Bifurcations of a Fractional-Order Four-Neuron Recurrent Neural Network with Multiple Delays

Abstract

This paper investigates the bifurcation issue of fractional-order four-neuron recurrent neural network with multiple delays. First, the stability and Hopf bifurcation of the system are studied by analyzing the associated characteristic equations. It is shown that the dynamics of delayed fractional-order neural networks not only depend heavily on the communication delay but also significantly affects the applications with different delays. Second, we numerically demonstrate the effect of the order on the Hopf bifurcation. Two numerical examples illustrate the validity of the theoretical results at the end.

1. Introduction

Recurrent neural network (RNN) is a type of recursive neural network that takes sequence data as input, recurses in the evolution direction of the sequence, and all nodes (recurrent units) are connected in a chain. Till now, several recurrent neural networks (RNNs) have been widely considered in various fields such as signal processing, optimizations control, image processing, robotics, pattern recognitions, and automatic control, so they have attracted extensive attention of researchers in recent years [1–7]. Since the applications of RNNs depend more heavily on dynamical neural networks, quite a few efforts have been undertaken to study their dynamical properties and a large number of useful results have been investigated, including oscillation, stability, bifurcation, synchronization, and chaos of various RNNs [8–14].

As the matter of fact, for some applications of nonlinear dynamical models, time delay has a significant impact, and in addition to affecting stability, it causes oscillations and other unstable phenomena, such as chaos [15]. Communication delays and the response times of neurons are considered key factors in the performance of neural networks, and this is caused by the finite switching speed of amplifiers and the noninstantaneous signal transmission between neurons [16]. In recently years, many scholars have been interested in studying the dynamics of neural networks with such time delays [17–19]. It must be pointed out that exponential stabilization of memristor-based RNNs with disturbance and mixed time delays by periodically intermittent control has been considered by Wang et al. [20]. Using the appropriate Lyapunov–Krasovski functionals and applying matrix inequality approach methods, Zhou [21] discussed the passivity of a class of recurrent neural networks with impulse and multiproportional delays. Zhou and Zhao [22] investigated the exponential synchronization and polynomial synchronization of recurrent neural networks with and without proportional delays. Robust stability analysis of recurrent neural networks is studied in Refs. [23, 24]. Furthermore, time delays are ubiquitous and unavoidable in the real world. Due to the existence of delays, the system can become unstable, and the dynamic behavior of nonlinear systems becomes more difficult. Moreover, since the solution space of the delay dynamical is infinite, it makes the systems more complex and bifurcation occurs. Hence, it is necessary to consider the properties and dynamics of neural networks via delays, such as time delay [25, 26], multiple delays [27, 28] time-varying delays [29, 30], and so on. In 2013, Zhang and Yang [31] studied a four-neuron recurrent neural network with multiple delays, described as follows:

$$\begin{array}{c}\left\{\begin{array}{c}\stackrel{.}{x_1}\left(t\right)=-x_1\left(t\right)+f\left(x_2\left(t-{\tau }_1\right)\right)\text{,}\\ \stackrel{.}{x_2}\left(t\right)=-x_2\left(t\right)+f\left(x_3\left(t-{\tau }_1\right)\right)\text{,}\\ \stackrel{.}{x_3}\left(t\right)=-x_3\left(t\right)+f\left(x_4\left(t-{\tau }_1\right)\right)\text{,}\\ \stackrel{.}{x_4}\left(t\right)=-x_4\left(t\right)+{\omega }_{1}f\left(x_1\left(t-{\tau }_2\right)\right)+{\omega }_{2}f\left(x_2\left(t-{\tau }_2\right)\right)+{\omega }_{3}f\left(x_3\left(t-{\tau }_2\right)\right)\text{,}\end{array}\right.\end{array}$$ (1)

C5(α₁β₁+α₂β₂)/(α₁²+α₂²) ≠ 0where x_i(t)(i=1,2,3,4) stand for state of the ith neuron at time t, ω_k ∈ R(k=1,2,3) are the network parameters or weight, f(·) is the connection function between neurons, and τ_j ≥ 0(j=1,2) are the communication time delay. By using the distribution of the solutions of the associated characteristic equation, the Hopf bifurcation and local stability of the four-dimensional RNNs with two delays are studied. For more recurrent neural network research results, see references [5, 10, 12, 20].

In more than three centuries, fractional calculus has developed into a classical mathematical concept. Nonlinear dynamics systems have shown that it has an exceptionally important role in generalizing ordinary differentiation and integration to arbitrary noninteger order. Therefore, if we study the effects of the memory and genetics factors, fractional neural network is sometimes more realistic and more general than integer neural networks. In recent years, the application of fractional order neural networks has developed rapidly, and the complex dynamical behaviors of fractional neural networks has become a very important research hot points, such as stability or multistability, Hopf bifurcation, synchronization, chaos, and so on. For instance, in Ref. [32], the multistability of a fractional-order competitive neural networks with delay is investigated by using the fractional calculus and partitioning of state space. In Lu and Xue [33] study, adaptive synchronization is investigated for fractional delayed stochastic neural networks. Yuan and Huang [34] considered the quantitative analysis of fractional-order neural networks with time delay. Udhayakumar and Rajan [35] discussed Hopf bifurcation of a delayed fractional-order octonion-valued neural networks.

We also know that Hopf bifurcations, which include subcritical and supercritical ones, can be used to efficiently design biochemical oscillators. Furthermore, fractional order neural networks with same delay cannot accurately describe the dynamical properties of real world neural networks compared with the ones with different delays. In recent years, some researchers have considered the dynamical behavior of fractional models with time delay [36–48]. In 2019 [49], we also investigated the existence of Hopf bifurcation for four-neuron fractional neural networks with leakage delays. To the best of our knowledge, so far there are few results on the Hopf bifurcation of four-dimensional fractional-order recurrent neural network with multiple delays are reported, and therefore, the study of Hopf bifurcation of fractional-order dynamical systems with multiple delays remains an open problem.

Based on the above motivations, we are dedicated to presenting a theoretical exploration of stability and Hopf bifurcation for a four-neuron fractional-order recurrent neural network with multiple delays in this work. The main contributions can be highlighted as follows:

1. A novel delayed fractional-order recurrent neural network with four-neuron and two different delays is studied

2. Double main dynamical properties of the fractional-order recurrent neural network with two delays are investigated: stability and oscillation

3. The Hopf bifurcation is discussed in terms of delays and order

In the article, we shall give some some lemmas and definitions of fractional-order calculus in Section 2, and models description in Section 3. In Section 4, the local stability of the trivial steady state of delayed fractional-order RNNs is examined by applying the associated characteristic equation. In addition, the authors will care about the Hopf bifurcation of fractional-order RNNs with multiple delays. In Section 5, two numerical examples are provided to demonstrate the theoretical results. The last section gives some conclusions.

2. Preliminaries

This section we will give some Caputo definitions and lemma for fractional calculus as a basis for the theoretical analysis and simulation proofs.

Definition 1 . —

(see [50]). The fractional integral of order ϕ for a function f(x) is defined as follows:

$$\begin{array}{c}{I}^{\varphi }f\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(\varphi \right)}{\int }_{x_0}^x{\left(x-s\right)}^{\varphi -1}f\left(s\right)d\mathrm{s}\text{,}\end{array}$$ (2)

where ϕ > 0, and Γ(·) is the Gamma function satisfying Γ(s)=∫₀^∞x^s−1e^−xdx.

