Brief papersAsymptotical synchronization analysis of fractional-order complex neural networks with non-delayed and delayed couplings
Introduction
Neural networks have been extensively researched over the past few decades because of their wide applications in pattern recognition, signal processing, electrical circuits, knowledge engineering, optimization combination, robot control problems and so on [1], [2]. The applications of neural networks heavily depend on their dynamical behaviors. Stability and synchronization are two kinds of important dynamical behaviors of neural networks. Various kinds of stability problems of neural networks have been extensively investigated [3], [4], [5]. The essence of synchronization control is to force a system to track another one by imposing a suitable controller. Up to now, synchronization of neural networks has been successfully applied to secure communication, public channel cryptography, neural cryptography and many other areas. Synchronization of chaotic system and neural networks has become an important area of study. Since Pecora and Carroll [6] proposed the concept of synchronization for the drive-response systems, various kinds of synchronization including phase synchronization, anomalous synchronization, complete synchronization, exponential synchronization and robust synchronization have been widely studied. Besides the classical drive-response mechanism for chaos synchronization, a wide variety of synchronization approaches such as adaptive feedback controlling method, impulsive control method and adaptive design control are developed. Basing on parameter identification and via output or state coupling, Lou et al. [7] study the synchronization of neural networks with time-varying delay. Based on Lyapunov theory, some new sufficient conditions are derived to guarantee the global synchronization of the weighted cellular neural network in [8]. Synchronization problem of chaotic neural networks with mixed time delays is investigated in [9]. A sufficient condition for global asymptotical synchronization of master–slave chaotic neural networks via output feedback impulsive control is established in [10]. In 2020, by constructing Lyapunov functionals and applying some inequality analysis techniques, several delay-dependent exponential synchronization criteria and polynomial synchronization criteria of the drive-response systems are obtained in [11]. Many results on the synchronization of systems of integer order have been obtained.
It is well known that a network may be constituted by two or more circuits. If the current or voltage in one circuit changes, it can affect other circuits and make other circuits to have similar changes. This network is a coupling circuit. Since the capacitor has the characteristic of isolating direct current and alternating current. It can couple the AC signal to the next stage amplifier. Time delays are inevitable due to the limited switching speed of the amplifiers. So coupling exists widely in circuits. Coupling may transfer the energy from one circuit to other circuits. However, there are not many results on synchronization of the coupled neural networks [12], [13].
As a special class of complex neural networks, the coupled neural networks may exhibit more complicated and unpredictable behaviors. Synchronization of coupled neural networks can achieve state consistency for all nodes of networks. Synchronization can play a positive role to avoid negative effects. Since synchronization of a coupled dynamical network can well explain many natural phenomena observed, the theoretical research and practical applications of synchronization for the coupled neural networks is with profound significance. By employing Lyapunov functional method and Kronecker product technique, sufficient conditions are derived to guarantee the global synchronization for static neural networks with hybrid couplings and time delays in [13]. In order to cope with a large number of highly interconnected dynamical units in coupled neural networks, basing on the similar idea in [13], we will construct an appropriate Lyapunov function by using the Kronecker product of matrices and investigate the synchronization of coupled neural networks.
As a generalization of the integer order integration and differentiation to arbitrary real order, fractional calculus was first proposed by Leibniz in 17th century, it can deal with the differential and integral of any arbitrary order. Naturally, in order to better describe the neurons’ dynamical behaviors, a class of fractional-order neural networks is formed when the fractional calculus is incorporated into neural networks. The main characteristics of fractional-order systems are with infinite memories, hereditary and more degrees of freedom [14], [15], [16]. So fractional-order neural networks have more complex dynamic behaviors [17], [18], [19] than the classical integer order systems. Compared with the traditional integer order calculus, fractional-order calculus is a good tool for describing memory and genetic characteristics in processing [20], [21].
One more fundamental difference between fractional-order neural network and integer order neural network is non-locality. The classical differential operators are local operators, the fractional derivatives are not. In order to calculate the n-th order derivative , it is sufficient to know in an arbitrarily small neighbourhood of t. However, to calculate fractional-order derivative for , we need to know throughout the entire interval . Fractional-order neural network can make use of information of all states from the initial time to the current time. So the reality can be accurately described by fractional-order neural networks due to the fractional-order derivatives’ memory and heredity. Different from the classical Leibniz’ formula and the chain rule, Leibniz’ formula and the chain rule for the fractional differential operator correspond a infinite sum, respectively. Since the classical Leibniz’ formula and the chain rule fail to work, more difficulties will lie in the construction of Lyapunov functions and the estimation for their fractional-order derivatives. So most of results on synchronization of complex neural networks focus on the integer order case in the existing literature.
