Pinning multisynchronization of delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations
Introduction
The memristor (Chua, 1971), was first defined by Chua in 1971 as the fourth fundamental circuit element (i.e. the resistor, the capacitor and the inductor). In 2008, the first prototype of memristor was produced in the Hewlett–Packard Labs (Strukov et al., 2008), bringing memristors from virtual to reality. The memristor owns the similar property of pinched hysteresis of neurons, which is considered as an ideal successor to simulate biological synapses in neural networks. Over the past few decades, memristor-based neural networks (MNNs) have drawn increasing attention from scientific researchers in various fields. In MNNs, the synaptic connection between two different neurons is implemented by memristors (Thomas, 2013). Different from conventional resistor-based neural networks, the synaptic connection weights in MNNs are not fixed, but vary in accordance with the characteristics of the memristors. Specifically, the connection weights change abruptly if the memristors used in MNNs are of voltage threshold type.
Considering a real capacitor is usually characterized by fractional calculus, it is reasonable and practical to introduce the fractional capacitor into MNNs to enhance the accuracy of network model and the efficiency of information processing. Recently, the fractional-order MNNs (FMNNs) were proposed (Chen et al., 2014), where the connection weights are realized by memristors and the common capacitors are replaced with fractance. Due to the superior memory capability, FMNNs have various potential applications in scientific and engineering areas, including reservoir computing, neuromorphic computing, brain science, and programmable logic devices (He, Ling et al., 2018, Prezioso et al., 2018, Wen et al., 2018, Wu et al., 2020, Xia and Yang, 2019, Yang et al., 2019, Zhang et al., 2019). These applications heavily rely on the dynamics of neural networks, such as passivity, attractivity, stability and synchronization. Hence, the qualitative analysis of these dynamical behaviors is an important step for practical realization of FMNNs.
In the past few years, some pioneering research has been devoted to investigating the stability and synchronization of neural networks (He, Zhou et al., 2018, Syed Ali et al., 2019, Wu et al., 2019, Zhang et al., 2015, Zhang et al., 2017). For instance, Chen et al. (2014) investigated the existence of unique equilibrium point of FMNNs. By using the Lyapunov-like function method, the criteria for global Mittag–Leffler stabilization and synchronization of FMNNs were established. In Wu and Zeng (2017), aided by a novel fractional derivative inequality and fractional-order differential inclusions, some new criteria for global Mittag-Leffler stabilization of FMNNs were obtained. Besides, projective synchronization criteria for FMNNs with a hybrid controller were given in Bao and Cao (2015). By utilizing a delayed state feedback control law, the quasi-synchronization problems for FMNNs with parameter mismatches were investigated in Huang et al. (2017). In Gu et al. (2016), several sufficient conditions for robust synchronization of FMNNs with parameter uncertainty were proposed via comparison method and Lyapunov theory. There are other interesting and up-to-data results in the stability and synchronization analysis of neural networks, which are shown in Arslan et al., 2020, Luo et al., 2019, Miao et al., 2021, Sun et al., 2020, Wang et al., 2020, Wei et al., 2020, Zeng et al., 2021 and Zhou et al. (2021).
However, the above mentioned results are obtained under the assumption that the addressed networks have only one equilibrium state or chaotic state. In fact, many real systems, such as genetic oscillators, memory systems and biological systems, are characterized by multiple equilibria. Moreover, in some practical applications like associative memory, image processing and pattern recognition, the multistability of neural networks is required. Hence, some research on multistability analysis of neural networks has been carried out very recently (Guo et al., 2020, Liu et al., 2016, Liu et al., 2017, Nie et al., 2019, Qin et al., 2020, Wan et al., 2020, Wan and Wu, 2018). Furthermore, a new concept named dynamical and static multisynchronization was proposed recently to tackle the synchronization problems of neural networks with multiple equilibria (Wang et al., 2017). On this basis, Lv et al. (2018) proposed a robust impulsive control scheme to guarantee the multisynchronization of CNNs with parametric uncertainties, and the multisynchronization of stochastic coupled multistable neural networks was studied in Li et al. (2019). Yao et al. (2019) discussed the hybrid multisynchronization of coupled multistable memristive neural networks. In Peng et al. (2020), authors investigated the -stable multisynchronization issue for memristor-based neural networks with impulsive coupling. However, the aforementioned literatures are related to coupled integer-order neural networks, while, to the authors’ knowledge, studies about multisynchronization of coupled fractional-order memristor-based neural networks (CFMNNs) are still few and far between. The methods used for coupled integer-order neural networks cannot apply to CFMNNs directly. The topic about multisynchronization of CFMNNs still needs further investigation. Moreover, the coupling terms of neural networks involved in these literatures (Li et al., 2019, Lv et al., 2018, Peng et al., 2020, Wang et al., 2017, Yao et al., 2019) are linear coupling. Generally, nonlinearly coupled neural networks are more practical, because state variables in the system may not be observable, and what one can observe is the corresponding nonlinear function. The study about multisynchronization of nonlinearly coupled fractional-order memristor-based neural networks is still an open challenge. In addition, periodicity and almost periodicity of the state variables in neural networks are frequently encountered in practical scenarios. Similarly, it is inevitable to encounter certain disturbances in the circuit implementation of neural networks, which may result in the properties of almost periodicity for connection weights and external inputs (Wang et al., 2009). Hence, it is practical and valuable to consider the almost periodicity when analyzing the dynamics of neural networks.