Definition 2 . —

(see [50]). Caputo fractional derivative of order ϕ for a function ψ(x) ∈ C^k[x₀, ∞), R) is defined by

$$\begin{array}{c}{D}^{\varphi }\psi \left(x\right)=\frac{1}{\mathrm{\Gamma }\left(n-\varphi \right)}{\int }_{x_0}^x\frac{{\psi }^{\left(k\right)}\left(s\right)}{{\left(x-s\right)}^{\varphi -k+1}}d\mathrm{s}\text{,}\end{array}$$ (3)

where x ≥ x₀ and k − 1 ≤ ϕ < k, k ∈ N⁺.

Moreover, when ϕ ∈ (0,1), then

$$\begin{array}{c}{D}^{\varphi }\psi \left(x\right)=\frac{1}{\mathrm{\Gamma }\left(1-\varphi \right)}{\int }_{x_0}^x\frac{{\psi }^{\prime }\left(s\right)}{{\left(x-s\right)}^{\varphi }}d\mathrm{s}\text{.}\end{array}$$ (4)

Lemma 1 (see [51]). —

Consider the following fractional order autonomous model.

$$\begin{array}{c}{D}^{\varphi }u=Ju,u\left(0\right)=u_0\text{,}\end{array}$$ (5)

in which 0 < ϕ ≤ 1, u ∈ R^k, and J ∈ R^k×k. Then the zero solution of the system (5) is asymptotically stable in the Lyapunov sense if all roots λ_i are the system (5) of character equation satisfy |arg(λ_i)| > ϕπ/2(i=1,2,…, k), and then each component of the states decays towards 0 like t^−ϕ. In addition, this model is stable if and only if |arg(λ_i)| ≥ ϕπ/2 and those critical eigenvalues that satisfy |arg(λ_i)|=ϕπ/2 have geometric multiplicity one.

3. Mathematics Model Elaboration

This article considers the following four-neuron fractional-order recurrent neural network with two delays:

$$\begin{array}{c}\left\{\begin{array}{c}{D}^{\varphi }{x}_1\left(t\right)=-x_1\left(t\right)+f\left(x_2\left(t-{\tau }_1\right)\right)\text{,}\\ D^{\varphi }{x}_2\left(t\right)=-x_2\left(t\right)+f\left(x_3\left(t-{\tau }_1\right)\right)\text{,}\\ D^{\varphi }{x}_3\left(t\right)=-x_3\left(t\right)+f\left(x_4\left(t-{\tau }_1\right)\right)\text{,}\\ D^{\varphi }{x}_4\left(t\right)=-x_4\left(t\right)+{\omega }_{1}f\left(x_1\left(t-{\tau }_2\right)\right)+{\omega }_{2}f\left(x_2\left(t-{\tau }_2\right)\right)+{\omega }_{3}f\left(x_3\left(t-{\tau }_2\right)\right)\text{,}\end{array}\right.\end{array}$$ (6)

where ϕ ∈ (0,1] are fractional order; x_i(t)(i = 1,2,3,4) stand for state variables; ω_i(i = 1,2,3) denote the connection weights; the function of connecting neurons is denoted by f(x(·)); and τ₁ and τ₂ are the communication time delays.

Remark 1 . —

In fact, if ϕ=1, the fractional delayed neural networks (6) changes into the general neural network (1).

Accordingly, the main purpose of this article is to investigate the stability and the application of Hopf bifurcations of the neural networks (6) taking different time delays τ₁ and τ₂ as the bifurcation parameters by the method of stability analysis [52]. In addition, the effects of the order on the creation of the Hopf bifurcation for the proposed fractional order neural network with multiple delays are also numerically discussed.

Throughout of this paper, assume that the following condition holds true:

(C1)f(·) ∈ C(R, R), f(0)=0, xf(x) > 0, for x ≠ 0.

4. Main Results

This section chooses τ₁ or τ₂ as a bifurcation parameter to study the stability analysis and Hopf bifurcation for the fractional order RNNs (6) and to study the bifurcation points accurately.

4.1. Bifurcation Depending on τ₁ in Equation (6)

In this subsection, we first study the effects of τ₁ on bifurcations of system (6) by establishing τ₂.

Applying Taylor series formula, the following form of equation (6) at the origin is

$$\begin{array}{c}\left\{\begin{array}{c}{D}^{\varphi }{x}_1\left(t\right)=-x_1\left(t\right)+m_1{x}_2\left(t-{\tau }_1\right)\text{,}\\ D^{\varphi }{x}_2\left(t\right)=-x_2\left(t\right)+m_2{x}_3\left(t-{\tau }_1\right)\text{,}\\ D^{\varphi }{x}_3\left(t\right)=-x_3\left(t\right)+m_3{x}_4\left(t-{\tau }_1\right)\text{,}\\ D^{\varphi }{x}_4\left(t\right)=-x_4\left(t\right)+m_4{x}_1\left(t-{\tau }_2\right)+m_5{x}_2\left(t-{\tau }_2\right)+m_6{x}_3\left(t-{\tau }_2\right)\text{.}\end{array}\right.\end{array}$$ (7)

By applying Laplace transformation, its characteristic equation is given as

$$\begin{array}{c}\mathrm{d}\mathrm{e}\mathrm{t}\left(\begin{array}{cccc}{s}^{\varphi }+1& -m_1{e}^{s{\tau }_1}& 0& 0\\ 0& s^{\varphi }+1& -m_2{e}^{-s{\tau }_1}& 0\\ 0& 0& s^{\varphi }+1& -m_3{e}^{-s{\tau }_1}\\ -m_4{e}^{-s{\tau }_2}& -m_5{e}^{-s{\tau }_2}& -m_6{e}^{-s{\tau }_2}& s^{\varphi }+1\end{array}\right)=0\text{,}\end{array}$$ (8)

where m_k=f′(0)(k=1,2,3), m_k=ω_jf′(0)(j=1,2,3, k=4,5,6).

From (8), we have

$$\begin{array}{c}{K}_1\left(s\right)+K_2\left(s\right)e^{-s{\tau }_1}+K_3\left(s\right)e^{-2s{\tau }_1}+K_4\left(s\right)e^{-3s{\tau }_1}=0\text{,}\end{array}$$ (9)

where

$$\begin{array}{c}{K}_1\left(s\right)=s^{4\varphi }+4s^{3\varphi }+6s^{2\varphi }+4s^{\varphi }+1\text{,}\\ K_2\left(s\right)=-m_3{m}_6\left(s^{2\varphi }+2s^{\varphi }+1\right)e^{-s{\tau }_2}\text{,}\\ K_3\left(s\right)=-m_2{m}_3{m}_5\left(s^{\varphi }+1\right)e^{-s{\tau }_2}\text{,}\\ K_4\left(s\right)=-m_1{m}_2{m}_3{m}_4{e}^{-s{\tau }_2}\text{.}\end{array}$$ (10)

Multiplying e^sτ₁ and e^2sτ₁ on both sides of equation (9), respectively, we can obtain

$$\begin{array}{c}\left\{\begin{array}{cc}{K}_1\left(s\right)e^{2s{\tau }_1}& +K_2\left(s\right)e^{s{\tau }_1}+K_3\left(s\right)+K_4\left(s\right)e^{-s{\tau }_1}=0\text{,}\\ K_1\left(s\right)e^{s{\tau }_1}& +K_2\left(s\right)+K_3\left(s\right)e^{-s{\tau }_1}+K_4\left(s\right)e^{-2s{\tau }_1}=0\text{.}\end{array}\right.\end{array}$$ (11)

