# Results for Chaos Synchronization with New Multi-Fractional Order of Neural Networks by Multi-Time Delay

Authors: Fatin Nabila Abd Latiff; Wan Ainun Mior Othman
Journal: Complexity
 (2021)

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## Abstract

A new finding is proposed for multi-fractional order of neural networks by multi-time delay (MFNNMD) to obtain stable chaotic synchronization. Moreover, our new result proved that chaos synchronization of two MFNNMDs could occur with fixed parameters and initial conditions with the proposed control scheme called sliding mode control (SMC) based on the time-delay chaotic systems. In comparison, the fractional-order Lyapunov direct method (FLDM) is proposed and is implemented to SMC to maintain the systems’ sturdiness and assure the global convergence of the error dynamics. An extensive literature survey has been conducted, and we found that many researchers focus only on fractional order of neural networks (FNNs) without delay in different systems. Furthermore, the proposed method has been tested with different multi-fractional orders and time-delay values to find the most stable MFNNMD. Finally, numerical simulations are presented by taking two MFNNMDs as an example to confirm the effectiveness of our control scheme.


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## Body

## 1. Introduction

Although the concept and the knowledge of fractional-order calculus have existed since L’Hôpital’s and Leibniz’s contribution in 1695s [1, 2], its applications to the real world of mathematics, physics engineering, and biology are only of interest recently [3, 4]. As a result, it has captured the attention of physicists and engineers over the last decade and has been a new trend up until this point. In addition, due to its universal application in different fields, such as brain networks, fractional-order dynamical systems have generated significant study attention. As opposed to integer-order systems, fractional-order systems exhibit long-term memory effects, making them ideal for representing varied materials and processing more precisely [5]. Fractional-order modeling has the capacity to explain real-world phenomena more realistically and precisely compared to the traditional integer-order calculus [6–8], making it particularly well adapted to study nonlinear systems, predominantly in biology and physics.The fractional-order chaotic system synchronization has recently garnered attention because of its capability of application in a diversity of physics and engineering science fields, including encryption and secure communication. As a result, there is considerable interest in properly synchronizing two chaotic systems with fractional order. Synchronization behavior is a fundamental natural phenomenon that frequently occurs in both the natural world and engineering. Chaos synchronization has garnered substantial interest and research over the last few years because of its tenacity in numerous fields, including secure communication, physical science, chemical reactors, and information processing. Numerous intriguing discoveries have been found thus far on chaotic synchronization [9–13]. However, the widely held research is based on synchronization asymptotically. Alternatively, we can say that synchronization of chaos can occur individually when the time approaches infinity.As we all know, it is more useful in reality for chaotic systems to be synchronized within finite time preferably compared to infinite time. As a result, much study has been conducted on fixed-time synchronization [14–17]. Fixed-time synchronization denotes the possibility of achieving chaos synchronization within a finite time window for any initial value of the examined system. In comparison to synchronization asymptotically, fixed-time synchronization has several advantages, including a faster convergence rate and enhanced anti-interference capacity. The error system must be stable for a defined period to maintain chaos synchronization within a fixed time. Even though certain conclusions have been recommended in the literature, the study of chaotic systems’ fixed temporal stability is still in its infancy. Thus, it is critical to continue developing some innovative and effective fixed-time stability criteria.Following Pecora and Carroll’s first work on chaotic synchronization [18], synchronization of neural networks (NNs) and complex networks has become an important topic. Due to the enormous advantages of controlling a problem and addressing complex nonlinear system analysis, NN-based modeling has been an active study area in recent years. Besides, there are some disadvantages in synchronization among NN systems, and the results of numerous control methods have been proven to deal with drawbacks, including adaptive [19], generalized projective [20], linear feedback [21], and SMC [22]. The adaptive SMC approach is an efficient technique for synchronizing chaotic systems with a fractional order among the strategies outlined above. The essential characteristics of this technique are its rapid reaction, resistance to perturbations, sensitivity to parameter disturbances, good momentary performance, and effortlessness of application in real-world applications. We discovered that projective synchronization could achieve remarkable performance in secure communication due to its proportional function.According to prior studies, synchronization of identical or nonidentical FNNs occurs without delay [19, 22–24]. Indeed, there is always some noise and disturbances that might impair performance and impair the synchronization’s output. However, from the standpoint of practical engineering, the work presented in the preceding publications is insufficient since they ignore the influence of time delays on the system, which is another crucial component contributing to the loss of system stability. Time delays are inevitable as a result of information’s finite propagation velocity [25, 26], the potential of feedback loops, and the finite switching durations. In the real world, many channels of data alternation, multiple switching mechanisms, and two or more feedback systems may exist [27]. Conversely, multi-time delay systems are frequently more accurate representations of interacting complex systems than single-time-delay systems.FNNs with multi-time-delay synchronization are extremely important to study. In the later years, delayed differential equations have been broadly studied as in [28, 29]. Recently, chaotic systems have been developed with temporal and time-varying delays; for example, see [30–33]. In addition, the first study on the significance of time delay on chaotic behavior was in [34] based on the previous literature. Nevertheless, most of these papers concern about ordinary differential equations or integer orders, and there are preliminary results concerning chaos synchronization of FNNs with multi-time-delay systems. Numerous researchers have attempted to employ the FNNs with multi-time-delay systems to research secure communication, and they have come up with some exciting results in [35–38].There are two widely used techniques for dealing with the degradation of system stability caused by time delays: Lyapunov–Razumikhin functions (LRFs) and Lyapunov–Krasovskii functions (LKFs) [37, 38]. In addition, the separation technique is frequently used to decompose unknown time-delayed functions into numerous continuous functions [39]. In [37], a Lyapunov–Razumikhin approach was created, and theoretical results established that systems with varying time delays might be stable in infinite time.Nonetheless, to the best of the authors’ understanding, no researcher has yet examined the theoretical investigation of MFNNMD. Indeed, this fascinating subject remains an open challenge, which prompted our investigation. As a result, this study analyses the dynamic properties of MFNNMD systems. We introduced the global Mittag–Leffler MFNNMD model of projective synchronization in our work and designed a new multidelay SMC controller. Following that, a new extended concept for cryptography and safe communication is introduced for further research. The following are the primary aspects of this article’s significant contributions:(1)
For the first time, MFNNMD is investigated.(2)
We are constructing appropriate proposed multi-time-delay SMC controller and FLDM of the error system for global Mittag–Leffler synchronization of MFNNMD to maintain their stability and ensure global convergence of the error dynamics.(3)
To determine the practical significance of the theoretical outcomes, a numerical example is provided with simulations. In addition, other examples are included in the table.(4)
Finally, we hope that our research will provide significant results to the real world, resulting in increased security in secure communication networks.Our proposed technique is particularly well-suited for secure communication. The transmission signal is transferred from the transmitter to the recipient by a chaotic motion across an analogue network in a normal chaotic synchronization communication technique. Numerous methods for masking the data in transmission signals using chaotic signals have been developed, including chaotic-switching and chaotic-modulation approaches. However, it has been demonstrated that the majority of these schemes are insecure as the encrypted message signal may be hacked or extracted from the transmitted chaotic signal using several unmasking techniques. By employing our proposed synchronization strategy to broadcast the signal, the transmitted signal can be added to MFNNMD. We believe that our proposed method can help chaotic communication become more secure.The following is the organization of this article: Section2 contains an explanation of the system and some relevant prerequisites. In Section 3, we examine the synchronization of chaotic systems with MFNNMD. In Section 4, we carry out numerical simulations to confirm and validate our theoretical results’ practicality and discuss the implications of varying fractional orders and time delays on synchronization. Finally, in Section 5, we conclude the article.