Recently, in [22], out lag synchronization criterion of fractional-order complex neural networks with non-delayed and delayed couplings (FOCNNNDC) is obtained by the Kronecker product knowledge. Cheng et al. [23] study the memristor-based fractional-order neural networks by using Lyapunov method. Some new sufficient conditions on the global Mittag–Leffler stability and synchronization are established in [23]. By taking information on activation functions into account, a fractional-order-dependent Lyapunov function and a new quadratic Lyapunov function are constructed [24]. Based on a fractional-order differential inequality, a Mittag–Leffler synchronization criterion in terms of LMIs is presented for drive-response fractional-order Hopfield neural networks under linear control. Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay is investigated in [25]. Gu et al. [26] study the projective synchronization of fractional-order memristive neural networks with switching jumps mismatch. So far, many excellent results on stability and synchronization of fractional differential systems have been obtained in [27], [28], [29], [30]. Obviously, fractional-order Lyapunov direct method is one of important tools for the dynamic behaviors analysis of complex dynamical networks. How to construct a suitable Lyapunov function is the key to carry out synchronization analysis of fractional-order complex dynamical networks. Time delay widely exists in real networks due to finite information transmission and processing speeds among the units of networks. It has significant effect on dynamical behavior of neural networks. It may cause the degradation of the system performance, even leads to instability [31], [32], [33], [34]. Time delay systems are very important both in theoretical research and practical design of neural networks. Both internal delay and coupling delay should be considered in dynamical networks. Recently, by applying fractional Razumikhin theorem, the global synchronization of fractional complex networks with non-delayed and delayed couplings is dealt in [35]. Zhang et al. [36] discuss the synchronization stability problem of the Riemann–Liouville fractional coupled complex interconnected delayed neural networks. Several synchronization stability criteria are established in terms of Lyapunov functional approach and linear matrix inequality technique. As far as we know, in Caputo sense, how to obtain useful algebraic criteria on synchronization for a fractional nonlinear delayed system still faces serious challenge.
Motivated by the above discussion, this paper aims to investigate Caputo fractional complex networks with non-delayed and delayed couplings. By applying the fractional-order Lyapunov direct method, Kronecker product method, Laplace transform and the comparison principle, several novel sufficient conditions on synchronization are proposed. The main contributions of this paper are summarized as follows:
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A suitable Lyapunov function is constructed by using Kronecker product technique. The fractional-order differential inequality for convex function is used to estimate the fractional-order derivative of Lyapunov function, which can avoid directly computing the Caputo fractional-order derivative of Lyapunov function.
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Different from method used in [36], by using Laplace transform and characteristic equation, together with Kronecker product method, two novel synchronization stability conditions for FOCNNNDC are obtained in terms of matrix inequalities.
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Based on the fractional-order Lyapunov direct method and the comparison principle, a weighted Lyapunov function and a new sufficient condition are also presented to achieve the asymptotical synchronization between fractional-order drive system and the response system.
The rest of the paper is organized as follows. In Section 2, some main definitions such as Caputo fractional calculus and Laplace transform are presented. Some useful lemmas are given. In Section 3, the fractional-order complex neural network model with non-delayed and delayed couplings is introduced. A necessary assumption is given. In Section 4, three asymptotical synchronization criteria for FOCNNNDC are given. In Section 5, three numerical examples are presented to verify the validity of the proposed theorems. The paper is concluded in Section 6.
Notations: In this paper, R is the set of real numbers. denotes the set of positive real numbers. denotes the set of positive integer numbers. is the n-dimensional Euclidean space. denotes the space of real matrices. denotes Banach space of all continuous and n-order differentiable functions. denotes the set of absolute integrable functions on . The superscript T stands for the transpose of matrix or vector. stands for the Kronecker product of matrices A and B. For a vector and denote 1-norm and 2-norm of z, respectively. , . For a matrix denotes 1-norm of matrix Q, . means that matrix Q is symmetric and negative definite. represents that matrix Q is negative semi-definite. represents the n-dimensional identity matrix. denotes a diagonal matrix with elements . The symmetric term in a symmetric matrix is denoted by .
Section snippets
Preliminaries and system description
In this section, we present some important definitions and lemmas, which will be used in later discussion.
Asymptotical synchronization analysis
In this section, we will develop several new asymptotical synchronization criteria. Theorem 1 Suppose . Assume that Assumption 1 holds. Then system (15) and system (16) achieve globally asymptotical synchronization if there exist positive definite diagonal matrices and such thatandwhere . Proof Construct the following Lyapunov
Numerical examples
In this section, three examples are provided to demonstrate effectiveness of our main results. Example 1 Consider the following 2-dimensional fractional-order complex neural networks with non-delayed and delayed couplings:where , . . For the drive system (69),
Conclusions
In this paper, the synchronization is investigated for a class of fractional-order complex neural networks with non-delayed and delayed couplings. Based on Kronecker product method, a new Lyapunov function is constructed. By using Laplace transform and the comparison principle of fractional-order system, two novel sufficient conditions on synchronization of fractional-order complex neural networks with non-delayed and delayed couplings are derived in terms of matrix inequalities. A new
CRediT authorship contribution statement
Li Li: Computation, Visualization, Validation. Xinge Liu: Supervision, Conceptualization, Funding acquisition. Meilan Tang: Doctor, Methodology, Investigation. Shuailei Zhang: Visualization. Xianming Zhang: Writing - review & editing.