Let K₁(s)=A₁+iB₁, K₂(s)=A₂+iB₂, K₃(s)=A₃+iB₃, K₄(s)=A₄+iB₄, and from equation (9), we have

$$\begin{array}{c}\left\{\begin{array}{c}\left(A_1+i{B}_1\right)e^{2s{\tau }_1}+\left(A_2+i{B}_2\right)e^{s{\tau }_1}+\left(A_3+i{B}_3\right)+\left(A_4+i{B}_4\right)e^{-s{\tau }_1}=0\text{,}\\ \left(A_1+i{B}_1\right)e^{s{\tau }_1}+\left(A_2+i{B}_2\right)+\left(A_3+i{B}_3\right)e^{-s{\tau }_1}+\left(A_4+i{B}_4\right)e^{-2s{\tau }_1}=0\text{.}\end{array}\right.\end{array}$$ (12)

Take s=iw=w(cosπ/2+i sinπ/2)(ω > 0) be a purely imaginary root of equation (11). Apply inserting s into equation (11) and separating the imaginary and real parts yields the following equations:

$$\begin{array}{c}\left\{\begin{array}{c}{A}_1\cos \left(2\omega {\tau }_1\right)-B_1\sin \left(2\omega {\tau }_1\right)+\left(A_2+A_4\right)\cos \left(\omega {\tau }_1\right)+\left(B_4-B_2\right)\sin \left(\omega {\tau }_1\right)=-A_3\text{,}\\ B_1\cos \left(2\omega {\tau }_1\right)+A_1\sin \left(2\omega {\tau }_1\right)+\left(B_2+B_4\right)\cos \left(\omega {\tau }_1\right)+\left(A_2-A_4\right)\sin \left(\omega {\tau }_1\right)=-B_3\text{,}\\ A_1\cos \left(2\omega {\tau }_1\right)-B_1\sin \left(2\omega {\tau }_1\right)+\left(A_2+A_4\right)\cos \left(\omega {\tau }_1\right)+\left(B_4-B_2\right)\sin \left(\omega {\tau }_1\right)=-A_3\text{,}\\ B_1\cos \left(2\omega {\tau }_1\right)+A_1\sin \left(2\omega {\tau }_1\right)+\left(B_2+B_4\right)\cos \left(\omega {\tau }_1\right)+\left(A_2-A_4\right)\sin \left(\omega {\tau }_1\right)=-B_3\text{.}\end{array}\right.\end{array}$$ (13)

Evidently,

$$\begin{array}{c}\left\{\begin{array}{c}\cos \left(\omega {\tau }_1\right)=\frac{F_{12}\left(\omega \right)}{F_{11}\left(\omega \right)}=F_{c1}\left(\omega \right)\text{,}\\ \sin \left(\omega {\tau }_1\right)=\frac{F_{22}\left(\omega \right)}{F_{21}\left(\omega \right)}=F_{s1}\left(\omega \right)\text{,}\end{array}\right.\end{array}$$ (14)

where A₁, A₂, A₃, A₄, B₁, B₂, B₃, B₄, F₁₁, F₁₂, F_21, and F₂₂ are given Appendix A. Obviously, from fist to second equation of system (14), it can be implied that

$$\begin{array}{c}{F}_{c1}^2\left(\omega \right)+F_{s1}^2\left(\omega \right)=1\text{.}\end{array}$$ (15)

From equation (13), one can obtain

$$\begin{array}{c}{\tau }_1^{\left(l\right)}=\frac{1}{w}\left[\arccos \frac{F_{12}\left(w\right)}{F_{11}\left(w\right)}+2l\pi \right],l=\mathrm{0,1,2},\dots \text{.}\end{array}$$ (16)

Remark 2 . —

This is an inhomogeneous system of linear equations (13), and the independent variables are cos (2ωτ₁), sin (2ωτ₁), cos (ωτ₁), sin (ωτ₁), respectively. According Cramer's rule of linear equation, if the coefficient determinant of the system of linear equations is not equal to 0, we can easily solve solutions of linear equations (13). That is, we can obtain cos (ωτ₁) and sin (ωτ₁) or cos (ω2τ₁) and sin (2ωτ₁).

Define the bifurcation point of fractional neural network with multiple delays (6) as

$$\begin{array}{c}{\tau }_{10}^{\ast }=\min \left\{{\tau }_1^{\left(l\right)}\right\},l=\mathrm{0,1,2},\dots \text{.}\end{array}$$ (17)

If τ₁ vanishes, then equation (9) becomes

$$\begin{array}{c}{H}_1\left(s\right)+H_2\left(s\right)e^{-s{\tau }_2}=0\text{,}\end{array}$$ (18)

where

$$\begin{array}{c}{H}_1\left(s\right)=s^{\mathit{4}\varphi }+{\mathit{4}s}^{\mathit{3}\varphi }+{\mathit{6}s}^{\mathit{2}\varphi }+{\mathit{4}s}^{\varphi }+1\text{,}\\ H_2\left(s\right)={{-m}_{\mathit{3}}{m}_{\mathit{6}}s}^{\mathit{2}\varphi }-{{\mathit{2}m}_{\mathit{3}}{m}_{\mathit{6}}s}^{\varphi }-m_{\mathit{3}}{m}_{\mathit{6}}-{m_{\mathit{2}}{m}_{\mathit{3}}{m}_{\mathit{5}}s}^{\varphi }-m_{\mathit{2}}{m}_{\mathit{3}}{m}_{\mathit{5}}-m_{\mathit{1}}{m}_{\mathit{2}}{m}_{\mathit{3}}{m}_{\mathit{4}}\text{.}\end{array}$$ (19)

If τ₂=0, then the equation (18) becomes

$$\begin{array}{c}0=s^{4\varphi }+4s^{3\varphi }+6s^{2\varphi }+4s^{\varphi }+1-m_3{m}_6{s}^{2\varphi }-2m_3{m}_6{s}^{\varphi }-m_3{m}_6\\ -m_2{m}_3{m}_5{s}^{\varphi }-m_2{m}_3{m}_5-m_1{m}_2{m}_3{m}_4\text{.}\end{array}$$ (20)

Suppose that all roots s of the equation (18) obey Lemma 1, then we get that both roots λ_i in equation (18) have negative real parts.

The imaginary and real parts of H_j(s)(j=1,2) can be denoted by H_j^I an d H_j^R, respectively. Multiplying e^2sτ₂ on both sides of equation (18), we can obtain

$$\begin{array}{c}{H}_1\left(s\right)e^{s{\tau }_2}+H_2\left(s\right)=0\text{.}\end{array}$$ (21)

Also, let s=iv=v(cosπ/2+i sinπ/2)(v > 0) be a purely imaginary root of equation (11) if and only if

$$\begin{array}{c}\left\{\begin{array}{c}{H}_1^R\cos \left(v{\tau }_2\right)-H_1^I\sin \left(v{\tau }_2\right)=-H_2^R\text{,}\\ H_1^I\cos \left(v{\tau }_2\right)+H_1^R\sin \left(v{\tau }_2\right)=-H_2^I\text{.}\end{array}\right.\end{array}$$ (22)

This leads to form

$$\begin{array}{c}\left\{\begin{array}{c}\cos \left(v{\tau }_2\right)=-\frac{H_2^R{H}_2^R+H_1^I{H}_2^I}{{H_1^R}^2+{H_1^I}^2}=f_{c1}\left(v\right)\text{,}\\ \sin \left(v{\tau }_1\right)=-\frac{-H_2^R{H}_1^I+H_1^R{H}_2^I}{{H_1^R}^2+{H_1^R}^2}=f_{s1}\left(v\right)\text{.}\end{array}\right.\end{array}$$ (23)

It is not difficult to see that

$$\begin{array}{c}{f}_{c1}^2\left(w\right)+f_{s1}^2\left(w\right)=1\text{.}\end{array}$$ (24)

Additionally, we will give the following assumptions which hold true.