## 2. Preliminaries and Description of the Model

Two definitions of the most conventional fractional-order calculus derivatives used are mentioned in the literature: the Caputo and the Riemann–Liouville. Then, the Caputo derivative is implemented in this work because its initial conditions may indeed be described in terms of integer-order derivatives, which is more practical in practice.Definition 1.
(see [40]). For the u (t) function, the fractional derivative of Caputo order α is described as(1)Dαut=1Γq−α∫0tt−τq−α−1uqτdτ,where t≥t0, q is the integer, q−1<α<q, Γ⋅ is a Gamma function, and Γp=∫0∞tp−1e−tdt.
In this research, we study the following NNs, which we designate to as MFNNMD, where the DS is denoted as(2)DαiXit=−ciXit+∑j=1xaijfjXjt+∑j=1xgijfjXjt−τj+φi,where 0<αi<1,i,j=1,2,…,x, x represents the unit number in a MFNNMD, Xit signifies the i-th neuron’s pseudo-state variable of the DS, and ci>0 is the ith unit parameters. The external input of the ith unit is φi, aij and gij denote the relation of the unit jth with the i th at time t and t−τj, respectively, τj>0 is the delayed transmission, and fjXjt and fjXjt−τJ represent the j-th unit activation function output at time t and t−τj, respectively.
Also, the vector form(3)DαiXt=−CXt+AfcXt+GfcXt−τj+φ.
The corresponding RS of MFNNMD is denoted as(4)Dαiyit=−diyit+∑j=1xbijgjyjt+∑j=1xhijgjyjt−τj+μi+ϑit,where 0<αi<1,i,j=1,2,…,x, x represents the unit number in a MFNNMD, yit represents the ith unit’s pseudo-state variable of the RS, and di>0 is the ith neuron’s parameters. μi represents the external input of the ith unit, bij and hij denote the relation of the unit jth with the ith at time t and t−τj , respectively, τj>0 is the delayed transmission, and gjyjt and gjyjt−τ denote the jth unit activation function output at time t and t−τj, respectively. However, ϑit is an appropriate controller. For simplifications, in the next part, definition, lemmas, and theorems are given.Definition 2.
By establishing the one-parameter Mittag–Leffler function by settingα>0 and u∈C, then(5)Eαu=∑k=0∞ukΓkα+1.and by establishing the two-parameter Mittag–Leffler function by setting α>0,β>0, and u∈C, then(6)Eα,βu=∑k=0∞ukΓkα+β.
According to Definition2, it is not difficult to see Eαu=Eα,1u and E1,1u=eu.Definition 3.
(see [19]). Suppose there is a constant of nonzero, ρ, such that sufficiently for every result of X (t) and Y (t) for systems (2) and (4) using distinct initial condition, you may obtain(7)limt⟶+∞Υt−ρXt=0.Therefore, DS (2) and RS (4) can accomplish globally projective synchronization asymptotically, i representing the vector’s Euclidean norm.Lemma 1.
(see [41]). Assume Xt∈R is a continuous function of differentiable vector value. Hence, for whichever time prompt ≥t0, we get the following:(8)DαXTtXt≤2XTtDαXt,where 0<α<1.


## 3. Controller Design and Stability Analysis

Two identical MFNNMD synchronization conditions are achieved in this section by proposing an appropriate controller. Define the errors of synchronization ast=ϒit−ρXit (i=1,2,…,x). Based on DS (2) and RS (4), the control function ϑit (i=1,2,…,x) can be designated as follows:(9)ϑit=−ci+Πjlx×m⋅ωlim×x−diEjt−Πjlx×m⋅εlim×xEjt−τj−ρci−diXjt−∑j=1ℵbijgjρXjt−∑j=1ℵhijgjρXjt−τj+ρ∑j=1ℵaijfjXjt+ρ∑j=1ℵgijfjXjt−ρφi+μi,where i,j,ℵ,m=1,2,…,ℵ and ρ is the projective coefficient. From (2), (4), and (9), you can get the DαEt error system as follows:(10)DαiEt=dieit+∑j=1xbijgjΥjt−gjρXjt+∑j=1xhijgjΥjt−τj−gjρXjt−τj+ρci−diXjt+∑j=1xbijgjXjt+∑j=1xhijgjρXjt−τj−ρ∑j=1xaijfjXjt−ρ∑j=1xgijfjXjt−τj−ρφi+μi+ϑit.Now, (9) and (10) are combined together and the error system of delayed sliding mode dynamics (DDMDEs), DαEt, can be defined by(11)DαiEt=−ci+Πjlx×m.ωlim×xEit−Πjlx×m.εlim×xEjt−τj+∑j=1xbijgjϒjt−gjρXjt+∑j=1xhijgjϒjt−τj−gjρXjt−τj.Then, it follows from (11) that(12)DαiEt=−cieit−∑j=1x∑l=1mΠjlωliEit−∑j=1x∑l=1mΠjlεliEjt−τj+∑j=1xbijgjϒjt−gjρXjt+∑j=1xhijgjϒjt−τj−gjρXjt−τj.To investigate the stability of DDMDEs (11) in this paper, Theorem 1 is stated.Theorem 1.
(see [42]). Assume that condition (H) is met and there are positive constants called mi and li in such a way that mi=maxmi−,mi+ and li=maxli−,li+ and constant matrices that Πjlℵ×i, ωlim×ℵ, εlii×ℵ, and(13)Ψ1∶=min1≤i≤xci−∑j=1x∑l=1mΠjlωli−∑j=1xbjimi>0,Ψ2∶=max1≤i≤x∑j=1x∑l=1mΠjlεli+∑j=1xhjili>0,Ψ1−Ψ2>0.Then, equation (11) is stable.Proof of Theorem 1.
According toH and Theorem 1, we have(14)mj−≤gjϒjt−gjρXjtϒjt−ρXjt≤mj+,(15)lj−≤gjϒjt−τj−gjρXjt−τjϒjt−τj−ρXjt−τj≤lj+.Next, we obtain that(16)gjϒjt−gjρXjt≤mjϒjt−ρXjt,gjϒjt−τj−gjρXjt−τj≤ljϒjt−τj−ρXjt−τj.Generate the following function of Lyapunov:(17)VEt=∑i=1xEit.
As a result of Lemma1, by applying the Caputo derivative’s upper right D+αEt along the trajectories, equation (17) can be obtained as follows.
We defined the Lyapunov function by(18)D+αiVEit=∑i=1xD+αiEit≤−∑i=1xsgnEitD+αEit=−∑i=1xsgnEitci+∑j=1x∑l=1mΠjlωliEjt+∑j=1x∑l=1mΠjlεliEjt−τj+∑j=1xbijgjϒjt−gjρXjt+∑j=1xhijgjϒjt−τj−gjρXjt−τ≤−∑i=1xciEit+∑i=1x∑j=1x∑l=1mΠjlωliEjt+∑i=1x∑j=1x∑l=1mΠjlεliEjt−τj+∑i=1x∑j=1xbijmjϒjt−ρXjt+∑i=1x∑j=1xhijljϒjt−τj−ρXjt−τj,=−∑i=1xciEit+∑i=1x∑j=1x∑l=1xΠjlωliEit+∑i=1x∑j=1x∑l=1mΠjlεliEjt−τj+∑i=1x∑j=1xbijmjϒjt−ρXjt+∑i=1x∑j=1xhijljϒjt−τj−ρXjt−τj,=−∑i=1xci−∑j=1x∑l=1mΠjlωli−∑j=1xbijmjϒjt−ρXjtEit+∑i=1x∑j=1x∑l=1mΠjlεli+∑i=1x∑j=1xhijljϒjt−τj−ρXjt−τjEjt−τj,=max1≤i≤x∑j=1x∑l=1mΠjlεli+∑j=1xhjiliEjt−τj−min1≤i≤xci−∑j=1x∑l=1mΠjlωli−∑j=1xbjimiEit≤Ψ1VEit+Ψ2supt−τj≤u≤tVEiu.
Note that(19)supt−τj≤u≤tVEiu≤VEit.
Then, assuming that there is a constant Δ greater than zero in conjunction with (18) and (19) and Theorem 1, one has(20)D+αiVEit≤−Ψ1−Ψ2VEit,Ψ1−Ψ2≥Δ.
We construct that (18) as(21)D+αiVEit≤ΔVEit,so(22)Eit=ϒjt−ρXjt=∑i=1ℵϒjt−ρXjt.
Finally, we can conclude that DDMDEs (9) are stable as Eit⟶0 as t approaches infinity.
This completes the proof.Remark 1.
When=1, projective synchronization is standardized by multiple time delays for the complete synchronization of MFNNMD.Remark 2.
When=−1, the projective synchronization is standardized to the complete globally antisynchronization of MFNNMD with multiple time delays.Remark 3.
No study on the topic of MFNNMD chaotic synchronization has been published so far. As a result of addressing these inadequacies, we developed for the first time new necessary conditions for ensuring the chaos synchronization of MFNNMD systems is archived. As a result, the paper’s primary findings are significantly new compared to those found in the prior literature.


