9) is the following:

$$\det \left(\lambda I_{ij}-A-B{e}^{-\lambda \tau }\right)=0$$

(13)

where $I_{ij}$ is the identity matrix of 2$n$ by 2$n$. Noting that 0, ${\beta }_1, 0, {\beta }_2, { \cdots , 0, \beta }_n$ are characteristic values of matrix $B$, from (13), we have

$$\left\{\begin{array}{c}\lambda -{\alpha }_{2i-1}=0,\\ \lambda -{\alpha }_{2i}-{\beta }_i e^{-\lambda {\tau }_i}=0, i=\mathrm{1,2},\cdots , n. \end{array}\right.$$

(14)

Thus, we are led to an investigation of the nature of the roots for some $k, k\in \{1, 2, \cdots , n$}:

$$\lambda -{\alpha }_{2k-1}=0,$$

(15)

or

$$\lambda -{\alpha }_{2k}-{\beta }_k e^{-\lambda {\tau }_k}=0$$

(16)

For Equation (15), if Re(${\alpha }_{2k-1}$) > 0, this implies that there is a positive real part characteristic value. So, the trivial solution to System (9) is unstable. Equation (16) is a transcendental equation for which it is hard to find all the solutions. However, we showed that the trivial solution to System (9) is unstable under the assumption of Theorem 1. Let $\lambda ={ \lambda }_1+{i\lambda }_2, {\alpha }_{2k}={\alpha }_{2k,1}+i{\alpha }_{2k,2 }, {\beta }_k={\beta }_{k,1}+i{\beta }_{k,2},$ where ${ \lambda }_1=\mathrm{R}\mathrm{e}\left(\lambda \right),{ \lambda }_2=\mathrm{I}\mathrm{m}\left(\lambda \right),{\alpha }_{2k,1}=Re \left({\alpha }_{2k}\right),{\alpha }_{2k,2}=Im \left({\alpha }_{2k}\right). { \beta }_{k,1}=Re \left({\beta }_k\right), { \beta }_{k,2 }=Im \left({\beta }_k\right),$ respectively. By separating the real part and imaginary part of Equation (16), we obtain the following:

$${ \lambda }_1-{\alpha }_{2k,1}-{\beta }_{k,1} e^{-{ \lambda }_1{\tau }_k}\mathit{cos}{ (\lambda }_2{\tau }_k)+{\beta }_{k,2} e^{-{ \lambda }_1{\tau }_k}\mathit{sin}{ (\lambda }_2{\tau }_k)=0$$

(17)

$${ \lambda }_2-{\alpha }_{2k,2}-{\beta }_{k,2} e^{-{ \lambda }_1{\tau }_k}\mathit{cos}{ (\lambda }_2{\tau }_k)-{\beta }_{k,1} e^{-{ \lambda }_1{\tau }_k}\mathit{sin}{ (\lambda }_2{\tau }_k)=0$$

(18)

From the assumptions, there exists one ${\alpha }_j$, and Re(${\alpha }_j$) > 0 with $\mathrm{R}\mathrm{e}\left({\alpha }_j\right)>\left|Re\left({\beta }_j\right)\right|+|Im \left({\beta }_j\right)|$; we showed that Equation (17) has a positive real part root. Let

$$\varphi \left({ \lambda }_1\right)={ \lambda }_1-{\alpha }_{2j,1}-{\beta }_{j,1} e^{-{ \lambda }_1{\tau }_j}\mathit{cos}{ (\lambda }_2{\tau }_j)+{\beta }_{j,2} e^{-{ \lambda }_1{\tau }_j}\mathit{sin}{ (\lambda }_2{\tau }_j)$$

(19)

Obviously, $\varphi \left({ \lambda }_1\right)$ is a continuous function of ${ \lambda }_1$. Noting that ${\alpha }_{2j,1}>{|\beta }_{j,1}|+\left|{\beta }_{j,2}\right|,$ then $\varphi \left(0\right)=-{\alpha }_{2j,1}-{\beta }_{j,1}\mathit{cos}{ (\lambda }_2{\tau }_j)+{\beta }_{j,2} \mathit{sin}\begin{array}{c}{ (\lambda }_2{\tau }_j)\le -{\alpha }_{2j,1}+\left|{\beta }_{j,1}\right|+\left|{\beta }_{j,2}\right|<0. \\ \end{array}$ Since $e^{-{ \lambda }_1{\tau }_k}\to 0$ as ${ \lambda }_1\to +\infty ,$ there exists a suitably large ${ \lambda }_1$, say ${\lambda }_1^{*}$, such that $\varphi \left({\lambda }_1^{*}\right)={\lambda }_1^{*}-{\alpha }_{2j,1}-{\beta }_{j,1} e^{-{\lambda }_1^{*}{\tau }_j}\mathit{cos}{ (\lambda }_2{\tau }_j)+{\beta }_{j,2} e^{-{\lambda }_1^{*}{\tau }_j}\mathit{sin}{ (\lambda }_2{\tau }_j)>0.$ □