(C2) The equation (24) has at least a positive real root.

From equation (24), the values ofv can be obtained according to Mathematics software Mathematica 10.0, and then the Hopf bifurcation point τ₂₀ of fractional order recurrent neural network (6) with τ₁ = 0 can be derived.To demonstrate our main results, we further present the following hypothesis: (C3)Υ₁Ω₁ + Υ₂Ω₂/Ω₁² + Ω₂² ≠ 0, where

$$\begin{array}{c}{{\rm Y}}_1=w_0\left[\begin{array}{c}{A}_2\sin w_0{\tau }_{10}-B_2\cos w_0{\tau }_{10}+2\left(A_3\sin 2w_0{\tau }_{10}-B_2\cos 2w_0{\tau }_{10}\right)\\ \left.+3\left(A_4\cos 3w_0{\tau }_{10}+B_4\sin 3w_0{\tau }_{10}\right.\right],\end{array}\right.\\ {{\rm Y}}_2=w_0\left[\begin{array}{c}{A}_2\cos w_0{\tau }_{10}+B_2\sin w_0{\tau }_{10}+2\left(A_3\cos 2w_0{\tau }_{10}+B_2\sin 2w_0{\tau }_{10}\right)\\ \left.+3\left(A_4\cos 3w_0{\tau }_{10}+B_4\sin 3w_0{\tau }_{10}\right.\right],\end{array}\right.\\ {\Omega }_1=A_1^{\prime }+\left[A_2^{\prime }-{\tau }_1{A}_2\right]\cos w_0{\tau }_{10}+\left[B_2^{\prime }-{\tau }_1{B}_2\right]\sin w_0{\tau }_{10}+\left[A_3^{\prime }-2{\tau }_1{A}_3\right]\cos 2w_0{\tau }_{10}+\left[B_2^{\prime }-{\tau }_1{B}_3\right]\sin 2w_0{\tau }_{10}+\left[A_4^{\prime }-3{\tau }_1{A}_4\right]\cos 3w_0{\tau }_{10}+\left[B_4^{\prime }-3{\tau }_1{B}_4\right]\sin 3w_0{\tau }_{10}\text{,}\\ {\Omega }_2=B_1^{\prime }+\left[B_2^{\prime }-{\tau }_1{B}_2\right]\cos w_0{\tau }_{10}-\left[A_2^{\prime }-{\tau }_1{A}_2\right]\sin w_0{\tau }_{10}+\left[B_3^{\prime }-2{\tau }_1{B}_3\right]\cos 2w_0{\tau }_{10}-\left[A_2^{\prime }-{\tau }_1{A}_3\right]\sin 2w_0{\tau }_{10}+\left[B_4^{\prime }-3{\tau }_1{B}_4\right]\cos 3w_0{\tau }_{10}-\left[A_4^{\prime }-3{\tau }_1{A}_4\right]\sin 3w_0{\tau }_{10}\text{.}\end{array}$$ (25)

Lemma 2 . —

Let s(τ₁)=ν(τ₁₀)+iw(τ₁) be a root of equation (9) near τ₁=τ_1j satisfying ν(τ_1j)=0, w(τ_1j)=w₀, then the following transversality condition is satisfied.

$$\begin{array}{c}\text{Re}\left[\frac{ds}{d{\tau }_1}\right]{|}_{\left(w=w_0,{\tau }_1={\tau }_{10}\right)}\ne 0\text{.}\end{array}$$ (26)

Proof —

With implicit function theorem, we can differentiate equation (9) with respect to τ₁, and thus we get

$$\begin{array}{c}0=K_1^{\prime }\left(s\right)\frac{ds}{d{\tau }_1}+K_2^{\prime }\left(s\right)e^{-s{\tau }_1}\frac{ds}{d{\tau }_1}+K_2\left(s\right)e^{-s{\tau }_1}\left(-{\tau }_1\frac{ds}{d{\tau }_1}-s\right)+K_3^{\prime }\left(s\right)e^{-2s{\tau }_1}\frac{ds}{d{\tau }_1}\\ +K_3\left(s\right)e^{-2s{\tau }_1}\left(-2{\tau }_1\frac{ds}{d{\tau }_1}-2s\right)+K_4^{\prime }\left(s\right)e^{-3s{\tau }_1}\frac{ds}{d{\tau }_1}+K_4\left(s\right)e^{-3s\tau -1}\left(-3\tau \frac{ds}{d{\tau }_1}-3s\right)\text{.}\\ \frac{ds}{d{\tau }_1}=\frac{{\rm Y}\left(s\right)}{\Omega \left(s\right)}\text{,}\end{array}$$ (27)

where

$$\begin{array}{c}{\rm Y}\left(s\right)=s\left[K_2\left(s\right)e^{-s{\tau }_1}+2K_3\left(s\right)e^{2-s{\tau }_1}+3K_4\left(s\right)e^{-3s{\tau }_1}\right]\text{,}\\ \Omega \left(s\right)=K_1^{\prime }\left(s\right)+\left[K_2^{\prime }\left(s\right)-{\tau }_1{K}_2\left(s\right)\right]e^{-2s{\tau }_1}+\left[K_3^{\prime }\left(s\right)-2{\tau }_1{K}_3\left(s\right)\right]e^{-2s{\tau }_1}\\ +\left[K_4^{\prime }\left(s\right)-3{\tau }_1{K}_4\left(s\right)\right]e^{-3s{\tau }_1}\text{.}\end{array}$$ (28)

We further suppose that Υ₁ and Υ₂ are the real and imaginary parts of Υ(s), respectively, and Ω₁ and Ω₂ are the real and imaginary parts of Ω(s), respectively, then

$$\begin{array}{c}\text{Re}\left[\frac{ds}{d\tau }\right]{|}_{\left(\tau ={\tau }_0^{\ast },w=w_0^{\ast }\right)}=\frac{{{\rm Y}}_1{\Omega }_1+{{\rm Y}}_2{\Omega }_2}{{\Omega }_1^2+{\Omega }_2^2}\text{.}\end{array}$$ (29)

From (C3), we conclude that the transversality condition holds true. This completes the proof of Lemma 2.

From the above investigation, we can obtain the following results.

Theorem 1 . —

assumptions (C1)–(C3) hold true, then the following results can be given:

1. The zero equilibrium point of fractional order four-neuron recurrent neural network with multiple delays (6) is asymptotically stable when τ₁ ∈ [0, τ₁₀^∗).

2. If τ₁ ∈ [0, τ₁₀^∗), then fractional order four neurons recurrent neural network with multiple delays (6) causes Hopf bifurcation at the origin when τ₁=τ₁₀^∗. That is, a branch of periodic solutions can bifurcate from the zero equilibrium point at τ₁=τ₁₀^∗.