## 4. Numerical Experiments

The following example demonstrates that the control method proposed in this study is capable of adequately managing the time-delay system’s stability while guaranteeing that all state variables remain within the specified ranges. Here, two types of experiments of time delay with alpha are introduced.


### 4.1. Projective Chaos Synchronization of MFNNMD Systems

#### 4.1.1. Experiment A: Synchronization of MFNNMD Systems with Fixed Time Delay,τ = 0.11

Assuming that two MFNNMD systems are synchronized with each other, define the drive system as(23)Dα1X1t=−c1X1+a11f1X1t+a12f2X2t+a13f3X3t+a14f4X4t+g11f1X1t−τ1+g12f2X2t−τ1+g13f3X3t−τ1+g14f4X4t−τ1+φ1,Dα2X2t=−c2X2+a21f1X2t+a22f2X2t+a23f3X3t+a24f4X4t+g21f1X1t−τ2+g22f2X2t−τ2+g23f3X3t−τ2+g24f4X4t−τ2+φ2,Dα3X3t=−c3X3+a31f1X1t+a32f2X2t+a33f3X3t+a34f4X4t+g31f1X1t−τ3+g32f2X2t−τ3+g33f3X3t−τ3+g34f4X4t−τ3+φ3,Dα4X4t=−c4X4+a41f1X1t+a42f2X2t+a43f3X3t+a44f4X4t+g41f1X1t−τ4+g42f2X2t−τ4+g43f3X3t−τ4+g44f4X4t−τ4+φ4.And, define the response system as(24)Dα5ϒ1t=−d1ϒ1+b11f1ϒ1t+b12f2ϒ2t+b13f3ϒ3t+a14f4ϒ4t+h11f1ϒ1t−τ1+h12f2ϒ2t−τ5+h13f3ϒ3t−τ5+h14f4ϒ4t−τ5+μ1+ϑ1t,Dα6ϒ2t=−d2ϒ2+b21f1ϒ2t+b22f2ϒ2t+b23f3ϒ3t+b24f4ϒ4t+h21f1ϒ1t−τ6+h22f2ϒ2t−τ6+h23f3ϒ3t−τ6+h24f4ϒ4t−τ6+μ2+ϑ2t,Dα7ϒ3t=−d3ϒ3+b31f1ϒ1t+b32f2ϒ2t+b33f3ϒ3t+b34f4ϒ4t+h31f1ϒ1t−τ7+h32f2ϒ2t−τ7+h33f3ϒ3t−τ7+h34f4ϒ4t−τ7+μ3+ϑ3t,Dα8ϒ4t=−d4ϒ4+b41f1ϒ1t+b42f2ϒ2t+b43f3ϒ3t+b44f4ϒ4t+h41f1ϒ1t−τ8+h42f2ϒ2t−τ8+h43f3ϒ3t−τ8+h44f4ϒ4t−τ8+μ4+ϑ4t,with the following parameters: α1=0.985,α2=0.980,α3=0.975,α4=0.970,α5=0.965,α6=0.960,α7=0.955,α8=0.950,c1=c4=−9.5,c2=−10.5,c3=−3.7,a11=2.0,a12=0.5,a13=5.5,a14=2,a21=0.5,a22=0.5,a23=5.1,a24=0.5,a31=0.5,a32=1.0,a33=−5.5,a34=5.1,a41=0.5,a42=1.0,a43=−9.5,a44=0.5g11=7,g12=7,g13=4.1,g14=7,g21=1,g22=1,g23=2.5,g24=1,g31=0.1,g32=−10.1,g33=4.5,g34=0.1,g41=7,g42=7,g43=4.1,g44=7,d1=d4=−1,d2=−3,d3=−7.7, b11=0.01,b12=0.3,b13=7.5,b14=0.01,b21=0.01,b22=7.5,b23=0.01,b24=0.01,b31=0.01,b32=3,b33=−5.5,b34=−5.5,b41=0.01,b42=3,b43=3,b44=−5.5, h11=0.3, h12=5.5, h13=5.5, h14=0.3, h21=0.5, h22=7.5, h23=0.01, h24=0.5, h31=0.1, h32=−1.5, h33=1.5, h34=0.1, h41=0.1, h42=5.5, h43=5.5, h44=0.1, f1.=f2.=f3.=f4.=tanhX, φ1=φ2=φ3=φ4=0, μ1=μ2=μ3=μ4=0, and ϑ1t=ϑ2t=ϑ3t=ϑ4t=1.With the above initial values, parameters, and controller gain, we solve the numerical solutions of DS (21), RS (22), and error system (11) using MATLAB tools. Furthermore, we use the step-by-step iteration method to solve the fractional differential-difference equations part. Then, they are illustrated in Figures 1–4.Figure 1
Experiment A: phase portrait of the chaotic attractor of MFNNMD systems (21) and (22).Figure 2
Experiment A: phase portrait of theX and Y of MFNNMD systems (21) and (22), when τ=0.11 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960].Figure 3
Experiment A: state trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)Figure 4
Experiment A: error trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)A graphical representation in phase portrait of the consequences of MFNNMD synchronization is shown in Figure1 using X and ϒ, as the dependent variables. At the same time, t is an independent variable function. Figure 2 shows a phase portrait of synchronization of MFNNMD, when τ = 0.11 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960]. Chaos synchronization of the MFNNMD system with and without the controller activation is shown in Figure 3. As illustrated in Figure 4, the dynamic error is unable to converge to zero when the controller is not engaged. However, when the controller is activated, the dynamic error does converge to zero in a limited time. As illustrated in Figure 4, master-slave systems (21) and (22) exhibit global Mittag–Leffler asymptotic synchronization, confirming the validity and effectiveness of the criteria specified in Theorem 1.


#### 4.1.2. Experiment B: Synchronization of Two MFNNMD Systems with Multi-Time Delay,τ = [0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17]

The numerical experiment of synchronization of two MFNNMD systems is demonstrated in this section. In this experiment, we consider system changes at different fractional orders and different time delays. The same systems of DS (21) and (22) are used. Figure 5 shows that the systems have chaotic characteristics, while Figure 6 shows a phase portrait of synchronization of MFNNMD, when τ=0.10,0.11,0.12,0.13,0.14,0.15,0.16,0.17 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960]. The synchronization of the MFNNMD system with and without the controller activation is shown in Figure 7. Furthermore, for the error dynamics in Figure 8, we can see that the dynamic error is unable to converge to zero when the controller is not engaged. However, when the controller is activated, the dynamic error does converge to zero in a limited time.Figure 5
Experiment B: phase portrait of the chaotic attractor of MFNNMD systems (21) and (22).Figure 6
Experiment B: phase portrait ofX and Y of MFNNMD systems (21) and (22), when τ=0.10,0.11,0.12,0.13,0.14,0.15,0.16,0.17 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960].Figure 7
Experiment B: state trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)Figure 8
Experiment B: error trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)


