Global exponential periodicity and stability of a class of memristor-based recurrent neural networks with multiple delays
Introduction
The first memristor (as a contraction of memory and resistor), originally theorized by Dr. Chua in 1971, was identified by a team at HP Labs in 2008. The memristor is a two-terminal passive device whose value depends on the magnitude and polarity of the voltage applied to it and the length of time that the voltage has been applied. From the previous work [3], [6], [7], [17], [22], [25], [27], [28], [29], [31], we know that the potential applications of this device is in next generation computer and powerful brain-like neural computer.
As we know, the recurrent neural networks are very important nonlinear circuit networks because of their wide applications in associative memory, pattern recognition, signal processing and so on, for reference, see [4], [5], [9], [11], [13], [14], [15], [16], [18], [19], [20], [21], [23], [24], [26], [32], [33], [34]. And the Hopfield neural network model can be implemented in a circuit where the connection weights are implemented by resistors, motivated by these facts, recently, by using memristors instead of resistors, Bao and Zeng [3], Wu et al. [27], [28], [29] and Zhang et al. [31] have studied a new model where the connection weights change according to its state, i.e., a state-dependent switching recurrent neural networks.
Different from the previous works [3], [27], [28], [29], [31], in this paper, we will deal with the problem of existence and global exponential stability of periodic solution for a class of memristor-based recurrent neural networks with multiple delays as follows:wherein which switching jumps , are all constant numbers, and are bounded continuous functions, is a continuous -periodic external input function.
The organization of this paper is as follows. Some preliminaries are introduced in Section 2. The main results are given in Section 3. And then, numerical simulations are given to demonstrate the effectiveness of the proposed approach in Section 4. Finally, this paper ends by a conclusion. Remark 1 The authors in [27], [29] have given a clear exposition about the relation between memristances and the coefficients of switching system (1), so readers can consult [27], [29] to get more explanation.
Section snippets
Preliminaries
Throughout this paper, solutions of all the systems considered in the following are intended in Filippov’s sense (see [12]). Let , we define , for , denotes the convex hull of . For a continuous function is called the upper right dini derivative and defined as . System (1) has the following form initial conditions:
Main results
Theorem 1 Under assumption (H1) and (H2), if there exists constants and , such thatwhere , . Then, system (1) exists exactly one -periodic solution and all other solutions of system (1) converge exponentially to it as . Proof For , we define the norm ,
Numerical simulations
Now, we perform some numerical simulations to illustrate our analysis by using MATLAB (7.0) programming.
System 1. Consider two-dimensional memristor-based neural networkswhere
Conclusion
In this paper, under the framework of Filippov’s solution, we made an effort to deal with the problem of the existence and global exponential stability analysis of periodic solution for a class of memristor recurrent neural networks with multiple delays. Through building a useful Lyapunov functional and the inequality techniques, some new testable algebraic criteria are obtained for ensuring the existence and global exponential stability of periodic solution of the system. The model based on
Acknowledgements
The authors gratefully acknowledge anonymous referees’ comments and patient work. This work is supported by the National Science Foundation of China under Grant No. 11271146, the Key Program of National Natural Science Foundation of China under Grant No. 61134012, and the 973 Program of China under Grant 2011CB710606.
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