4.2. Bifurcation Depending on τ₂ in Equation (6)

As in the previous subsection, next we change another delay τ₂ to the bifurcation parameter to account for the bifurcation of the model (6). It is hard to point out that equation (8) changes as follows:

$$\begin{array}{c}{q}_1\left(s\right)+q_2\left(s\right)e^{-s{\tau }_2}=0\text{,}\end{array}$$ (30)

where

$$\begin{array}{c}{q}_1\left(s\right)=1+4s^{2\varphi }+6s^{2\varphi }+4s^{3\varphi }+s^{4\varphi }\text{,}\\ q_2\left(s\right)=-m_3{m}_6\left(1+2s^{\varphi }+s^{2\varphi }\right)e^{-s{\tau }_1}-m_2{m}_3{m}_5\left(1+s^{\varphi }\right)e^{-2s{\tau }_1}-m_1{m}_2{m}_3{m}_4{e}^{-3s{\tau }_1}\text{.}\end{array}$$ (31)

Multiplying e^2sτ₂ on both sides of equation (30), we can obtain

$$\begin{array}{c}{q}_1\left(s\right)e^{s{\tau }_2}+q_2\left(s\right)=0\text{.}\end{array}$$ (32)

Suppose q₁(s)=a₁+ib₁ and q₂(s)=a₂+ib₂, and from equation (32), we have

$$\begin{array}{c}\left(a_1+i{b}_1\right)e^{s{\tau }_2}+a_2+i{b}_2=0\text{,}\end{array}$$ (33)

where a₁, a₂, b₁, b₂ are given in Appendix B.

Take $s=i\tilde{w}=\tilde{w}\left(\cos \pi /\mathit{2}+i\sin \pi /\mathit{2}\right)\left(\tilde{\omega }>0\right)$ as a root of equation (33) if and only if

$$\begin{array}{c}\left\{\begin{array}{c}{a}_1\cos \left(\tilde{w}{\tau }_2\right)-b_1\sin \left(\tilde{w}{\tau }_2\right)=-a_2\text{,}\\ b_1\cos \left(\tilde{w}{\tau }_2\right)+a_1\sin \left(\tilde{w}{\tau }_2\right)=-b_2\text{,}\end{array}\right.\end{array}$$ (34)

that is,

$$\begin{array}{c}\left\{\begin{array}{c}\cos \tilde{w}{\tau }_2=-\frac{a_1{a}_2+b_1{b}_2}{a_1^2+b_1^2}=\rho \left(\tilde{w}\right)\text{,}\\ \sin \tilde{w}{\tau }_2=-\frac{-a_2{b}_1+a_1{b}_2}{a_1^2+b_1^2}=\mathrm{\varrho }\left(\tilde{w}\right)\text{.}\end{array}\right.\end{array}$$ (35)

It is simple to derive the following equation.

$$\begin{array}{c}{\rho }^2\left(\tilde{w}\right)+{\mathrm{\varrho }}^2\left(\tilde{w}\right)=1\text{.}\end{array}$$ (36)

From (35), one can obtain

$$\begin{array}{c}{\tau }_2^{\left(l\right)}=\frac{1}{\tilde{w}}\left[\arccos \mathrm{\varrho }\left(\tilde{w}\right)+2l\pi \right],l=\mathrm{0,1,2},\dots \text{.}\end{array}$$ (37)

The bifurcation point is defined by ω_k(k=1,2,3)(C3)(Υ₁Ω₁+Υ₂Ω₂)(Ω₁²+Ω₂²) ≠ 0

$$\begin{array}{c}{\tau }_{20}^{\ast }=\min \left\{{\tilde{\tau }}_2^{\left(l\right)}\right\},l=\mathrm{0,1,2},\dots \text{.}\end{array}$$ (38)

C5(α₁β₁+α₂β₂)/(α₁²+α₂²) ≠ 0 here τ₂^lis defined by equation (38)

If τ₂=0, then the equation (32) becomes

$$\begin{array}{c}{M}_1\left(s\right)+M_2\left(s\right)e^{-s{\tau }_1}+M_3\left(s\right)e^{-2s{\tau }_1}+M_4\left(s\right)e^{-3s{\tau }_1}=0\text{,}\end{array}$$ (39)

where

$$\begin{array}{c}{M}_1\left(s\right)=1+4s^{\varphi }+6s^{2\varphi }+4s^{3\varphi }+s^{4\varphi }\text{,}\\ M_2\left(s\right)=-m_3{m}_6\left(1+2s^{\varphi }+s^{2\varphi }\right)\\ M_3\left(s\right)=-m_2{m}_3{m}_5\left(1+s^{\varphi }\right)\text{,}\\ M_4\left(s\right)=-m_1{m}_2{m}_3{m}_4\text{.}\end{array}$$ (40)

Assume that all roots s of equation (39) observe Lemma 1, then we get that both roots of equation (39) have negative real parts.

The imaginary and real parts of M_i(s)(i = 1,2,3,4) can be expressed as M_i^l an d M_i^R, respectively. Multiplying both sides of the equation (39) by e^2sτ₁ and e^sτ₁ yields

$$\begin{array}{c}\left\{\begin{array}{c}{M}_1\left(s\right)e^{2s{\tau }_1}+M_2\left(s\right)e^{s{\tau }_1}+M_3\left(s\right)+M_4\left(s\right)e^{-s{\tau }_1}=0\text{,}\\ M_1\left(s\right)e^{s{\tau }_1}+M_2\left(s\right)+M_3\left(s\right)e^{-s{\tau }_1}+M_4\left(s\right)e^{-2s{\tau }_1}=0\text{.}\end{array}\right.\end{array}$$ (41)

Let $s=i\tilde{v}=\tilde{v}\left(\cos \pi /\mathit{2}+i\sin \pi /\mathit{2}\right)\left(\tilde{v}>0\right)$ be a solution of equation (41). Substituting s into equation (41) and separating the imaginary and real units yields the following equations:

$$\begin{array}{c}\left\{\begin{array}{c}{M}_1^R\cos \left(2\tilde{v}{\tau }_1\right)-M_1^I\sin \left(2\tilde{v}{\tau }_1\right)+\left(M_2^R+M_4^R\right)\cos \left(\tilde{v}{\tau }_1\right)+\left(M_4^I-P_2^I\right)\sin \left(\tilde{v}{\tau }_1\right)=-M_3^R\text{,}\\ M_1^I\cos \left(2\tilde{v}{\tau }_1\right)+M_1^R\sin \left(2\tilde{v}{\tau }_1\right)+\left(M_2^I+M_4^I\right)\cos \left(\tilde{v}{\tau }_1\right)+\left(M_2^R-P_4^R\right)\sin \left(\tilde{v}{\tau }_1\right)=-M_3^I\text{,}\\ P_1^R\cos \left(2\tilde{v}{\tau }_1\right)-M_1^I\sin \left(2\tilde{v}{\tau }_1\right)+\left(M_2^R+P_4^R\right)\cos \left(\tilde{v}{\tau }_1\right)+\left(M_4^I-P_2^I\right)\sin \left(\tilde{v}{\tau }_1\right)=-M_3^R\text{,}\\ P_1^I\cos \left(2\tilde{v}{\tau }_1\right)+M_1^R\sin \left(2\tilde{v}{\tau }_1\right)+\left(M_2^I+P_4^R\right)\cos \left(\tilde{v}{\tau }_1\right)+\left(M_2^R-P_4^R\right)\sin \left(\tilde{v}{\tau }_1\right)=-M_3^I\text{,}\end{array}\right.\end{array}$$ (42)

which lead to

$$\begin{array}{c}\left\{\begin{array}{c}\cos \tilde{v}{\tau }_1=\frac{E_{12}\left(\tilde{v}\right)}{E_{11}\left(\tilde{v}\right)}=\mathfrak{C}{\left(\tilde{v}\right)}^2\text{,}\\ \sin \tilde{v}{\tau }_1=\frac{E_{22}\left(\tilde{v}\right)}{E_{21}\left(\tilde{v}\right)}=\mathfrak{S}{\left(\tilde{v}\right)}^2\text{.}\end{array}\right.\end{array}$$ (43)

Obviously, from first and second equation of system (43), we get

$$\begin{array}{c}\mathfrak{C}{\left(\tilde{v}\right)}^2+\mathfrak{S}{\left(\tilde{v}\right)}^2=1\text{.}\end{array}$$ (44)

To theoretically gain the sufficient conditions for the Hopf bifurcation, we assume that the following assumptions hold true:

(C4) Equation (36) has at least a positive real root.