### 4.2. Effects on Chaos Synchronization ofα-Order and τ-Time Delay

According to the literature,α-order affects the performance of fractional-order chaotic dynamical systems. As previously demonstrated, time delay affects the performance of chaotic dynamical systems. In the actual world, time delays are pervasive and unavoidable. The time-delay dynamic system exhibits complicated dynamic characteristics that are frequently utilized to illustrate time-delayed engineering issues. In this paper, two NN chaotic systems that act as DS and RS are used to examine the dynamic performance from the effect of multi-fractional order and multi-time delay on nonlinear dynamic systems. Furthermore, the capability of the SMC design in this research is studied to prove that it can control the MFNNMD system.From Figure2, we consider system changes at different fractional orders with the alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960] while fixing delay τ = 0.11. We can learn from Figure 2 that the system becomes chaotic and the system trajectory becomes periodic. The system continues to be chaotic as different values of alpha are applied, for example, alpha = [0.895, 0.890, 0.885, 0.880, 0.875, 0.870, 0.865, 0.860] and alpha = [0.795, 0.790, 0.785, 0.780, 0.775, 0.770, 0.765, 0.760]. From Figures 9(a) and 9(b), if we continue decreasing the value of fractional orders, we can see that the systems slowly produce the intermittent chaos phenomenon. Finally, when the value of alpha = [0.695, 0.690, 0.685, 0.680, 0.675, 0.670, 0.665, 0.660], as shown in Figure 9(c), the system will be unstable and lose its chaotic characteristic.Figure 9
Phase portrait of synchronization of MFNNMD systems (21) and (22) when τ=0.11: (a) alpha = [0.895, 0.890, 0.885, 0.880, 0.875, 0.870, 0.865, 0.860], (b) alpha = [0.795, 0.790, 0.785, 0.780, 0.775, 0.770, 0.765, 0.760], and (c) alpha = [0.695, 0.690, 0.685, 0.680, 0.675, 0.670, 0.665, 0.660].
(a)(b)(c)This section addresses the effects of altering theα-order and τ-time delay on chaotic synchronization. We summarised some observations about the error functions for various values of α at specified time values in Table 1. As shown in Table 1, the synchronization error reduces as the order of α is raised. In other words, the synchronization process begins earlier with a greater value of α.Table 1
Dynamic errors for various values of alpha (α) and τ=0.11.
Value of alphate1e2e3e40.20.1246582.168321−2.763986−3.362894α1=0.9950.4−2.401059−1.517229−3.991367−4.667138α2=0.9900.6−4.364460−2.5578561.480212−1.060209α3=0.9850.8−3.045370−1.1418782.9168640.961262α4=0.9801.0−0.8892771.0387212.0092670.697992α5=0.9751.22.8487534.206538−2.248520−1.196980α6=0.9701.4−0.405743−0.055542−5.719650−5.521897α7=0.9651.6−5.144471−3.832519−0.562004−3.244713α8=0.9601.8−3.941395−2.2449552.7784120.5208822.0−2.816576−1.1767823.2467071.5024490.20.1210471.547881−3.908083−3.985490α1=0.8950.4−3.456733−1.6627450.398536−1.882546α2=0.8900.6−0.8669101.0643611.021344−0.378369α3=0.8850.81.6289602.425840−4.054332−3.501186α4=0.8801.0−3.634861−2.180439−0.506693−2.636411α5=0.8751.2−1.890809−0.2274391.540518−0.197372α6=0.8701.41.4143752.816090−1.083584−0.738211α7=0.8651.6−1.924365−1.241058−3.434225−4.093808α8=0.8601.8−2.284485−0.8426591.033461−0.8406262.0−0.1940501.2228200.563513−0.2297260.2−2.602491−0.940676−1.924003−3.521095α1=0.7950.40.2094121.830452−1.281248−1.679577α2=0.7900.6−1.4367650.027426−1.132682−2.296571α3=0.7850.80.0587711.260416−1.964939−2.089092α4=0.7801.0−0.8820700.379702−1.004009−1.839316α5=0.7751.2−0.3489530.717825−1.659654−1.952206α6=0.7701.4−0.5233440.531580−1.125355−1.656581α7=0.7651.6−0.3552090.581332−1.355911−1.682493α8=0.7601.8−0.3794220.508523−1.144286−1.5260372.0−0.2921120.519874−1.172738−1.4683460.2−0.8368290.855342−1.077397−2.142311α1=0.6950.4−0.7606790.667618−1.321963−2.095273α2=0.6900.6−0.6768700.599926−1.329555−1.973587α3=0.6850.8−0.5980960.566648−1.282252−1.845552α4=0.6801.0−0.5360080.536812−1.225057−1.731554α5=0.6751.2−0.4847280.508685−1.171231−1.632134α6=0.6701.4−0.4397690.483483−1.121826−1.543438α7=0.6651.6−0.3996540.460903−1.075778−1.462798α8=0.6601.8−0.3637130.440376−1.032476−1.3887362.0−0.3314080.421524−0.991581−1.320280Based on Experiment A, Table1 contains some results on the dynamic error for different order values in system (21) and system (22). Here, trials are conducted with a fixed delay value of τ = 0.11. It is evident that as the order drops, the synchronization error increases first and subsequently reduces, which is not the case in [43]. Simultaneously, we sum up some data related to the dynamic error for a different combination of fractional orders with the fixed time delay in systems (21) and (22). As seen in Table 1, the dynamic errors e1t,e2t,e3t, and e4t converge to zero as time increases. Precisely, the error function reduces first, then increases, and eventually decreases over the time points for all sets of fractional orders, and we show the evidence as in Figure 9.For Experiment B, we consider system changes at different fractional orders and different time delays. Whenτ1=τ2=⋯=τn=τ, MFNNMD with multidelay can be treated as a single MFNNMD, as explained in Experiment A. Indeed, neurons may have varying communication delays, necessitating the consideration of varied delays in neural networks. Three examples with different values of fractional orders are given with different values of time delays. Figures 10(a) and 10(b) show that the system becomes chaotic, and Figure 10(c) shows that the systems start to become unstable and lose their chaotic characteristics. The summarization for Experiment B is shown in Table 2. All the dynamic errors with different fractional orders and time delay values show the same performance as with the fixed time delays. The error function reduces first, then increases, and eventually decreases over time. Finally, detailed time taken for dynamic errors converging to zero for Experiments A and B are shown in Tables 3 and 4.Figure 10
Phase portrait of synchronization of MFNNMD systems (21) and (22) when τ=0.10,0.11,0.12,0.13,0.14,0.15,0.16,0.17: (a) alpha = [0.895, 0.890, 0.885, 0.880, 0.875, 0.870, 0.865, 0.860], (b) alpha = [0.795, 0.790, 0.785, 0.780, 0.775, 0.770, 0.765, 0.760], and (c) alpha = [0.695, 0.690, 0.685, 0.680, 0.675, 0.670, 0.665, 0.660].
(a)(b)(c)Table 2
Dynamic errors for various values of alpha (α) and τ = [0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17].
Value of alphate1e2e3e40.2−2.022495−0.0656862.8530491.205545α1=0.9950.4−4.080703−2.2932182.6875970.311466α2=0.9900.6−1.320927−1.122006−5.013653−5.232768α3=0.9850.8−0.2143951.0824451.6866270.853142α4=0.9801.0−2.728065−1.6137152.4296910.460877α5=0.9751.2−2.354545−1.618750−1.338532−2.756335α6=0.9701.4−0.4100800.051628−1.815041−1.838976α7=0.9651.6−0.1348540.233707−1.195806−1.213489α8=0.9601.80.0025740.298025−0.846826−0.8351302.00.0518010.297613−0.650303−0.6302230.20.9403632.652079−0.323689−0.373100α1=0.8950.41.0327232.446092−0.593181−0.486812α2=0.8900.61.5165372.179133−2.695674−2.141750α3=0.8850.8−1.193599−0.160199−1.124218−1.989497α4=0.8801.0−0.0950080.720590−0.875975−1.115168α5=0.8751.2−0.1875240.466583−1.086851−1.240937α6=0.8701.4−0.1507880.407471−0.926858−1.065040α7=0.8651.6−0.1025130.370933−0.828564−0.929539α8=0.8601.8−0.0671050.336450−0.742594−0.8144202.0−0.0408250.304890−0.665046−0.7144290.2−0.6284860.752827−1.065626−1.705763α1=0.7950.4−0.6166810.508169−1.175186−1.674920α2=0.7900.6−0.5014800.455924−1.084421−1.493109α3=0.7850.8−0.4131070.413356−0.992929−1.334976α4=0.7801.0−0.3419050.377650−0.910913−1.198937α5=0.7751.2−0.2839420.346661−0.836579−1.080232α6=0.7701.4−0.2364180.319153−0.768815−0.975700α7=0.7651.6−0.1972440.294346−0.706797−0.883013α8=0.7601.8−0.1648140.271723−0.649864−0.8003622.0−0.1378680.250926−0.597470−0.7263000.2−0.7264360.450155−0.988908−1.560065α1=0.6950.4−0.6077640.387185−0.887064−1.357983α2=0.6900.6−0.5207290.344249−0.808443−1.210932α3=0.6850.8−0.4517190.310852−0.741817−1.091060α4=0.6801.0−0.3950140.283278−0.683254−0.988966α5=0.6751.2−0.3474100.259682−0.630746−0.899911α6=0.6701.4−0.3068540.238995−0.583070−0.821019α7=0.6651.6−0.2719200.220533−0.539396−0.750353α8=0.6601.8−0.2415670.203835−0.499119−0.6865142.0−0.2150070.188574−0.461779−0.628448Table 3
Error dynamics of MFNNMD time taken for the system to converge to zero at 12.5 s.
te1e2e3e40.5−4.543203−3.139601−0.945464−3.2805291.0−0.8892771.0387212.0092670.6979921.5−3.304613−2.685071−3.987378−5.3158202.0−2.816576−1.1767823.2467071.5024492.52.7215073.151517−4.530053−3.4817343.0−3.859544−2.4098182.6816770.4325673.5−0.2546571.1635671.6981790.8398444.0−3.870729−3.410456−2.468871−4.2846814.5−2.467705−1.3378393.1050341.4495275.02.65719463.142837−2.195461−1.0821265.5−3.088547−2.1284261.827998−0.2978066.0−0.6976810.1664801.8939560.9111066.5−1.516066−1.314321−2.716662−3.1848047.0−1.203530−0.6671531.4280960.1322087.51.1193431.305388−1.469884−0.7397668.0−0.711474−0.424856−0.134413−0.8057768.50.2809830.492073−0.270894−0.1410059.00.0005350.142655−0.714078−0.6680829.5−0.0254670.097831−0.166414−0.24552010.00.1016260.181401−0.299729−0.22646311.50.0287090.062857−0.127231−0.10889912.00.0200330.045478−0.101927−0.08857212.50.0142750.033330−0.074386−0.06529413.00.0118210.025848−0.056715−0.04851613.50.0092020.019420−0.043685−0.03673214.00.0068960.014234−0.032887−0.02725214.50.0052360.010356−0.024558−0.01981315.00.0039510.007358−0.018224−0.01413715.50.0029160.004999−0.013279−0.00973016.00.0021050.003155−0.009405−0.00627116.50.0014710.001710−0.006375−0.00355617.00.0009690.000569−0.003990−0.00141317.50.000570−0.000338−0.0020980.00029118.00.000251−0.001064−0.0005890.00165818.5−5.147413−0.0016510.0006240.00276219.0−0.000213−0.0021280.0016090.00366319.5−0.000384−0.0025210.0024150.00440520.0−0.000525−0.0028470.0030820.005023Table 4
Error dynamics of MFNNMD time taken for the system to converge to zero at 42.0 s.
te1e2e3e42.0−2.022495−0.0656862.8530491.2055454.0−4.080703−2.2932182.6875970.3114666.0−1.320927−1.122006−5.013653−5.2327688.0−0.2143951.0824451.6866270.85314210.0−2.728065−1.6137152.4296910.46087712.0−2.354545−1.618750−1.338532−2.75633514.0−0.4100800.051628−1.815041−1.83897616.0−0.1348540.233707−1.195806−1.21348918.00.0025740.298025−0.846826−0.83513020.00.0518010.297613−0.650303−0.63022322.00.0599290.260617−0.534643−0.51234224.00.0602170.222786−0.450020−0.42467126.00.0574510.189368−0.378857−0.35250928.00.0529520.160436−0.318578−0.29302730.00.0476830.135605−0.267669−0.24396932.00.0422460.114409−0.224808−0.20348334.00.0370000.096395−0.188799−0.17003136.00.0321380.081131−0.158585−0.14234738.00.0277460.068226−0.133245−0.11939740.00.0238480.057329−0.111995−0.10033342.00.0204280.048135−0.094166−0.08446744.00.0174550.040379−0.079199−0.07123546.00.0148860.033835−0.066623−0.06017848.00.0126740.028311−0.056043−0.05091850.00.0107770.023645−0.047133−0.04314952.00.0091520.019700−0.039617−0.03661754.00.0077630.016360−0.033269−0.03111356.00.0065750.013529−0.027897−0.02646558.00.0055590.011126−0.023343−0.02253260.00.0046900.009082−0.019476−0.01919762.00.0039460.007340−0.016184−0.01636064.00.0033080.005853−0.013377−0.01394366.00.0027600.004580−0.010976−0.01187768.00.0022890.003488−0.008918−0.01010770.00.0018830.002548−0.007148−0.008585We choose alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960] because based on the previous study, these values are most stable compared to other values of alpha. Besides, a fractional-order chaotic system will show chaotic characteristics when the alpha value is 0.95 and above. We decided to choose it and run the test on MATLAB to verify it, and we found that the error synchronization approaches zero when the systems are synchronized with each other. We also tried to run MATLAB with different alpha values, summarized the system’s error synchronization in Table2, and provided the figures as the systems start to lose their chaotic characteristics when the value of alpha decreases. These are shown in Figures 6 and 10. Besides, the main objective of our paper is to prove that with different values of alpha and time delay, there exists a chaos synchronization. Moreover, we prove that chaos synchronization occurs with the proposed control function introduced in the paper.Besides, we chooseτ = [0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17] for practicability of calculation. In fact, the value of τ can be of any amount as long as the chaos characteristics are not lost. Some problems are hard for computers because of the number of combinations or the size of the data. Hence, they are chosen to solve them efficiently and avoid incorrect computations in results. We also show that these parameters show the chaotic behavior from Figures 5 and 6.