According to the intermediate value theorem, there exists a ${ \lambda }_1,$ say ${ \lambda }_{10}\in$ (0, ${\lambda }_1^{*}$), such that $\varphi \left({ \lambda }_{10}\right)=0,$ implying that there is a positive real part characteristic value of Equation (17). This means that the trivial solution to System (9) is unstable. Note that $f\left(x\left(t\right)\right)$ of System (8) is a higher-order infinitesimal as ${ x}_i\to 0 \left(i=\mathrm{1,2},\cdots , 2n\right).$ Therefore, the instability of the trivial solution to System (9) implies the instability of the trivial solution to nonlinear System (8). This suggests that the unique trivial equilibrium point of System (7) is unstable. This instability of the unique trivial equilibrium point, together with the boundedness of the solutions, will force System (7) (or (8)) to generate an oscillatory solution [26,27].

Now we adopt the following norms of the vector and matrix [28]:

$$‖x‖={\displaystyle {\sum }_{i=1}^{2n}}\left|{ x}_i\right|, ‖B‖=\underset{1\le j\le 2n}{\mathit{max}}{\displaystyle {\sum }_{i=1}^{2n}} \left|{ b}_{ij}\right|,\mathrm{and} \mathrm{let} s=\underset{1\le i\le n}{max}\left\{-{\gamma }_i, 1+{\mu }_i\right\}.$$

Note that ${\mu }_i>0$; thus, $s>0$ and we have the following:

Theorem 2. 

Assume that the conditions of Lemma 1 and Lemma 2 hold. Determine if the following inequality is satisfied:

$$7s ‖B‖{\tau }_{*}^2>4 e^{s{\tau }_{*}},$$

(20)

where${\tau }_{*}=\mathit{min}\left\{{\tau }_1,{\tau }_2, \cdots , {\tau }_n\right\}.$ If so, then the trivial solution to System (9) is unstable, implying that System (7) (or (8)) has an oscillatory solution.

Proof of Theorem 2. 

To prove the instability of the trivial solution to System (9), let $y\left(t\right)={\displaystyle {\sum }_{i=1}^{2n}}\left|x_i\left(t\right)\right|.$ Therefore, $y\left(t\right)>0$ and

$$y\prime \left(t\right)\le s y\left(t\right)+‖B‖ y(t-\tau )$$

(21)

Specifically, consider a scalar equation:

$$z\prime \left(t\right)=s z\left(t\right)+‖B‖ z(t-\tau )$$

(22)

According to the comparison theory of differential equations, we have $y\left(t\right)\le z\left(t\right).$ If the trivial solution to Equation (22) is unstable, then the trivial solution to (21) is still unstable. The characteristic equation associated with Equation (22) is given by

$$\lambda =s+‖B‖e^{-\lambda \tau }$$

(23)

If the trivial solution to Equation (22) is stable, then Equation (23) must have a real negative root, say ${\lambda }_{*}$, and from (23), we have

$$\left|{\lambda }_{*}\right|>‖B‖e^{\left|{\lambda }_{*}\tau \right|}-s\ge ‖B‖e^{\left|{\lambda }_{*}{\tau }_{*}\right|}-s$$

(24)

One can prove that the inequality $e^x\ge {\displaystyle \frac{7}{4}}{x}^2(x>0)$ holds. So, we have

$$1\ge {\displaystyle \frac{‖B‖e^{\left|{\lambda }_{*}{\tau }_{*}\right|}}{\left|{\lambda }_{*}\right|+s}}={\displaystyle \frac{‖B‖ {\tau }_{*}{e}^{\left(\left|{\lambda }_{*}\right|+s\right){\tau }_{*}}}{e^{s{\tau }_{*}}\left(\left|{\lambda }_{*}\right|+s\right){\tau }_{*}}}\ge {\displaystyle \frac{7‖B‖ {\tau }_{*}^2\left(\left|{\lambda }_{*}\right|+s\right)}{4e^{s{\tau }_{*}}}}>{\displaystyle \frac{7s‖B‖ {\tau }_{*}^2}{4e^{s{\tau }_{*}}}}$$