By means of equation (36), the values of $\tilde{\omega }$ can be obtained according to mathematical software Mathematica 10.0, and then the bifurcation point τ₁₀ of recullrent fractional four-neuron neural networks (6) with τ₂ = 0 can be derived.As a summary of our main results, we provide the following assumption: (C5)α₁β₁ + α₂β₂/α₁² + α₂² ≠ 0, where

$$\begin{array}{c}{\alpha }_1=a_1^{\prime }+\left(a_2^{\prime }-{\tau }_{20}{a}_2\right)\cos {\tilde{w}}_0{\tau }_{20}+\left(b_2^{\prime }-{\tau }_2{b}_2\right)\sin {\tilde{w}}_0{\tau }_{20}\text{,}\\ {\alpha }_2=b_1^{\prime }+\left(b_2^{\prime }-{\tau }_{20}{b}_2\right)\cos {\tilde{w}}_0{\tau }_{20}-\left(a_2^{\prime }-{\tau }_2{a}_2\right)\sin {\tilde{w}}_0{\tau }_{20}\text{,}\\ {\beta }_1={\tilde{w}}_0\left(a_2\sin {\tilde{w}}_0{\tau }_{20}-b_2\cos {\tilde{w}}_0{\tau }_{20}\right)\text{,}\\ {\beta }_2={\tilde{w}}_0\left(a_2\cos {\tilde{w}}_0{\tau }_{20}+b_2\sin {\tilde{w}}_0{\tau }_{20}\right)\text{.}\end{array}$$ (45)

Lemma 3 . —

Let $\mathrm{s}\left({\tau }_2\right)=\eta \left({\tau }_2\right)+\mathrm{i}\tilde{\mathrm{w}}\left({\tau }_2\right)$ be a root of equation (9) near τ₂=τ_2j satisfying η(τ_2j)=0, $w\left({{\tau }_2}_{\mathrm{j}}\right)={\tilde{w}}_0$, then we get the following transversality condition

$$\begin{array}{c}\text{Re}\left[\frac{ds}{d{\tau }_2}\right]{|}_{\left(\tilde{w}={\tilde{w}}_0,{\tau }_2={\tau }_{20}\right)}\ne 0\text{.}\end{array}$$ (46)

Proof —

Similar to Lemma 2, by utilizing the implicit function theorem and differentiating (9) with respect to τ₂, we get

$$\begin{array}{c}0=q_1^{\prime }\left(s\right)\frac{ds}{d{\tau }_2}+q_2^{\prime }\left(s\right)e^{-s{\tau }_2}\frac{ds}{d{\tau }_2}+q_2\left(s\right)e^{-s{\tau }_2}\left(-{\tau }_2\frac{ds}{d{\tau }_2}-s\right),\frac{ds}{d{\tau }_2}=\frac{\beta \left(s\right)}{\alpha \left(s\right)}\text{,}\end{array}$$ (47)

where

$$\begin{array}{c}\beta \left(s\right)=s{q}_2\left(s\right)e^{-s{\tau }_2}\text{,}\\ \alpha \left(s\right)=q_1^{\prime }\left(s\right)+q_2^{\prime }\left(s\right)e^{-s{\tau }_2}-{\tau }_2{q}_2\left(s\right)e^{-s{\tau }_2}\text{.}\end{array}$$ (48)

We further suppose that α₁ and α₂ are the real and imaginary units of α(s), respectively, and β₁ and β₂ are the real and imaginary parts of β(s), respectively, then we get

$$\begin{array}{c}\text{Re}\left[\frac{ds}{d{\tau }_2}\right]{|}_{\left(\tilde{w}={\tilde{w}}_0,{\tau }_2={\tau }_{20}\right)}=\frac{{\alpha }_1{\beta }_1+{\alpha }_2{\beta }_2}{{\alpha }_1^2+{\alpha }_2^2}\text{.}\end{array}$$ (49)

As a direct consequence of (C5), we can conclude that the transversality condition is satisfied. Then the proof of Lemma 3 is complete.

Based on the above analysis, the following conclusions can be drawn.

Theorem 2 . —

By assuming that assumptions (C1), (C4), and (C5) are valid, the following conditions can be inferred:

1. The zero equilibrium point of fractional order four-neuron recurrent neural network with multiple delays (6) is asymptotically stable when τ₂ ∈ [0, τ₂₀^∗))

2. The fractional order four-neuron recurrent neural network with multiple delays (6) experiences a Hopf bifurcation at its origin when τ₂=τ₂₀^∗; that is, a family of periodic solutions can bifurcate from the zero equilibrium point near τ₂=τ₂₀^∗

5. Numerical Examples

To demonstrate the validity and feasibility of the conclusions reached in this paper, we provide two examples. The simulations were based on a prediction and correction scheme [53] of Adama–Bashforth–Moulton and step-size h=0.01.

5.1. Example 1

Consider the four-neuron fractional recurrent neural networks with multiple delays as

$$\begin{array}{c}\left\{\begin{array}{c}{D}^{\varphi }{x}_1\left(t\right)=-x_1\left(t\right)+f\left(x_2\left(t-{\tau }_1\right)\right)\text{,}\\ D^{\varphi }{x}_2\left(t\right)=-x_2\left(t\right)+f\left(x_3\left(t-{\tau }_1\right)\right)\text{,}\\ D^{\varphi }{x}_3\left(t\right)=-x_3\left(t\right)+f\left(x_4\left(t-{\tau }_1\right)\right)\text{,}\\ D^{\varphi }{x}_4\left(t\right)=-x_4\left(t\right)+{\omega }_{1}f\left(x_1\left(t-{\tau }_2\right)\right)+{\omega }_{2}f\left(x_2\left(t-{\tau }_2\right)\right)+{\omega }_{3}f\left(x_3\left(t-{\tau }_2\right)\right)\text{.}\end{array}\right.\end{array}$$ (50)

Choose parameters ϕ=0.9, ω₁=2, ω₂=ω₃=−2, action function f(·)=tanh(·); therefore, f(0)=tanh(0)=0, f′(0)=1.