## 4.1. Projective Chaos Synchronization of MFNNMD Systems

### 4.1.1. Experiment A: Synchronization of MFNNMD Systems with Fixed Time Delay,τ = 0.11

Assuming that two MFNNMD systems are synchronized with each other, define the drive system as(23)Dα1X1t=−c1X1+a11f1X1t+a12f2X2t+a13f3X3t+a14f4X4t+g11f1X1t−τ1+g12f2X2t−τ1+g13f3X3t−τ1+g14f4X4t−τ1+φ1,Dα2X2t=−c2X2+a21f1X2t+a22f2X2t+a23f3X3t+a24f4X4t+g21f1X1t−τ2+g22f2X2t−τ2+g23f3X3t−τ2+g24f4X4t−τ2+φ2,Dα3X3t=−c3X3+a31f1X1t+a32f2X2t+a33f3X3t+a34f4X4t+g31f1X1t−τ3+g32f2X2t−τ3+g33f3X3t−τ3+g34f4X4t−τ3+φ3,Dα4X4t=−c4X4+a41f1X1t+a42f2X2t+a43f3X3t+a44f4X4t+g41f1X1t−τ4+g42f2X2t−τ4+g43f3X3t−τ4+g44f4X4t−τ4+φ4.And, define the response system as(24)Dα5ϒ1t=−d1ϒ1+b11f1ϒ1t+b12f2ϒ2t+b13f3ϒ3t+a14f4ϒ4t+h11f1ϒ1t−τ1+h12f2ϒ2t−τ5+h13f3ϒ3t−τ5+h14f4ϒ4t−τ5+μ1+ϑ1t,Dα6ϒ2t=−d2ϒ2+b21f1ϒ2t+b22f2ϒ2t+b23f3ϒ3t+b24f4ϒ4t+h21f1ϒ1t−τ6+h22f2ϒ2t−τ6+h23f3ϒ3t−τ6+h24f4ϒ4t−τ6+μ2+ϑ2t,Dα7ϒ3t=−d3ϒ3+b31f1ϒ1t+b32f2ϒ2t+b33f3ϒ3t+b34f4ϒ4t+h31f1ϒ1t−τ7+h32f2ϒ2t−τ7+h33f3ϒ3t−τ7+h34f4ϒ4t−τ7+μ3+ϑ3t,Dα8ϒ4t=−d4ϒ4+b41f1ϒ1t+b42f2ϒ2t+b43f3ϒ3t+b44f4ϒ4t+h41f1ϒ1t−τ8+h42f2ϒ2t−τ8+h43f3ϒ3t−τ8+h44f4ϒ4t−τ8+μ4+ϑ4t,with the following parameters: α1=0.985,α2=0.980,α3=0.975,α4=0.970,α5=0.965,α6=0.960,α7=0.955,α8=0.950,c1=c4=−9.5,c2=−10.5,c3=−3.7,a11=2.0,a12=0.5,a13=5.5,a14=2,a21=0.5,a22=0.5,a23=5.1,a24=0.5,a31=0.5,a32=1.0,a33=−5.5,a34=5.1,a41=0.5,a42=1.0,a43=−9.5,a44=0.5g11=7,g12=7,g13=4.1,g14=7,g21=1,g22=1,g23=2.5,g24=1,g31=0.1,g32=−10.1,g33=4.5,g34=0.1,g41=7,g42=7,g43=4.1,g44=7,d1=d4=−1,d2=−3,d3=−7.7, b11=0.01,b12=0.3,b13=7.5,b14=0.01,b21=0.01,b22=7.5,b23=0.01,b24=0.01,b31=0.01,b32=3,b33=−5.5,b34=−5.5,b41=0.01,b42=3,b43=3,b44=−5.5, h11=0.3, h12=5.5, h13=5.5, h14=0.3, h21=0.5, h22=7.5, h23=0.01, h24=0.5, h31=0.1, h32=−1.5, h33=1.5, h34=0.1, h41=0.1, h42=5.5, h43=5.5, h44=0.1, f1.=f2.=f3.=f4.=tanhX, φ1=φ2=φ3=φ4=0, μ1=μ2=μ3=μ4=0, and ϑ1t=ϑ2t=ϑ3t=ϑ4t=1.With the above initial values, parameters, and controller gain, we solve the numerical solutions of DS (21), RS (22), and error system (11) using MATLAB tools. Furthermore, we use the step-by-step iteration method to solve the fractional differential-difference equations part. Then, they are illustrated in Figures 1–4.Figure 1
Experiment A: phase portrait of the chaotic attractor of MFNNMD systems (21) and (22).Figure 2
Experiment A: phase portrait of theX and Y of MFNNMD systems (21) and (22), when τ=0.11 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960].Figure 3
Experiment A: state trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)Figure 4
Experiment A: error trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)A graphical representation in phase portrait of the consequences of MFNNMD synchronization is shown in Figure1 using X and ϒ, as the dependent variables. At the same time, t is an independent variable function. Figure 2 shows a phase portrait of synchronization of MFNNMD, when τ = 0.11 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960]. Chaos synchronization of the MFNNMD system with and without the controller activation is shown in Figure 3. As illustrated in Figure 4, the dynamic error is unable to converge to zero when the controller is not engaged. However, when the controller is activated, the dynamic error does converge to zero in a limited time. As illustrated in Figure 4, master-slave systems (21) and (22) exhibit global Mittag–Leffler asymptotic synchronization, confirming the validity and effectiveness of the criteria specified in Theorem 1.