(25)

A contradiction with Inequality (20) implies that the trivial solution to Equation (22) is unstable. This suggests that the trivial solution to Equation (21) is unstable, implying that the trivial solution to System (8) is unstable. Similar to Theorem 1, System (7) (or (8)) generates an oscillatory solution. The proof is completed. □

4. Simulation Result

Firstly, consider $n=5$ in System (6) as the following:

$$\left\{\begin{array}{c}{x}_1^{\prime }=x_2,\\ x_2^{\prime }={\mu }_1\left(1-x_1^2+{\alpha }_1 x_1^4-{\beta }_1 x_1^6\right)x_2-{\gamma }_1{x}_1+k_1 x_1^3+a_{11}{x}_2\left(t-{\tau }_1\right)\\ \begin{array}{c}+\cdots +a_{15}{x}_{10}\left(t-{\tau }_5\right)+{\displaystyle \sum _{k=2}^5}{p}_{2k-\mathrm{1,1}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_1\left(t-{\tau }_1\right)\right],\\ x_3^{\prime }=x_4,\\ \begin{array}{c}{x}_4^{\prime }={\mu }_2\left(1-x_3^2+{\alpha }_2 x_3^4-{\beta }_2 x_3^6\right)x_4-{\gamma }_2{x}_3+k_2 x_3^3+a_{22}{x}_4\left(t-{\tau }_2\right)\\ +a_{23}{x}_6\left(t-{\tau }_3\right)+a_{24}{x}_8\left(t-{\tau }_4\right)+\cdots +a_{21}{x}_2\left(t-{\tau }_1\right)\\ \begin{array}{c}+{\displaystyle \sum _{k=1, k\ne 2}^5}{p}_{2k-\mathrm{1,3}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_3\left(t-{\tau }_2\right)\right],\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \begin{array}{c}{x}_9^{\prime }=x_{10},\\ x_{10}^{\prime }={\mu }_5\left(1-x_9^2+{\alpha }_5 x_9^4-{\beta }_5 x_9^6\right)x_{10}-{\gamma }_5{x}_9+k_5 x_9^3+a_{55}{x}_{10}\left(t-{\tau }_5\right)\\ \begin{array}{c}+a_{51}{x}_2\left(t-{\tau }_1\right)+a_{52}{x}_4\left(t-{\tau }_2\right)+\cdots +a_{54}{x}_8\left(t-{\tau }_4\right)\\ +{\displaystyle \sum _{k=1}^4}{p}_{2k-\mathrm{1,9}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_9\left(t-{\tau }_5\right)\right],\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right.$$

(26)

The parameter values are shown in

Table 1. The parameter values in System (26).

μ1	μ2	μ3	μ4	μ5	α1	α2	α3	α4	α5
0.0045	0.0048	0.0044	0.0046	0.0042	0.75	0.74	0.72	0.78	0.82
β1	β2	β3	β4	β5	γ1	γ2	γ3	γ4	γ5
0.65	0.68	0.62	0.64	0.66	0.95	0.96	0.92	0.98	0.90
k1	k2	k3	k4	k5	a11	a12	a13	a14	a15
0.0015	0.0018	0.0012	0.0016	0.0014	0.82	−0.88	0.85	−0.86	0.84
a21	a22	a23	a24	a25	a31	a32	a33	a34	a35
1.16	1.15	−1.10	1.18	−1.14	0.84	−0.87	0.86	−0.85	0.84
a41	a42	a43	a44	a45	a51	a52	a53	a54	a55
0.92	−0.98	0.94	−0.96	0.95	0.75	0.72	0.78	−0.76	0.71
p31	p51	p71	p91		p13	p53	p73	p93
0.76	−0.78	0.72	0.90		0.35	−0.32	0.34	0.30
p15	p35	p75	p95		p17	p37	p57	p97
0.24	−0.25	0.28	0.32		0.70	−0.52	0.48	0.46
p19	p39	p59	p79
0.52	−0.45	0.48	0.42

.

