Results for Chaos Synchronization with New Multi-Fractional Order of Neural Networks by Multi-Time Delay
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.
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 . 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 , 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 , generalized projective , linear feedback , and SMC . 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 . 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  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 . In , 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: Section 2 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 ). For the u (t) function, the fractional derivative of Caputo order is described aswhere t, is the integer, , is a Gamma function, and .
In this research, we study the following NNs, which we designate to as MFNNMD, where the DS is denoted aswhere , represents the unit number in a MFNNMD, signifies the -th neuron’s pseudo-state variable of the DS, and is the th unit parameters. The external input of the ith unit is , and denote the relation of the unit jth with the th at time and , respectively, is the delayed transmission, and and represent the j-th unit activation function output at time and , respectively.
Also, the vector formThe corresponding RS of MFNNMD is denoted aswhere , represents the unit number in a MFNNMD, represents the th unit’s pseudo-state variable of the RS, and is the th neuron’s parameters. represents the external input of the ith unit, and denote the relation of the unit jth with the ith at time and , respectively, is the delayed transmission, and and denote the jth unit activation function output at time and , respectively. However, 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 setting and , thenand by establishing the two-parameter Mittag–Leffler function by setting , and , thenAccording to Definition 2, it is not difficult to see and .
Definition 3. (see ). Suppose there is a constant of nonzero, , such that sufficiently for every result of (t) and Y (t) for systems (2) and (4) using distinct initial condition, you may obtainTherefore, DS (2) and RS (4) can accomplish globally projective synchronization asymptotically, representing the vector’s Euclidean norm.
Lemma 1. (see ). Assume is a continuous function of differentiable vector value. Hence, for whichever time prompt , we get the following:where .
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 as (). Based on DS (2) and RS (4), the control function () can be designated as follows:where and is the projective coefficient. From (2), (4), and (9), you can get the error system as follows:
Now, (9) and (10) are combined together and the error system of delayed sliding mode dynamics (DDMDEs), , can be defined by
Then, it follows from (11) that
To investigate the stability of DDMDEs (11) in this paper, Theorem 1 is stated.
Theorem 1. (see ). Assume that condition (H) is met and there are positive constants called and in such a way that and and constant matrices that , , , andThen, equation (11) is stable.
Proof of Theorem 1. According to and Theorem 1, we haveNext, we obtain thatGenerate the following function of Lyapunov:As a result of Lemma 1, by applying the Caputo derivative’s upper right along the trajectories, equation (17) can be obtained as follows.
We defined the Lyapunov function byNote thatThen, assuming that there is a constant Δ greater than zero in conjunction with (18) and (19) and Theorem 1, one hasWe construct that (18) assoFinally, we can conclude that DDMDEs (9) are stable as as approaches infinity.
This completes the proof.
Remark 1. When , projective synchronization is standardized by multiple time delays for the complete synchronization of MFNNMD.
Remark 2. When , 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 asAnd, define the response system aswith the following parameters: , , , , , , , , , , , , , , , , , , , , , and .
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.
A graphical representation in phase portrait of the consequences of MFNNMD synchronization is shown in Figure 1 using and , as the dependent variables. At the same time, 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 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.
4.2. Effects on Chaos Synchronization of -Order and -Time Delay
According to the literature, -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 Figure 2, 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.
This section addresses the effects of altering the -order and -time delay on chaotic synchronization. We summarised some observations about the error functions for various values of at specified time values in Table 1. As shown in Table 1, the synchronization error reduces as the order of is raised. In other words, the synchronization process begins earlier with a greater value of .
Based on Experiment A, Table 1 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 . 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 , and 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. When , 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.
We 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 Table 2, 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 choose τ = [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.
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.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
All authors were partially funded by Universiti Malaya under grant no. IIRG001C-2019.
The annex contains a MATLAB code named MFNNMDNocontrolnew. (Supplementary Materials)
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