### 4.1.2. Experiment B: Synchronization of Two MFNNMD Systems with Multi-Time Delay,τ = [0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17]

The numerical experiment of synchronization of two MFNNMD systems is demonstrated in this section. In this experiment, we consider system changes at different fractional orders and different time delays. The same systems of DS (21) and (22) are used. Figure 5 shows that the systems have chaotic characteristics, while Figure 6 shows a phase portrait of synchronization of MFNNMD, when τ=0.10,0.11,0.12,0.13,0.14,0.15,0.16,0.17 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960]. The synchronization of the MFNNMD system with and without the controller activation is shown in Figure 7. Furthermore, for the error dynamics in Figure 8, we can see that the dynamic error is unable to converge to zero when the controller is not engaged. However, when the controller is activated, the dynamic error does converge to zero in a limited time.Figure 5
Experiment B: phase portrait of the chaotic attractor of MFNNMD systems (21) and (22).Figure 6
Experiment B: phase portrait ofX and Y of MFNNMD systems (21) and (22), when τ=0.10,0.11,0.12,0.13,0.14,0.15,0.16,0.17 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960].Figure 7
Experiment B: state trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)Figure 8
Experiment B: error trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)


## 4.1.1. Experiment A: Synchronization of MFNNMD Systems with Fixed Time Delay,τ = 0.11

Assuming that two MFNNMD systems are synchronized with each other, define the drive system as(23)Dα1X1t=−c1X1+a11f1X1t+a12f2X2t+a13f3X3t+a14f4X4t+g11f1X1t−τ1+g12f2X2t−τ1+g13f3X3t−τ1+g14f4X4t−τ1+φ1,Dα2X2t=−c2X2+a21f1X2t+a22f2X2t+a23f3X3t+a24f4X4t+g21f1X1t−τ2+g22f2X2t−τ2+g23f3X3t−τ2+g24f4X4t−τ2+φ2,Dα3X3t=−c3X3+a31f1X1t+a32f2X2t+a33f3X3t+a34f4X4t+g31f1X1t−τ3+g32f2X2t−τ3+g33f3X3t−τ3+g34f4X4t−τ3+φ3,Dα4X4t=−c4X4+a41f1X1t+a42f2X2t+a43f3X3t+a44f4X4t+g41f1X1t−τ4+g42f2X2t−τ4+g43f3X3t−τ4+g44f4X4t−τ4+φ4.And, define the response system as(24)Dα5ϒ1t=−d1ϒ1+b11f1ϒ1t+b12f2ϒ2t+b13f3ϒ3t+a14f4ϒ4t+h11f1ϒ1t−τ1+h12f2ϒ2t−τ5+h13f3ϒ3t−τ5+h14f4ϒ4t−τ5+μ1+ϑ1t,Dα6ϒ2t=−d2ϒ2+b21f1ϒ2t+b22f2ϒ2t+b23f3ϒ3t+b24f4ϒ4t+h21f1ϒ1t−τ6+h22f2ϒ2t−τ6+h23f3ϒ3t−τ6+h24f4ϒ4t−τ6+μ2+ϑ2t,Dα7ϒ3t=−d3ϒ3+b31f1ϒ1t+b32f2ϒ2t+b33f3ϒ3t+b34f4ϒ4t+h31f1ϒ1t−τ7+h32f2ϒ2t−τ7+h33f3ϒ3t−τ7+h34f4ϒ4t−τ7+μ3+ϑ3t,Dα8ϒ4t=−d4ϒ4+b41f1ϒ1t+b42f2ϒ2t+b43f3ϒ3t+b44f4ϒ4t+h41f1ϒ1t−τ8+h42f2ϒ2t−τ8+h43f3ϒ3t−τ8+h44f4ϒ4t−τ8+μ4+ϑ4t,with the following parameters: α1=0.985,α2=0.980,α3=0.975,α4=0.970,α5=0.965,α6=0.960,α7=0.955,α8=0.950,c1=c4=−9.5,c2=−10.5,c3=−3.7,a11=2.0,a12=0.5,a13=5.5,a14=2,a21=0.5,a22=0.5,a23=5.1,a24=0.5,a31=0.5,a32=1.0,a33=−5.5,a34=5.1,a41=0.5,a42=1.0,a43=−9.5,a44=0.5g11=7,g12=7,g13=4.1,g14=7,g21=1,g22=1,g23=2.5,g24=1,g31=0.1,g32=−10.1,g33=4.5,g34=0.1,g41=7,g42=7,g43=4.1,g44=7,d1=d4=−1,d2=−3,d3=−7.7, b11=0.01,b12=0.3,b13=7.5,b14=0.01,b21=0.01,b22=7.5,b23=0.01,b24=0.01,b31=0.01,b32=3,b33=−5.5,b34=−5.5,b41=0.01,b42=3,b43=3,b44=−5.5, h11=0.3, h12=5.5, h13=5.5, h14=0.3, h21=0.5, h22=7.5, h23=0.01, h24=0.5, h31=0.1, h32=−1.5, h33=1.5, h34=0.1, h41=0.1, h42=5.5, h43=5.5, h44=0.1, f1.=f2.=f3.=f4.=tanhX, φ1=φ2=φ3=φ4=0, μ1=μ2=μ3=μ4=0, and ϑ1t=ϑ2t=ϑ3t=ϑ4t=1.With the above initial values, parameters, and controller gain, we solve the numerical solutions of DS (21), RS (22), and error system (11) using MATLAB tools. Furthermore, we use the step-by-step iteration method to solve the fractional differential-difference equations part. Then, they are illustrated in Figures 1–4.Figure 1
Experiment A: phase portrait of the chaotic attractor of MFNNMD systems (21) and (22).Figure 2
Experiment A: phase portrait of theX and Y of MFNNMD systems (21) and (22), when τ=0.11 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960].Figure 3
Experiment A: state trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)Figure 4
Experiment A: error trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)A graphical representation in phase portrait of the consequences of MFNNMD synchronization is shown in Figure1 using X and ϒ, as the dependent variables. At the same time, t is an independent variable function. Figure 2 shows a phase portrait of synchronization of MFNNMD, when τ = 0.11 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960]. Chaos synchronization of the MFNNMD system with and without the controller activation is shown in Figure 3. As illustrated in Figure 4, the dynamic error is unable to converge to zero when the controller is not engaged. However, when the controller is activated, the dynamic error does converge to zero in a limited time. As illustrated in Figure 4, master-slave systems (21) and (22) exhibit global Mittag–Leffler asymptotic synchronization, confirming the validity and effectiveness of the criteria specified in Theorem 1.