Then, the characteristic values of matrices $A$ and $B$ in System (26) are $0.0021\pm 0.9487i,$0.0022$\pm$ 0.9747 $i,$ 0.0022 $\pm$ 0.9592 $i,$ 0.0023$\pm$ 0.9899 $i,$ 0.0024 $\pm$ 0.9798 $i,$ and $-0.0015,$0.0446, 1.4635, 0.5417$\pm$ 1.2873 $i,$ 0, 0, 0, 0, 0, respectively. The time delays are selected as ${\tau }_1=$1.15, ${\tau }_2=1$.16, ${\tau }_3=1.25,$ ${\tau }_4=1.18,$ ${\tau }_5=1.12,$ and ${\tau }_1=1.65,$ ${\tau }_2=1.66,$ ${\tau }_3=1.75,$ ${\tau }_4=1.68, { \tau }_5=1.62$, respectively. All the characteristic values of matrix $A$ are complex numbers, and the real parts are greater than zero. Matrix $B$ has a characteristic value of zero; thus the conditions of Theorem 1 are satisfied. There exists an oscillatory solution to System (26) (see Figure 1 and Figure 2). From Figure 1 and Figure 2, the curves of $x_{1 }\left(t\right)$, $x_{3 }\left(t\right)$, $x_{5 }\left(t\right)$, $x_{7 }\left(t\right)$, and $x_{9 }\left(t\right)$ are smooth and the amplitude of $x_{1 }\left(t\right)$ is only 4.0743. However, the curves of $x_{2 }\left(t\right)$, $x_{4 }\left(t\right)$, $x_{6 }\left(t\right)$, $x_{8 }\left(t\right)$, and $x_{10 }\left(t\right)$ are jagged due to the higher order of variables in their equations. The amplitude of curve $x_{2 }\left(t\right)$ is 14.1072. The oscillation frequency slows down as the time delay increases.

Then, consider $n=6$ in System (6) as the following:

$$\left\{\begin{array}{c}{x}_1^{\prime }=x_2,\\ x_2^{\prime }={\mu }_1\left(1-x_1^2+{\alpha }_1 x_1^4-{\beta }_1 x_1^6\right)x_2-{\gamma }_1{x}_1+k_1 x_1^3+a_{11}{x}_2\left(t-{\tau }_1\right)\\ \begin{array}{c}+\cdots +a_{16}{x}_{12}\left(t-{\tau }_6\right)+{\displaystyle \sum _{k=2}^6}{p}_{2k-\mathrm{1,1}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_1\left(t-{\tau }_1\right)\right],\\ x_3^{\prime }=x_4,\\ \begin{array}{c}{x}_4^{\prime }={\mu }_2\left(1-x_3^2+{\alpha }_2 x_3^4-{\beta }_2 x_3^6\right)x_4-{\gamma }_2{x}_3+k_2 x_3^3+a_{22}{x}_4\left(t-{\tau }_2\right)\\ +a_{23}{x}_6\left(t-{\tau }_3\right)+a_{24}{x}_8\left(t-{\tau }_4\right)+\cdots +a_{21}{x}_2\left(t-{\tau }_1\right)\\ \begin{array}{c}+{\displaystyle \sum _{k=1, k\ne 2}^6}{p}_{2k-\mathrm{1,3}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_3\left(t-{\tau }_2\right)\right],\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \begin{array}{c}{x}_{11}^{\prime }=x_{12},\\ x_{12}^{\prime }={\mu }_6\left(1-x_{11}^2+{\alpha }_6 x_{11}^4-{\beta }_6 x_{11}^6\right)x_{12}-{\gamma }_6{x}_{11}+k_6 x_{11}^3+a_{66}{x}_{12}\left(t-{\tau }_6\right)\\ \begin{array}{c}+a_{61}{x}_2\left(t-{\tau }_1\right)+a_{62}{x}_4\left(t-{\tau }_2\right)+\cdots +a_{65}{x}_{10}\left(t-{\tau }_5\right)\\ +{\displaystyle \sum _{k=1}^5}{p}_{2k-\mathrm{1,11}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_{11}\left(t-{\tau }_5\right)\right],\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right.$$

(27)

The parameters are shown in

Table 2. The parameter values in System (27).