Let the initial values be selected as (x₁(0), x₂(0), x₃(0), x₄(0))=(0.15, −0.14,0.1,0.2) for the system (50). First, taking fixed τ₂ such that τ₂=0.6 by complex computing, we get ω₁₀=5.23599, and then τ₁₀=0.312709. Obviously, it is easy to verify that the conditions in Theorem 1 are satisfied. The numerical simulations in Figures 1 and 2 that the zero equilibrium point of system (50) is locally asymptotically stable when τ₁=0.25 < τ₁₀=0.312709. Moreover, Figures 3 and 4 simulates that the zero equilibrium point of system (50) is unstable, and Hopf bifurcation occurs when τ₁=0.35 > τ₁₀=0.312709. The bifurcation diagrams are plotted in Figure 5, which illustrates the theoretical results.

Figure 1.

Time responses of system (50) with ϕ=0.9, τ₁=0.25 < τ₁₀=0.312709.

Figure 2.

Phase diagrams of system (50) with ϕ=0.9, τ₁=0.25 < τ₁₀=0.312709.

Figure 3.

Time responses of system (50) with ϕ=0.9, τ₁=0.36 > τ₁₀=0.312709.

Figure 4.

Phase diagrams of system (50) with ϕ=0.9, τ₁=0.36 > τ₁₀=0.312709.

Figure 5.

Bifurcation diagram of system (50) with ϕ=0.9, τ₁=0.36 > τ₁₀=0.312709.

5.2. Example 2

The same as example 1, let ϕ=0.95, and now we consider the following four-neurons fractional current network with double different delays:

$$\begin{array}{c}\left\{\begin{array}{c}{D}^{0.95}{x}_1\left(t\right)=-x_1\left(t\right)+f\left(x_2\left(t-{\tau }_1\right)\right)\text{,}\\ D^{0.95}{x}_2\left(t\right)=-x_2\left(t\right)+f\left(x_3\left(t-{\tau }_1\right)\right)\text{,}\\ D^{0.95}{x}_3\left(t\right)=-x_3\left(t\right)+f\left(x_4\left(t-{\tau }_1\right)\right)\text{,}\\ D^{0.95}{x}_4\left(t\right)=-x_4\left(t\right)+{\omega }_{1}f\left(x_1\left(t-{\tau }_2\right)\right)+{\omega }_{2}f\left(x_2\left(t-{\tau }_2\right)\right)+{\omega }_{3}f\left(x_3\left(t-{\tau }_2\right)\right)\text{.}\end{array}\right.\end{array}$$ (51)

Taking ω₁ = 1, ω₂ = ω₃ = −1.5, ϕ = 0.95, action function f(·) = tanh(·), then f(0) = tanh(0) = 0, f′(0) = 1, and we first also set τ₁ = 0.8, in the next step, we apply a complex calculation, and it obtains a ${\tilde{\omega }}_{20}=1.02089$ and τ₂₀ = 0.329454. Thus, Theorem 2 yields that the zero solution (0,0,0,0) of the system (51) is locally asymptotically stable when τ₂ = 0.22 < τ₂₀, which is simulated in Figures 6 and 7 which describes the impact of fractional order on τ₂₀. In addition, the zero equilibrium point of the system (51) is unstable, and Hopf bifurcation occurs when τ₂ = 0.38 > τ₂₀, as shown in Figures 8 and 9. Moreover, the bifurcation diagrams are plotted in Figure 10, which illustrates the theoretical results.

Figure 6.

Time responses of system (51) with ϕ=0.95, τ₁=0.22 < τ₂₀=0.329454.

Figure 7.

Phase diagrams of system (51) with ϕ=0.95, τ₁=0.22 < τ₂₀=0.329454.

Figure 8.

Time responses of system (51) with ϕ=0.95, τ₁=0.38 > τ₂₀=0.329454.

Figure 9.

Phase diagrams of system (51) with ϕ=0.95, τ₁=0.38 > τ₂₀=0.329454.

Figure 10.

Bifurcation diagrams of system (51) with ϕ=0.95, τ₁=0.38 > τ₂₀=0.329454.

Remark 3 . —

In fact, in order to better reflect the influence of different time delays at the bifurcation point of the systems (50) and (51), the corresponding bifurcation point τ₁₀ and τ₂₀ and τ₁₀^∗ and τ₂₀^∗ can be determined by changing the order of ϕ. This means that systems (50) and (51) involving different two delays are prone to earlier Hopf bifurcation for some fixed fractional order ϕ.

6. Conclusion

This paper examines the Hopf bifurcation problem of fractional recurrent neural networks with four neurons and two delays. Using time delay as the bifurcation parameter, several criteria are destabilized in order to ensure the Hopf bifurcation for the fractional four-neuron of recurrent neural networks. Based on our analysis, different communication time delays and order effects have quantitatively changed the dynamic behavior of the system (6). These results can contribute to our understanding of delayed fractional recurrent neural networks as a continuation of the previous work. The results of the simulations are illustrated by two numerical examples.