## 4.1.2. Experiment B: Synchronization of Two MFNNMD Systems with Multi-Time Delay,τ = [0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17]

The numerical experiment of synchronization of two MFNNMD systems is demonstrated in this section. In this experiment, we consider system changes at different fractional orders and different time delays. The same systems of DS (21) and (22) are used. Figure 5 shows that the systems have chaotic characteristics, while Figure 6 shows a phase portrait of synchronization of MFNNMD, when τ=0.10,0.11,0.12,0.13,0.14,0.15,0.16,0.17 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960]. The synchronization of the MFNNMD system with and without the controller activation is shown in Figure 7. Furthermore, for the error dynamics in Figure 8, we can see that the dynamic error is unable to converge to zero when the controller is not engaged. However, when the controller is activated, the dynamic error does converge to zero in a limited time.Figure 5
Experiment B: phase portrait of the chaotic attractor of MFNNMD systems (21) and (22).Figure 6
Experiment B: phase portrait ofX and Y of MFNNMD systems (21) and (22), when τ=0.10,0.11,0.12,0.13,0.14,0.15,0.16,0.17 and alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960].Figure 7
Experiment B: state trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)Figure 8
Experiment B: error trajectories of MFNNMD systems (21) and (22). (a) Without control activation. (b) With control activation.
(a)(b)


