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

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.


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][2][3][4][5][6][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][9][10][11][12][13][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][18][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: (i) A novel delayed fractional-order recurrent neural network with four-neuron and two different delays is studied (ii) Double main dynamical properties of the fractionalorder recurrent neural network with two delays are investigated: stability and oscillation (iii) e 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. e last section gives some conclusions.

Preliminaries
is section we will give some Caputo definitions and lemma for fractional calculus as a basis for the theoretical analysis and simulation proofs.
in which 0 < ϕ ≤ 1, u ∈ R k , and J ∈ R k×k . en the zero solution of the system (5)

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 τ 1 and τ 2 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. roughout of this paper, assume that the following condition holds true:

Main Results
is section chooses τ 1 or τ 2 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.

Bifurcation
Depending on τ 1 in Equation (6). In this subsection, we first study the effects of τ 1 on bifurcations of system (6) by establishing τ 2 .
Applying Taylor series formula, the following form of equation (6) at the origin is By applying Laplace transformation, its characteristic equation is given as de t where From (8), we have where K 1 (s) � s 4ϕ + 4s 3ϕ + 6s 2ϕ + 4s ϕ + 1, Multiplying e sτ 1 and e 2sτ 1 on both sides of equation (9), respectively, we can obtain Computational Intelligence and Neuroscience 4 , and from equation (9), we have Take s � iw � w(cos π/2 + i sin π/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: Evidently,  (14), it can be implied that From equation (13), one can obtain
Define the bifurcation point of fractional neural network with multiple delays (6) as If τ 1 vanishes, then equation (9) becomes where If τ 2 � 0, then the equation (18) becomes Suppose that all roots s of the equation (18) obey Lemma 1, then we get that both roots λ i in equation (18) Also, let s � iv � v(cos π/2 + i sinπ/2)(v > 0) be a purely imaginary root of equation (11) if and only if is leads to form It is not difficult to see that Additionally, we will give the following assumptions which hold true.
Proof. With implicit function theorem, we can differentiate equation (9) with respect to τ 1 , and thus we get where Υ(s) � s K 2 (s)e −sτ 1 + 2K 3 (s)e 2−sτ 1 + 3K 4 (s)e −3sτ 1 , We further suppose that Υ 1 and Υ 2 are the real and imaginary parts of Υ(s), respectively, and Ω 1 and Ω 2 are the real and imaginary parts of Ω(s), respectively, then Computational Intelligence and Neuroscience From (C3), we conclude that the transversality condition holds true. is 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: (i) e zero equilibrium point of fractional order fourneuron recurrent neural network with multiple delays (6) is asymptotically stable when τ 1 ∈ [0, τ * 10 ).
, then fractional order four neurons recurrent neural network with multiple delays (6) causes Hopf bifurcation at the origin when τ 1 � τ * 10 . at is, a branch of periodic solutions can bifurcate from the zero equilibrium point at τ 1 � τ * 10 .

Bifurcation
Depending on τ 2 in Equation (6). As in the previous subsection, next we change another delay τ 2 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: where q 1 (s) � 1 + 4s 2ϕ + 6s 2ϕ + 4s 3ϕ + s 4ϕ , Multiplying e 2sτ 2 on both sides of equation (30), we can obtain q 1 (s)e sτ 2 + q 2 (s) � 0.
Suppose q 1 (s) � a 1 + ib 1 and q 2 (s) � a 2 + ib 2 , and from equation (32), we have where a 1 , a 2 , b 1 , b 2 are given in Appendix B. Take s � iw � w(cos π/2 + i sin π/2)(ω > 0) as a root of equation (33) if and only if that is, It is simple to derive the following equation.

⎧ ⎨ ⎩ (41)
Let s � iv � v(cos π/2 + i sin π/2)(v > 0) be a solution of equation (41). Substituting s into equation (41) and separating the imaginary and real units yields the following equations: Obviously, from first and second equation of system (43), we get 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 ω can be obtained according to mathematical software Mathematica 10.0, and then the bifurcation point τ 10 of recullrent fractional four-neuron neural networks (6) with τ 2 � 0 can be derived.As a summary of our main results, we provide the following assumption: Lemma 3. Let s(τ 2 ) � η(τ 2 ) + iw(τ 2 ) be a root of equation (9) near τ 2 � τ 2j satisfying η(τ 2j ) � 0, w(τ 2j ) � w 0 , then we get the following transversality condition Proof. Similar to Lemma 2, by utilizing the implicit function theorem and differentiating (9) with respect to τ 2 , we get where β(s) � sq 2 (s)e −sτ 2 , We further suppose that α 1 and α 2 are the real and imaginary units of α(s), respectively, and β 1 and β 2 are the real and imaginary parts of β(s), respectively, then we get As a direct consequence of (C5), we can conclude that the transversality condition is satisfied. en the proof of Lemma 3 is complete.
Based on the above analysis, the following conclusions can be drawn.

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

Example 1.
Consider the four-neuron fractional recurrent neural networks with multiple delays as
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 τ 10 and τ 20 and τ * 10 and τ * 20 can be determined by changing the order of ϕ.
is means that systems (50) and (51) involving  different two delays are prone to earlier Hopf bifurcation for some fixed fractional order ϕ.

Conclusion
is 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). ese results can contribute to our understanding of delayed fractional recurrent neural networks as a continuation of the previous work. e results of the simulations are illustrated by two numerical examples.

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.