μ1	μ2	μ3	μ4	μ5	μ6	α1	α2	α3	α4	α5	α6
0.0145	0.0148	0.0144	0.0146	0.0142	0.0143	0.25	0.24	0.22	0.28	0.26	0.27
β1	β2	β3	β4	β5	β6	γ1	γ2	γ3	γ4	γ5	γ6
1.55	1.68	1.62	1.64	1.66	1.60	2.95	2.96	2.92	2.98	2.90	2.93
k1	k2	k3	k4	k5	k6	a11	a12	a13	a14	a15	a16
0.15	0.18	0.12	0.16	0.14	0.17	0.12	−0.38	0.15	−0.36	0.14	0.13
a21	a22	a23	a24	a25	a26	a31	a32	a33	a34	a35	a36
0.16	0.15	−0.10	0.18	−0.14	0.13	0.42	−0.28	0.46	−0.45	0.44	0.42
a41	a42	a43	a44	a45	a46	a51	a52	a53	a54	a55	a56
0.52	−0.58	0.54	−0.66	0.45	−0.66	0.45	0.42	0.48	−0.36	0.46	0.40
a61	a62	a63	a64	a65	a66	p31	p51	p71	p91	p111
0.72	−0.74	0.70	0.68	−0.62	0.66	0.78	−0.75	0.72	0.70	0.74
p13	p53	p73	p93	p113		p15	p35	p75	p95	p115
0.35	−0.62	0.34	−0.40	0.32		0.24	0.26	−0.38	0.34	0.37
p17	p37	p57	p97	p117		p19	p39	p59	p79	p119
0.70	−0.72	0.48	0.46	0.50		0.52	−0.45	0.48	0.42	0.46
p111	p311	p511	p711	p911
0.62	−0.78	0.68	0.64	−0.70

.

It is easy to see that $s=$1.0148 and $‖B‖=$ 4.74 in System (27). The time delays are selected as ${\tau }_1=$0.75, ${\tau }_2=0$.78, ${\tau }_3=0.74,$ ${\tau }_4=0.72,$ ${\tau }_5=0.73,$ ${\tau }_6=0.77,$ and ${\tau }_1=1.05,{\tau }_2=1.08,$ ${\tau }_3=1.04,$ ${\tau }_4=1.02,$ ${\tau }_5=1.03$, ${\tau }_6=1.07,$ respectively. Then, ${\tau }_{*}=0.72,$ and ${\tau }_{*}=1.02.$ Thus, $7s‖B‖{\tau }_{*}^2=7\times 1.0148\times 4.74\times 0.72\times 0.72=17.4551>8.3060=4e^{s{\tau }_{*}}$ and $‖B‖{\tau }_{*}^2=7\times 1.0148\times 4.74\times 1.02\times 1.02=35.0314>11.2616=4e^{s{\tau }_{*}}.$ The conditions of Theorem 2 are satisfied. There exists an oscillatory solution to System (27) (see Figure 3 and Figure 4). From Figure 3 and Figure 4, the curves for odd subscripts are even and almost the same shapes, and the amplitude of $x_1\left(t\right)$ is 2.4018. The curves for even subscripts are steep and uneven, and the amplitude of $x_2\left(t\right)$ is 10.1625. The amplitude of the less-even curves is greater than that of the smooth curves. When the time delays increase, the oscillation frequencies decrease slightly.

Then, consider $n=7$ in System (6) as the following:

$$\left\{\begin{array}{c}{x}_1^{\prime }=x_2,\\ x_2^{\prime }={\mu }_1\left(1-x_1^2+{\alpha }_1 x_1^4-{\beta }_1 x_1^6\right)x_2-{\gamma }_1{x}_1+k_1 x_1^3+a_{11}{x}_2\left(t-{\tau }_1\right)\\ \begin{array}{c}+\cdots +a_{17}{x}_{14}\left(t-{\tau }_7\right)+{\displaystyle \sum _{k=2}^7}{p}_{2k-\mathrm{1,1}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_1\left(t-{\tau }_1\right)\right],\\ x_3^{\prime }=x_4,\\ \begin{array}{c}{x}_4^{\prime }={\mu }_2\left(1-x_3^2+{\alpha }_2 x_3^4-{\beta }_2 x_3^6\right)x_4-{\gamma }_2{x}_3+k_2 x_3^3+a_{22}{x}_4\left(t-{\tau }_2\right)\\ +a_{23}{x}_6\left(t-{\tau }_3\right)+\cdots +a_{27}{x}_{14}\left(t-{\tau }_7\right)+a_{21}{x}_2\left(t-{\tau }_1\right)\\ \begin{array}{c}+{\displaystyle \sum _{k=1, k\ne 2}^7}{p}_{2k-\mathrm{1,3}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_3\left(t-{\tau }_2\right)\right],\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \begin{array}{c}{x}_{13}^{\prime }=x_{14},\\ x_{14}^{\prime }={\mu }_7\left(1-x_{13}^2+{\alpha }_7 x_{13}^4-{\beta }_7 x_{13}^6\right)x_{14}-{\gamma }_7{x}_{13}+k_7 x_{13}^3+a_{77}{x}_{14}\left(t-{\tau }_7\right)\\ \begin{array}{c}+a_{71}{x}_2\left(t-{\tau }_1\right)+a_{72}{x}_4\left(t-{\tau }_2\right)+\cdots +a_{76}{x}_{12}\left(t-{\tau }_6\right)\\ +{\displaystyle \sum _{k=1}^6}{p}_{2k-\mathrm{1,13}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_{13}\left(t-{\tau }_7\right)\right],\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right.$$

(28)

The parameters are shown in

Table 3. The parameter values in System (28).