Appendix

A

$$\begin{array}{c}{A}_1=6{\omega }^{2\varphi }\cos \left(\pi \varphi \right)+4{\omega }^{3\varphi }\cos \left(\frac{3\pi \varphi }{2}\right)+{\omega }^{4\varphi }\cos \left(2\pi \varphi \right)+4{\omega }^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)+1\text{,}\\ B_1=6{\omega }^{2\varphi }\sin \left(\pi \varphi \right)+4{\omega }^{3\varphi }\sin \left(\frac{3\pi \varphi }{2}\right)+{\omega }^{4\varphi }\sin \left(2\pi \varphi \right)+4{\omega }^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\text{,}\\ A_2=-m_3{m}_6{\omega }^{2\varphi }\sin \left(\pi \varphi \right)\sin \left(\omega {\tau }_2\right)-m_3{m}_6{\omega }^{2\varphi }\cos \left(\pi \varphi \right)\cos \left(\omega {\tau }_2\right)\\ -2m_3{m}_6{\omega }^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\sin \left(\omega {\tau }_2\right)-2m_3{m}_6{\omega }^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)\cos \left(\omega {\tau }_2\right)-m_3{m}_6\cos \left(\omega {\tau }_2\right)\text{,}\\ B_2=-m_3{m}_6{\omega }^{2\varphi }\sin \left(\pi \varphi \right)\cos \left(\omega {\tau }_2\right)+m_3{m}_6{\omega }^{2\varphi }\cos \left(\pi \varphi \right)\sin \left(\omega {\tau }_2\right)\\ -2m_3{m}_6{\omega }^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\cos \left(\omega {\tau }_2\right)+2m_3{m}_6{\omega }^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)\sin \left(\omega {\tau }_2\right)+m_3{m}_6\sin \left(\omega {\tau }_2\right)\text{,}\\ A_3=-m_2{m}_3{m}_5{\omega }^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\sin \left(\omega {\tau }_2\right)-m_2{m}_3{m}_5{\omega }^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)\cos \left(\omega {\tau }_2\right)-m_2{m}_3{m}_5\cos \left(\omega {\tau }_2\right)\text{,}\\ B_3=-m_2{m}_3{m}_5{\omega }^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\cos \left(\omega {\tau }_2\right)+m_2{m}_3{m}_5{\omega }^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)\sin \left(\omega {\tau }_2\right)+m_2{m}_3{m}_5\sin \left(\omega {\tau }_2\right)\text{,}\\ A_4=-m_1{m}_2{m}_3{m}_4\cos \left(\omega {\tau }_2\right)\text{,}\\ B_4=m_1{m}_2{m}_3{m}_4\sin \left(\omega {\tau }_2\right)\text{,}\\ F_{11}=-B_4^2\left(A_1{B}_4+A_4{B}_1\right)\left(A_1^4-A_1^2\left(A_3^2+2A_4^2-2B_1^2+B_3^2+2B_4^2\right)+2A_1\right.\left(A_2{A}_3{A}_4\right.\\ +A_2{B}_3{B}_4+A_3{B}_2{B}_4-A_4{B}_2\left.B_3\right)-A_2^2\left(A_4^2+B_4^2\right)+2A_2{B}_1\left(A_4{B}_3-A_3{B}_4\right)\\ -A_3^2{B}_1^2+2A_3{A}_4{B}_1{B}_2+A_4^4-2A_4^2{B}_1^2-A_4^2{B}_2^2+2A_4^2{B}_4^2+B_1^4-B_1^2{B}_3^2\\ -2B_1^2{B}_4^2+2B_1{B}_2{B}_3{B}_4-B_2^2{B}_4^2+\left.B_4^4\right)\text{,}\\ F_{12}=B_4\left(-\left(\left(A_4^2+B_4^2\right)\right.\left(A_2{B}_1+A_3{B}_4\right)-\left(A_1{B}_4+A_4{B}_1\right)\left(A_2{A}_4+B_2{B}_4\right)\right)\left(A_1^2\left(B_1-B_3\right)\right.\\ +\left.A_1\left(A_2{B}_4-A_4{B}_2\right)+B_1\left(A_2{A}_4-A_4^2+B_1^2-B_1{B}_3+B_2{B}_4-B_4^2\right)\right)-B_4\left(-A_1^2{A}_2\right.\\ \left.+A_1\left(A_3{A}_4+B_3{B}_4\right)-B_1\left(A_2{B}_1+A_3{B}_4-A_4{B}_3\right)\right)\left(B_4\left(A_1^2+B_1^2-B_1{B}_3\right.\right.\\ \left.\left.\left.+B_4\left(B_2-B_4\right)\right)-A_3\left(A_1{B}_4+A_4{B}_1\right)+A_1{A}_4{B}_3+A_4^2\left(B_2-B_4\right)\right)\right)\text{,}\\ F_{21}=A_1^4-A_1^2\left(A_3^2+2A_4^2-2B_1^2+B_3^2+2B_4^2\right)+2A_1\left(A_2{A}_3{A}_4+A_2{B}_3{B}_4+A_3{B}_2{B}_4\right.\\ -A_4{B}_2\left.B_3\right)-A_2^2\left(A_4^2+B_4^2\right)+2A_2{B}_1\left(A_4{B}_3-A_3{B}_4\right)-A_3^2{B}_1^2+2A_3{A}_4{B}_1{B}_2\\ +A_4^4-2A_4^2{B}_1^2-A_4^2{B}_2^2+2A_4^2{B}_4^2+B_1^4-B_1^2{B}_3^2-2B_1^2{B}_4^2+2B_1{B}_2{B}_3{B}_4-B_2^2{B}_4^2+B_4^4\text{,}\\ F_{22}=A_1^3\left(-B_2\right)+A_1^2\left(A_2\left(B_1+B_3\right)+A_3\left(B_4-B_2\right)-A_4{B}_3\right)+A_1\left(A_2^2\left(-B_4\right)+2A_2{A}_4{B}_2\right.\\ +A_3^2{B}_4-2A_3{A}_4{B}_3+A_4^2{B}_2-B_1^2{B}_2+B_2^2{B}_4+B_2{B}_4^2-B_3^2\left.B_4\right)-A_2^2{A}_4{B}_1\\ +A_2\left(A_4^2\left(B_3-B_1\right)+B_1^3+B_1^2{B}_3-B_1{B}_4\left(2B_2+B_4\right)+B_3{B}_4^2\right)+A_3^2{A}_4{B}_1\\ -A_3{A}_4^2{B}_2-A_3{A}_4^2{B}_4-A_3{B}_1^2{B}_2+A_3{B}_1^2{B}_4+2A_3{B}_1{B}_3{B}_4-A_3{B}_2{B}_4^2-A_3{B}_4^3\\ +A_4^3{B}_3-A_4{B}_1^2{B}_3+A_4{B}_1{B}_2^2-A_4{B}_1{B}_3^2+A_4{B}_3{B}_4^2\text{.}\end{array}$$ (A.1)

B

$$\begin{array}{c}{a}_1=6{\tilde{\omega }}^{2\varphi }\cos \left(\pi \varphi \right)+4{\tilde{\omega }}^{3\varphi }\cos \left(\frac{3\pi \varphi }{2}\right)+{\tilde{\omega }}^{4\varphi }\cos \left(2\pi \varphi \right)+4{\tilde{\omega }}^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)+1\text{,}\\ b_1=6{\tilde{\omega }}^{2\varphi }\sin \left(\pi \varphi \right)+4{\tilde{\omega }}^{3\varphi }\sin \left(\frac{3\pi \varphi }{2}\right)+{\tilde{\omega }}^{4\varphi }\sin \left(2\pi \varphi \right)+4{\tilde{\omega }}^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\text{,}\\ a_2=-m_2{m}_3\left(m_1{m}_4\cos \left(3\tilde{\omega }{\tau }_1\right)+m_5{\tilde{\omega }}^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\sin \left(2\tilde{\omega }{\tau }_1\right)\right)\\ -m_2{m}_3{m}_5\left({\tilde{\omega }}^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)\cos \left(2\tilde{\omega }{\tau }_1\right)+\cos \left(2\tilde{\omega }{\tau }_1\right)\right)\\ -m_3{m}_6\left({\tilde{\omega }}^{2\varphi }\sin \left(\pi \varphi \right)\sin \left(\tilde{\omega }{\tau }_1\right)+{\tilde{\omega }}^{2\varphi }\cos \left(\pi \varphi \right)\cos \left(\tilde{\omega }{\tau }_1\right)\right.\\ \left.+2{\tilde{\omega }}^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\sin \left(\tilde{\omega }{\tau }_1\right)+2{\tilde{\omega }}^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)\cos \left(\tilde{\omega }{\tau }_1\right)+\cos \left(\tilde{\omega }{\tau }_1\right)\right)\text{,}\\ b_2=m_2{m}_3\left(m_1{m}_4\sin \left(3\tilde{\omega }{\tau }_1\right)+m_5{\tilde{\omega }}^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\cos \left(2\tilde{\omega }{\tau }_1\right)\right)+m_2{m}_3{m}_5\left({\tilde{\omega }}^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)\right.\\ \times \left.\sin \left(2\tilde{\omega }{\tau }_1\right)+\sin \left(2\tilde{\omega }{\tau }_1\right)\right)-m_3{m}_6\left({\tilde{\omega }}^{2\varphi }\sin \left(\pi \varphi \right)\cos \left(\tilde{\omega }{\tau }_1\right)-{\tilde{\omega }}^{2\varphi }\cos \left(\pi \varphi \right)\sin \left(\tilde{\omega }{\tau }_1\right)\right.\\ \left.+2{\tilde{\omega }}^{\varphi }\sin \left(\frac{\pi \varphi }{2}\right)\cos \left(\tilde{\omega }{\tau }_1\right)-2{\tilde{\omega }}^{\varphi }\cos \left(\frac{\pi \varphi }{2}\right)\sin \left(\tilde{\omega }{\tau }_1\right)-\sin \left(\tilde{\omega }{\tau }_1\right)\right)\text{.}\end{array}$$ (B.1)

Data Availability

Data sharing not is applicable in this article as no datasets were generated or analysed during the current paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.