Motivated by the aforementioned discussions, this paper dedicates to investigate the multisynchronization of delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations. The main contributions of this paper are summarized as follows.
- (1)
A class of delayed nonlinear-coupled fractional-order memristor-based neural networks (NCFMNNs) is built in this paper, which consists of one master subnetwork without controller and identical slave subnetworks with nonlinear couplings and controllers. Besides, the values of memristor connection weights and the external input in the NCFMNNs are considered to be almost periodic. The model can be see as a class of coupled neural networks with subnetworks, where the weighted adjacency matrix is asymmetric. The advantages of this model lie in that all slave subnetworks can synchronize the master subnetwork under the designed controllers, and meanwhile the multistability of the isolated master subnetwork can be tackled conveniently.
- (2)
It is necessary to ensure the multistability of an isolated network in order to achieve synchronization of coupled networks. Therefore, the multistability of the isolated master subnetwork is analyzed first. Based on state-space decomposition, fractional-order Halanay inequality and Caputo derivative properties, a novel multistability criterion is derived such that the master subnetwork with arbitrary activation function will possess multiple locally stable almost periodic orbits or equilibrium points, and the number of equilibrium states is determined by the geometrical configuration of the employed activation function. The type of activation function is not specified here. It can be linear or nonlinear, monotonous or nonmonotonous. Compared with those works (Guo et al., 2020, Liu et al., 2016, Nie et al., 2019, Wan et al., 2020, Wan and Wu, 2018), the conditions given herein are milder and more general.
- (3)
By using graph theory and depth first search algorithm, an effective nodes selection scheme is introduced, and a pinning control law is devised to guarantee the multisynchronization of the considered NCFMNNs. Correspondingly, some sufficient conditions pertaining to the multisynchronization of the addressed NCFMNNs are derived.
The organization of this paper is outlined as follows. In Section 2, the problem description and some preliminaries are presented. Section 3 shows the main results including the multistability analysis for the isolated master subnetwork and the multisynchronization results. Section 4 presents the simulation results. The conclusion lies in Section 5.
Notations: denote the space of -dimensional real matrices. is the domain of positive real numbers. ( or ) indicates the matrix is negative definite (negative semidefinite, positive definite or positive semidefinite). is the set of continuous functions mapping into . represents the maximum eigenvalue of matrix , while denotes the minimum one. denotes the sign function. stands for the -dimensional identity matrix, and . The symbol stands for the operation of Kronecker product. The symbol in a symmetric block matrix represents the elements below the main diagonal. Without additional statements, the definition of symbols is consistent throughout the whole paper.
Section snippets
Preliminaries
In this section, we present the model description and list some necessary definitions and lemmas which will be used in the subsequent analysis.
Definition 1 The fractional integral of order for a function is defined as where and ; is the Gamma function with the formulation .Podlubny, 1999
Definition 2 Suppose that is a differentiable function on . The Caputo fractional derivative of order for the function is defined as Podlubny, 1999
Definition 3 AAlmost Periodicity Wang et al., 2009
Main results
This section presents the theoretical results of multisynchronization for NCFMNNs. As discussed before, when the NCFMNNs (1) reach synchronization, the states of the addressed networks will converge to the master subnetwork. First, we will ensure that the master subnetwork of NCFMNNs has multiple locally stable equilibrium states, which means the NCFMNNs can obtain multiple synchronization manifolds. Then we construct the controller to realize the multisynchronization of the controlled
Simulation results
In this section, we dedicate to demonstrate the effectiveness of the multistability criterion obtained in Theorem 1 and the multisynchronization criterion obtained in Theorem 2. Two simulation examples under different activation functions are carried out.
Case I: Consider a two-neuron NCFMNNs (1) with six subnetworks and time-vary delay. The model parameters are listed as follows:
Conclusion
In this paper, the multisynchronization issue of delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations has been investigated for the first time. The model of NCFMNNs consists of one master subnetwork without controller and N-1 identical slave subnetworks with nonlinear couplings and controllers is built to analyze the multistability and multisynchronization problems expediently. By using state-space decomposition, fractional-order
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 61971111, Grant 62027803, Grant 61601096, Grant 61801089, and Grant 61701095; and in part by the Sichuan Science and Technology Program, China under Grant 2020YFG0044, Grant 2021YFG0200 and Grant 2020YFG0046; and in part by Defense Industrial Technology Development Program, China under Grant JCKY2020110C041.
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