μ1

μ2

μ3

μ4

μ5

μ6

μ7

0.0284

0.0292

0.0295

0.0286

0.0288

0.0283

0.0281

α1

α2

α3

α4

α5

α6

α7

0.15

0.14

0.12

0.18

0.16

0.13

0.10

β1

β2

β3

β4

β5

β6

β7

0.65

0.68

0.62

0.64

0.66

0.60

0.65

γ1

γ2

γ3

γ4

γ5

γ6

γ7

3.15

3.16

3.12

3.18

3.10

3.14

3.17

k1

k2

k3

k4

k5

k6

k7

0.015

0.020

0.012

0.016

0.018

0.013

0.014

a11

a12

a13

a14

a15

a16

a17

a21

a22

a23

a24

a25

a26

0.34

−0.46

0.45

−0.26

0.42

0.33

−0.37

0.36

0.35

−0.39

0.38

−0.34

0.35

a27

a31

a32

a33

a34

a35

a36

a37

a41

a42

a43

a44

a45

0.37

0.42

−0.38

0.46

−0.45

0.44

0.43

0.47

0.53

−0.58

0.54

−0.66

0.55

a46

a47

a51

a52

a53

a54

a55

a56

a57

a61

a62

a63

a64

−0.54

0.56

0.76

0.72

0.75

−0.76

0.72

0.79

−0.68

0.65

0.63

0.66

−0.65

a65

a66

a67

a71

a72

a73

a74

a75

a76

a77

0.64

0.57

−0.68

0.65

0.61

0.68

−0.62

0.67

0.69

0.63

p31

p51

p71

p91

p111

p131

p13

p53

p73

p93

p113

p133

0.70

0.78

−0.76

0.73

0.74

0.68

0.36

−0.32

0.30

−0.40

0.36

−0.38

p15

p35

p75

p95

p115

p135

p17

p37

p57

p97

p117

p137

0.26

−0.52

−0.46

0.43

0.41

0.44

0.72

−0.64

0.63

0.65

0.64

0.53

p19

p39

p59

p79

p119

p139

p111

p311

p5,11

p7,11

p9,11

p13,11

0.55

−0.48

0.42

0.48

0.47

0.40

0.54

−0.58

0.46

0.52

−0.67

0.54

p1,13

p3,13

p5,13

p7,13

p9,13

p11,13

0.47

0.58

0.36

0.45

−0.54

0.41

.

The characteristic values of matrices $A$ and $B$ in System (28) are $0.0140\pm 1.7748i,$ $0.0140\pm 1.7719i,$ $0.0140\pm 1.7832i,$ $0.0145\pm 1.7776i,$ $0.0145\pm 1.7725i,$ $0.0145\pm 1.7714i,$ $0.1450\pm 1.7604i,$ and $0.0016,-0.1385,0.2426,$ $0.2438\pm 0.6337i,$ $1.3726\pm 0.7414i, 0, 0, 0, 0, 0, 0, 0,$ respectively. The time delays are selected as ${\tau }_1=$0.75, ${\tau }_2=0$.78, ${\tau }_3=0.80,$ ${\tau }_4=0.76,$ ${\tau }_5=0.74,$ ${\tau }_6=0.77,$ ${\tau }_7=0.72,$ and ${\tau }_1=0.85,$ ${\tau }_2=0.88,{\tau }_3=0.90, {\tau }_4=0.86, {\tau }_5=0.84, {\tau }_6=0.87, {\tau }_7=0.82,$ respectively. It is easy to see that the conditions of Theorem 1 are satisfied. There exists an oscillatory solution to System (28) (see Figure 5 and Figure 6). The curves for odd subscripts are even and the curves for even subscripts are uneven. In Figure 5, the amplitude of ${ x}_1\left(t\right)$ is just 3.3591, but that for $x_8\left(t\right)$ is 12.6238. Figure 5 and Figure 6 indicate that the time delay affects the oscillation frequency. A change in the time delays will cause a change in the oscillatory frequency.