## 4.2. Effects on Chaos Synchronization ofα-Order and τ-Time Delay

According to the literature,α-order affects the performance of fractional-order chaotic dynamical systems. As previously demonstrated, time delay affects the performance of chaotic dynamical systems. In the actual world, time delays are pervasive and unavoidable. The time-delay dynamic system exhibits complicated dynamic characteristics that are frequently utilized to illustrate time-delayed engineering issues. In this paper, two NN chaotic systems that act as DS and RS are used to examine the dynamic performance from the effect of multi-fractional order and multi-time delay on nonlinear dynamic systems. Furthermore, the capability of the SMC design in this research is studied to prove that it can control the MFNNMD system.From Figure2, we consider system changes at different fractional orders with the alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960] while fixing delay τ = 0.11. We can learn from Figure 2 that the system becomes chaotic and the system trajectory becomes periodic. The system continues to be chaotic as different values of alpha are applied, for example, alpha = [0.895, 0.890, 0.885, 0.880, 0.875, 0.870, 0.865, 0.860] and alpha = [0.795, 0.790, 0.785, 0.780, 0.775, 0.770, 0.765, 0.760]. From Figures 9(a) and 9(b), if we continue decreasing the value of fractional orders, we can see that the systems slowly produce the intermittent chaos phenomenon. Finally, when the value of alpha = [0.695, 0.690, 0.685, 0.680, 0.675, 0.670, 0.665, 0.660], as shown in Figure 9(c), the system will be unstable and lose its chaotic characteristic.Figure 9
Phase portrait of synchronization of MFNNMD systems (21) and (22) when τ=0.11: (a) alpha = [0.895, 0.890, 0.885, 0.880, 0.875, 0.870, 0.865, 0.860], (b) alpha = [0.795, 0.790, 0.785, 0.780, 0.775, 0.770, 0.765, 0.760], and (c) alpha = [0.695, 0.690, 0.685, 0.680, 0.675, 0.670, 0.665, 0.660].
(a)(b)(c)This section addresses the effects of altering theα-order and τ-time delay on chaotic synchronization. We summarised some observations about the error functions for various values of α at specified time values in Table 1. As shown in Table 1, the synchronization error reduces as the order of α is raised. In other words, the synchronization process begins earlier with a greater value of α.Table 1
Dynamic errors for various values of alpha (α) and τ=0.11.
Value of alphate1e2e3e40.20.1246582.168321−2.763986−3.362894α1=0.9950.4−2.401059−1.517229−3.991367−4.667138α2=0.9900.6−4.364460−2.5578561.480212−1.060209α3=0.9850.8−3.045370−1.1418782.9168640.961262α4=0.9801.0−0.8892771.0387212.0092670.697992α5=0.9751.22.8487534.206538−2.248520−1.196980α6=0.9701.4−0.405743−0.055542−5.719650−5.521897α7=0.9651.6−5.144471−3.832519−0.562004−3.244713α8=0.9601.8−3.941395−2.2449552.7784120.5208822.0−2.816576−1.1767823.2467071.5024490.20.1210471.547881−3.908083−3.985490α1=0.8950.4−3.456733−1.6627450.398536−1.882546α2=0.8900.6−0.8669101.0643611.021344−0.378369α3=0.8850.81.6289602.425840−4.054332−3.501186α4=0.8801.0−3.634861−2.180439−0.506693−2.636411α5=0.8751.2−1.890809−0.2274391.540518−0.197372α6=0.8701.41.4143752.816090−1.083584−0.738211α7=0.8651.6−1.924365−1.241058−3.434225−4.093808α8=0.8601.8−2.284485−0.8426591.033461−0.8406262.0−0.1940501.2228200.563513−0.2297260.2−2.602491−0.940676−1.924003−3.521095α1=0.7950.40.2094121.830452−1.281248−1.679577α2=0.7900.6−1.4367650.027426−1.132682−2.296571α3=0.7850.80.0587711.260416−1.964939−2.089092α4=0.7801.0−0.8820700.379702−1.004009−1.839316α5=0.7751.2−0.3489530.717825−1.659654−1.952206α6=0.7701.4−0.5233440.531580−1.125355−1.656581α7=0.7651.6−0.3552090.581332−1.355911−1.682493α8=0.7601.8−0.3794220.508523−1.144286−1.5260372.0−0.2921120.519874−1.172738−1.4683460.2−0.8368290.855342−1.077397−2.142311α1=0.6950.4−0.7606790.667618−1.321963−2.095273α2=0.6900.6−0.6768700.599926−1.329555−1.973587α3=0.6850.8−0.5980960.566648−1.282252−1.845552α4=0.6801.0−0.5360080.536812−1.225057−1.731554α5=0.6751.2−0.4847280.508685−1.171231−1.632134α6=0.6701.4−0.4397690.483483−1.121826−1.543438α7=0.6651.6−0.3996540.460903−1.075778−1.462798α8=0.6601.8−0.3637130.440376−1.032476−1.3887362.0−0.3314080.421524−0.991581−1.320280Based on Experiment A, Table1 contains some results on the dynamic error for different order values in system (21) and system (22). Here, trials are conducted with a fixed delay value of τ = 0.11. It is evident that as the order drops, the synchronization error increases first and subsequently reduces, which is not the case in [43]. Simultaneously, we sum up some data related to the dynamic error for a different combination of fractional orders with the fixed time delay in systems (21) and (22). As seen in Table 1, the dynamic errors e1t,e2t,e3t, and e4t converge to zero as time increases. Precisely, the error function reduces first, then increases, and eventually decreases over the time points for all sets of fractional orders, and we show the evidence as in Figure 9.For Experiment B, we consider system changes at different fractional orders and different time delays. Whenτ1=τ2=⋯=τn=τ, MFNNMD with multidelay can be treated as a single MFNNMD, as explained in Experiment A. Indeed, neurons may have varying communication delays, necessitating the consideration of varied delays in neural networks. Three examples with different values of fractional orders are given with different values of time delays. Figures 10(a) and 10(b) show that the system becomes chaotic, and Figure 10(c) shows that the systems start to become unstable and lose their chaotic characteristics. The summarization for Experiment B is shown in Table 2. All the dynamic errors with different fractional orders and time delay values show the same performance as with the fixed time delays. The error function reduces first, then increases, and eventually decreases over time. Finally, detailed time taken for dynamic errors converging to zero for Experiments A and B are shown in Tables 3 and 4.Figure 10
Phase portrait of synchronization of MFNNMD systems (21) and (22) when τ=0.10,0.11,0.12,0.13,0.14,0.15,0.16,0.17: (a) alpha = [0.895, 0.890, 0.885, 0.880, 0.875, 0.870, 0.865, 0.860], (b) alpha = [0.795, 0.790, 0.785, 0.780, 0.775, 0.770, 0.765, 0.760], and (c) alpha = [0.695, 0.690, 0.685, 0.680, 0.675, 0.670, 0.665, 0.660].
(a)(b)(c)Table 2
Dynamic errors for various values of alpha (α) and τ = [0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17].
Value of alphate1e2e3e40.2−2.022495−0.0656862.8530491.205545α1=0.9950.4−4.080703−2.2932182.6875970.311466α2=0.9900.6−1.320927−1.122006−5.013653−5.232768α3=0.9850.8−0.2143951.0824451.6866270.853142α4=0.9801.0−2.728065−1.6137152.4296910.460877α5=0.9751.2−2.354545−1.618750−1.338532−2.756335α6=0.9701.4−0.4100800.051628−1.815041−1.838976α7=0.9651.6−0.1348540.233707−1.195806−1.213489α8=0.9601.80.0025740.298025−0.846826−0.8351302.00.0518010.297613−0.650303−0.6302230.20.9403632.652079−0.323689−0.373100α1=0.8950.41.0327232.446092−0.593181−0.486812α2=0.8900.61.5165372.179133−2.695674−2.141750α3=0.8850.8−1.193599−0.160199−1.124218−1.989497α4=0.8801.0−0.0950080.720590−0.875975−1.115168α5=0.8751.2−0.1875240.466583−1.086851−1.240937α6=0.8701.4−0.1507880.407471−0.926858−1.065040α7=0.8651.6−0.1025130.370933−0.828564−0.929539α8=0.8601.8−0.0671050.336450−0.742594−0.8144202.0−0.0408250.304890−0.665046−0.7144290.2−0.6284860.752827−1.065626−1.705763α1=0.7950.4−0.6166810.508169−1.175186−1.674920α2=0.7900.6−0.5014800.455924−1.084421−1.493109α3=0.7850.8−0.4131070.413356−0.992929−1.334976α4=0.7801.0−0.3419050.377650−0.910913−1.198937α5=0.7751.2−0.2839420.346661−0.836579−1.080232α6=0.7701.4−0.2364180.319153−0.768815−0.975700α7=0.7651.6−0.1972440.294346−0.706797−0.883013α8=0.7601.8−0.1648140.271723−0.649864−0.8003622.0−0.1378680.250926−0.597470−0.7263000.2−0.7264360.450155−0.988908−1.560065α1=0.6950.4−0.6077640.387185−0.887064−1.357983α2=0.6900.6−0.5207290.344249−0.808443−1.210932α3=0.6850.8−0.4517190.310852−0.741817−1.091060α4=0.6801.0−0.3950140.283278−0.683254−0.988966α5=0.6751.2−0.3474100.259682−0.630746−0.899911α6=0.6701.4−0.3068540.238995−0.583070−0.821019α7=0.6651.6−0.2719200.220533−0.539396−0.750353α8=0.6601.8−0.2415670.203835−0.499119−0.6865142.0−0.2150070.188574−0.461779−0.628448Table 3
Error dynamics of MFNNMD time taken for the system to converge to zero at 12.5 s.
te1e2e3e40.5−4.543203−3.139601−0.945464−3.2805291.0−0.8892771.0387212.0092670.6979921.5−3.304613−2.685071−3.987378−5.3158202.0−2.816576−1.1767823.2467071.5024492.52.7215073.151517−4.530053−3.4817343.0−3.859544−2.4098182.6816770.4325673.5−0.2546571.1635671.6981790.8398444.0−3.870729−3.410456−2.468871−4.2846814.5−2.467705−1.3378393.1050341.4495275.02.65719463.142837−2.195461−1.0821265.5−3.088547−2.1284261.827998−0.2978066.0−0.6976810.1664801.8939560.9111066.5−1.516066−1.314321−2.716662−3.1848047.0−1.203530−0.6671531.4280960.1322087.51.1193431.305388−1.469884−0.7397668.0−0.711474−0.424856−0.134413−0.8057768.50.2809830.492073−0.270894−0.1410059.00.0005350.142655−0.714078−0.6680829.5−0.0254670.097831−0.166414−0.24552010.00.1016260.181401−0.299729−0.22646311.50.0287090.062857−0.127231−0.10889912.00.0200330.045478−0.101927−0.08857212.50.0142750.033330−0.074386−0.06529413.00.0118210.025848−0.056715−0.04851613.50.0092020.019420−0.043685−0.03673214.00.0068960.014234−0.032887−0.02725214.50.0052360.010356−0.024558−0.01981315.00.0039510.007358−0.018224−0.01413715.50.0029160.004999−0.013279−0.00973016.00.0021050.003155−0.009405−0.00627116.50.0014710.001710−0.006375−0.00355617.00.0009690.000569−0.003990−0.00141317.50.000570−0.000338−0.0020980.00029118.00.000251−0.001064−0.0005890.00165818.5−5.147413−0.0016510.0006240.00276219.0−0.000213−0.0021280.0016090.00366319.5−0.000384−0.0025210.0024150.00440520.0−0.000525−0.0028470.0030820.005023Table 4
Error dynamics of MFNNMD time taken for the system to converge to zero at 42.0 s.
te1e2e3e42.0−2.022495−0.0656862.8530491.2055454.0−4.080703−2.2932182.6875970.3114666.0−1.320927−1.122006−5.013653−5.2327688.0−0.2143951.0824451.6866270.85314210.0−2.728065−1.6137152.4296910.46087712.0−2.354545−1.618750−1.338532−2.75633514.0−0.4100800.051628−1.815041−1.83897616.0−0.1348540.233707−1.195806−1.21348918.00.0025740.298025−0.846826−0.83513020.00.0518010.297613−0.650303−0.63022322.00.0599290.260617−0.534643−0.51234224.00.0602170.222786−0.450020−0.42467126.00.0574510.189368−0.378857−0.35250928.00.0529520.160436−0.318578−0.29302730.00.0476830.135605−0.267669−0.24396932.00.0422460.114409−0.224808−0.20348334.00.0370000.096395−0.188799−0.17003136.00.0321380.081131−0.158585−0.14234738.00.0277460.068226−0.133245−0.11939740.00.0238480.057329−0.111995−0.10033342.00.0204280.048135−0.094166−0.08446744.00.0174550.040379−0.079199−0.07123546.00.0148860.033835−0.066623−0.06017848.00.0126740.028311−0.056043−0.05091850.00.0107770.023645−0.047133−0.04314952.00.0091520.019700−0.039617−0.03661754.00.0077630.016360−0.033269−0.03111356.00.0065750.013529−0.027897−0.02646558.00.0055590.011126−0.023343−0.02253260.00.0046900.009082−0.019476−0.01919762.00.0039460.007340−0.016184−0.01636064.00.0033080.005853−0.013377−0.01394366.00.0027600.004580−0.010976−0.01187768.00.0022890.003488−0.008918−0.01010770.00.0018830.002548−0.007148−0.008585We choose alpha = [0.995, 0.990, 0.985, 0.980, 0.975, 0.970, 0.965, 0.960] because based on the previous study, these values are most stable compared to other values of alpha. Besides, a fractional-order chaotic system will show chaotic characteristics when the alpha value is 0.95 and above. We decided to choose it and run the test on MATLAB to verify it, and we found that the error synchronization approaches zero when the systems are synchronized with each other. We also tried to run MATLAB with different alpha values, summarized the system’s error synchronization in Table2, and provided the figures as the systems start to lose their chaotic characteristics when the value of alpha decreases. These are shown in Figures 6 and 10. Besides, the main objective of our paper is to prove that with different values of alpha and time delay, there exists a chaos synchronization. Moreover, we prove that chaos synchronization occurs with the proposed control function introduced in the paper.Besides, we chooseτ = [0.10, 0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17] for practicability of calculation. In fact, the value of τ can be of any amount as long as the chaos characteristics are not lost. Some problems are hard for computers because of the number of combinations or the size of the data. Hence, they are chosen to solve them efficiently and avoid incorrect computations in results. We also show that these parameters show the chaotic behavior from Figures 5 and 6.


## 5. Conclusions

The chaos synchronization of MFNNMD is evaluated in this article. The conditions for synchronization of MFNNMD systems are examined using FLDM and sliding mode techniques. We have demonstrated that MFNNMD systems can archive chaos synchronization with distinct fractional orders when the parameters and beginning conditions are fixed. Numerical simulations are used to authenticate the suggested technique’s feasibility and effectiveness. It has been demonstrated that the provided strategy is capable of ensuring that all error signals converge to zero and remain within the system. Then, an example of graphical results is given to authenticate our proposed method’s effectiveness. Finally, the impacts of multi-fractional order and multi-time delay on synchronization are addressed in greater detail. In addition, we anticipate some applications to the actual world that relate to secure communication.



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*Source: 9398333-2021-12-30.xml*