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
The authors would like to thank the editor and anonymous reviewers for their valuable comments and suggestions. This work is partly supported by the National Science Foundation of China(NSFC) under Grant Nos.61773404 and 61271355.
Li Li was born in 1994. She received the B.S. degree from Central South University, Changsha, Hunan, China in 2016 in Information and Computing Science. She is currently pursuing the M.S. degree in Operational Research and Cybernetics, Central South University. Her current research interest is stability theory of dynamical systems.
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2023, NeurocomputingCitation Excerpt :However, compared with integer-order derivative, fractional-order derivative can more accurately describe numerous physical systems [19] for the reason that it has the characteristics of infinite memory and hereditary. Therefore, plenty of authors have expressed deep concern over the synchronization for coupled fractional-order neural networks (CFONNs), and a good deal of important results have been obtained [20–28]. In [20], two global synchronization criteria for coupled fractional-order recurrent neural networks with sequentially connected topologies were put forward.
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2022, Mathematics and Computers in SimulationCitation Excerpt :At the same time, synchronization as an important dynamical behaviour is favoured by scholars. It is worth mentioning that various synchronization methods have been studied such as phase synchronization [13], lag synchronization [40], anti-synchronization [29,31], finite-time synchronization [42], fixed-time synchronization [12], Mittag-Leffler synchronization [7,8,23,33,36], quasi-uniform synchronization [34,38], adaptive synchronization [5,14], asymptotical synchronization [2,18], projective synchronization [39] and so on, in which the factors including the delay, impulses, fractional-order derivative operator and quaternion-value have not been completely considered. In order to further realize the synchronization of FONNs, various types of control schemes had been extensively applied to arrive at the desired purpose including delay-feedback control [36], intermittent control [42], sliding mode control [7], linear feedback control [8], backstepping control [14], quantized output control [5] and pinning control [2].
Li Li was born in 1994. She received the B.S. degree from Central South University, Changsha, Hunan, China in 2016 in Information and Computing Science. She is currently pursuing the M.S. degree in Operational Research and Cybernetics, Central South University. Her current research interest is stability theory of dynamical systems.
Xinge Liu was born in 1969, received the B.S. and M.S. degrees from Hunan Normal University, Changsha, China and the Ph.D. degree from Central South University, Changsha, China, all in mathematics, in 1991, 1994, and 2001, respectively.
In May 1994, he joined the School of Mathematics and Statistics, Central South University, where he is currently a professor. From 2002 to 2004, he was a postdoctor of Automatic Control Engineering, Central South University. From October 2004 to September 2006, he was a postdoctor and visiting scholar in the School of Computer Science, Cardiff University, Cardiff, U.K. In 2008 he was approved to be a doctorial advisor. His research interests include neural networks, stability theory, functional differential (difference) equations, inequality theory, and harmonic analysis.
Meilan Tang was born in 1972, received the B.S. degree from Hunan Normal University, Changsha, China and the M.S. and Ph.D. degrees from Central South University, Changsha, China, all in mathematics, in 1997, 2005, and 2011, respectively. In August 2001, she joined the School of Mathematics and Statistics, Central South University, where she is currently an associate professor. Her research interests include functional differential (difference) equations and neural networks.
Shuailei Zhang was born in 1997. He received the B.S. degree from Central South University, Changsha, Hunan, China in 2018 in Information and Computing Science. He is currently pursuing the M.S. degree in Operational Research and Cybernetics, Central South University. His current research interest is stability theory of dynamical systems.
Xian-Ming Zhang (Senior Member, IEEE) received the M.Sc. degree in applied mathematics and the Ph.D. degree in control theory and engineering from Central South University, Changsha, China, in 1992 and 2006, respectively.
In 1992, he joined Central South University, where he was an Associate Professor with the School of Mathematics and Statistics. From 2007 to 2014, he was a Postdoctoral Research Fellow and a Lecturer with the School of Engineering and Technology, Central Queensland University, Rockhampton, QLD, Australia. From 2014 to 2016, he was a Lecturer with the Griffith School of Engineering, Griffith University, Gold Coast, QLD, Australia. In 2016, he joined the Swinburne University of Technology, Melbourne, VIC, Australia, where he is currently an Associate Professor with the School of Software and Electrical Engineering. His current research interests include H-infinity filtering, event-triggered control, networked control systems, neural networks, distributed systems, and time-delay systems.
Dr. Zhang was a recipient of the National Natural Science Award (Secondclass) in China in 2013, and the Hunan Provincial Natural Science Award (First-class) in Hunan Province in China in 2011, both jointly with Profs. M. Wu and Y. He. Dr. Zhang was also a recipient of the IEEE Transactions on Industrial Informatics Outstanding Paper Award 2020, the Andrew P. Sage Best Transactions Paper Award 2019, and the IET Control Theory and Applications Premium Award 2016. He is acting as an Associate Editor of several international journals, including the IEEE TRANSACTIONS ON CYBERNETICS, Neural Processing Letters, Journal of the Franklin Institute, International Journal of Control, Automation, and Systems, Neurocomputing.