Finally, we considered $n=8$ in System (6) as the following:

$$\left\{\begin{array}{c}{x}_1^{\prime }=x_2,\\ x_2^{\prime }={\mu }_1\left(1-x_1^2+{\alpha }_1 x_1^4-{\beta }_1 x_1^6\right)x_2-{\gamma }_1{x}_1+k_1 x_1^3+a_{11}{x}_2\left(t-{\tau }_1\right)\\ \begin{array}{c}+\cdots +a_{18}{x}_{16}\left(t-{\tau }_8\right)+{\displaystyle \sum _{k=2}^8}{p}_{2k-\mathrm{1,1}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_1\left(t-{\tau }_1\right)\right],\\ x_3^{\prime }=x_4,\\ \begin{array}{c}{x}_4^{\prime }={\mu }_2\left(1-x_3^2+{\alpha }_2 x_3^4-{\beta }_2 x_3^6\right)x_4-{\gamma }_2{x}_3+k_2 x_3^3+a_{22}{x}_4\left(t-{\tau }_2\right)\\ +a_{23}{x}_6\left(t-{\tau }_3\right)+\cdots +a_{28}{x}_{16}\left(t-{\tau }_8\right)+a_{21}{x}_2\left(t-{\tau }_1\right)\\ \begin{array}{c}+{\displaystyle \sum _{k=1, k\ne 2}^8}{p}_{2k-\mathrm{1,3}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_3\left(t-{\tau }_2\right)\right],\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \begin{array}{c}{x}_{15}^{\prime }=x_{16},\\ x_{16}^{\prime }={\mu }_8\left(1-x_{15}^2+{\alpha }_8 x_{15}^4-{\beta }_8 x_{15}^6\right)x_{16}-{\gamma }_8{x}_{15}+k_8 x_{15}^3+a_{88}{x}_{16}\left(t-{\tau }_8\right)\\ \begin{array}{c}+a_{81}{x}_2\left(t-{\tau }_1\right)+a_{82}{x}_4\left(t-{\tau }_2\right)+\cdots +a_{87}{x}_{14}\left(t-{\tau }_7\right)\\ +{\displaystyle \sum _{k=1}^7}{p}_{2k-\mathrm{1,15}}\left[x_{2k-1}\left(t-{\tau }_k\right)-x_{15}\left(t-{\tau }_8\right)\right],\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\right.$$

(29)

From

Table 4. The parameter values in System (29).

μ1

μ2

μ3

μ4

μ5

μ6

μ7

μ8

0.0015

0.0012

0.0013

0.0016

0.0018

0.0014

0.0019

0.0017

α1

α2

α3

α4

α5

α6

α7

α8

0.55

0.54

0.52

0.58

0.56

0.53

0.60

0.62

β1

β2

β3

β4

β5

β6

β7

β8

0.95

0.98

0.92

0.94

0.96

0.93

0.97

0.91

γ1

γ2

γ3

γ4

γ5

γ6

γ7

γ8

2.55

2.56

2.52

2.58

2.50

2.54

2.57

2.60

k1

k2

k3

k4

k5

k6

k7

k8

0.006

0.003

0.007

0.004

0.008

0.009

0.004

0.005

a11

a12

a13

a14

a15

a16

a17

a18

a21

a22

a23

a24

a25

0.41

0.48

0.45

0.46

0.44

0.43

0.47

0.50

0.46

−0.58

0.40

0.48

0.55

a26

a27

a28

a31

a32

a33

a34

a35

a36

a37

a38

a41

a42

−0.65

0.57

0.67

0.72

−0.78

0.76

−0.76

0.74

−0.72

0.78

0.75

0.62

−0.72

a43

a44

a45

a46

a47

a48

a51

a52

a53

a54

a55

a56

a57

0.34

−0.46

−0.55

−0.73

0.77

0.42

0.65

0.62

0.81

−0.56

0.82

0.47

−0.65

a58

a61

a62

a63

a64

a65

a66

a67

a68

a71

a72

a73

a74

0.54

0.55

0.52

0.58

−0.54

0.61

0.57

−0.86

0.63

0.45

0.42

0.48

−0.46

a75

a76

a77

a78

a81

a82

a83

a84

a86

a87

a88

0.52

0.47

0.57

0.50

0.65

0.64

0.61

−0.35

0.68

0.65

−0.54

p31

p51

p71

p91

p111

p131

p151

p13

p53

p73

p93

p113

p133

0.76

−0.78

0.82

0.80

0.81

0.75

0.71

0.45

0.32

0.41

−0.54

−0.54

0.46

p153

p15

p35

p75

p95

p115

p135

p155

p17

p37

p57

p97

p117

0.55

0.54

−0.65

0.48

0.52

0.57

−0.58

0.54

−0.72

0.78

−0.68

−0.76

0.72

p137

p157

p19

p39

p59

p79

p119

p139

p159

p11

p311

p511

p711

−0.46

0.38

0.62

−0.45

0.48

0.42

0.56

0.51

0.36

0.35

−0.48

0.58

0.54

p911

p1311

p1511

p1,13

p3,13

p5,13

p7,13

p9,13

p11,13

p15,13

p1,15

p3,15

p5,15

−0.68

0.44

0.48

0.55

0.42

−0.44

0.46

−0.48

0.50

0.50

0.55

−0.52

0.56

p7,15

p9,15

p11,15

p13,15

0.54

−0.44

0.48

0.40

, $s=$1.0019 and $‖B‖=$ 7.65. The time delays are selected as ${\tau }_1=$0.38, ${\tau }_2=0$.40, ${\tau }_3=0.42,$ ${\tau }_4=0.43,$ ${\tau }_5=0.36,$ ${\tau }_6=0.41,$ ${\tau }_8=$0.39, Thus, ${\tau }_{*}=0.35,$ $7s‖B‖{\tau }_{*}^2=7\times 1.0018\times 7.65\times 0.35\times 0.35=6.5723>5.5801=4e^{s{\tau }_{*}}.$ The conditions of Theorem 2 are satisfied. There exists an oscillatory solution to System (29) (see Figure 7 and Figure 8). Figure 7 and Figure 8 indicate that the values of ${\gamma }_i$ $\left(i=\mathrm{1,2},\cdots ,8\right)$ affect the oscillation frequency. The parameters in the two figures are almost the same. In Figure 8, only the parameters ${\gamma }_i$ are less than the values of ${\gamma }_i$ $\left(i=\mathrm{1,2},\cdots ,8\right)$ in Figure 7. The oscillation frequencies are different from each other.

The values of ${\gamma }_i \mathrm{were} \mathrm{selected} \mathrm{as} 2.55,2.56,2.52,2.58,2.50,2.54,2.57, \mathrm{and} 2.60.$

The values of ${\gamma }_i \mathrm{were} \mathrm{selected} \mathrm{as} 1.55,1.56,1.52,1.58,1.50,1.54,1.57, \mathrm{and} 1.60.$

5. Discussion

The present criteria only determine the existence of oscillatory solutions. They cannot provide the bifurcation values about the delays. It is pointed out that it is difficult to deal with the present models using the bifurcation method. Even if for $n=5,$ the bifurcation equation will be the following:

$$p_0^{10}\left(\lambda \right)+{\displaystyle \sum _{i=1}^5}{p}_i^9\left(\lambda \right)e^{-\lambda {\tau }_i}+\sum _{i\ne j}{p}_i^8\left(\lambda \right)e^{-\lambda ({\tau }_i+{\tau }_j)}+\cdots +p^0{e}^{-\lambda \left({\tau }_1+\cdots +{\tau }_5\right)}=0,$$

(30)

where $p_0^{10}\left(\lambda \right)$ is a ten-order polynomial,$p_i^9\left(\lambda \right) (i=\mathrm{1,2},\cdots , 5)$ are nine-order polynomials about $\lambda$, and $p^0$ is a constant. If ${\tau }_i (i=\mathrm{1,2},\cdots , 5)$ are different real numbers, then $\lambda {\tau }_i(i=\mathrm{1,2},\cdots , 5)$ are different variables. Equation (30) is a transcendental equation with five different variables. Theoretically, one can use Ruan’s method to find the bifurcation values [29]. However, finding the bifurcating values from Equation (30) is not easy work.

6. Conclusions

Gap differential equations of types related to the van der Pol and van der Pol–Duffing equations are difficult to study directly from the bifurcation conditions when the system involves many gaps. In this paper, it was shown that, alternatively, a study of these systems can be effective with a method of reduction to first-order, linearization, and the determination of the instability of the trivial solution, together with boundedness. This is shown by the periodic appearance of the graphs for simulations with randomly chosen parameters. Two theorems are proved in this paper that provide sufficient conditions.